LOCAL ESTIMATES OF EXPONENTIAL POLYNOMIALS

AND THEIR APPLICATIONS

TO INEQUALITIES OF UNCERTAINTY PRINCIPLE TYPE

F. L. Nazarov

28/DEC/1992

Abstract. The paper is devoted to certain inequalities (modelled on the Turan lemma) for exponential polynomials, and to applications of these inequalities to various uniqueness theorems in harmonic analysis, like the uncertainty principle. An estimate is obtained of the maximum of the absolute value of an exponential polynomial

p(t) = ∑_{k=1}^{n} c_{k}e^{λ_{k}t}

(c_{k} ∈ ℂ, λ_{k} ∈ ℂ) in terms of the maximum of its absolute value on a measurable set E ⊂ I of positive Lebesgue measure:

sup_{t∈I} |p(t)| ⩽ e^{max | Re λ_{k}|μ(I)}\{ \frac{Aμ(I)}{μ(E)} \}^{n−1} sup_{t∈E} |p(t)|.

As for applications, the inequality

∥f∥_{L^{2}(ℝ)}^{2} ⩽ Ae^{Aμ(E)μ(Σ)}(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2})

is proved (here f is an arbitrary function from L^{2}(ℝ) and E, Σ are arbitrary sets of finite Lebesgue measure), as well as a theorem on the integrability of small powers (less than 1/2) of the logarithm of the absolute value of a function f ∈ L^{2}(𝕋) with lacunary spectrum in the sense of Zygmund.

Introduction

\underline{Chapter 1.} Estimates of exponential polynomials

§1.1. Historical remarks

Key words and phrases. Exponential polynomial, uncertainty principle in harmonic analysis, lacunary trigonometric series.

Typeset by AMS-TEX 2 F. L. NAZAROV

§1.2. Two useful lemmas §1.3. The proof of estimate (1.1) in a special case §1.4. The proof of the Turan lemma in the general case \underline{Chapter 2.} Random periodization technique and the Morgan theorem §2.1. Lattice averaging lemma and random periodization operators §2.2. Strong annihilation of sets of finite measure §2.3. The Morgan theorem §2.4. The Turan lemma for H^{2}(𝔻) and a semi-one-sided uniqueness theorem §2.5. The one-sided Amrein-Berthier theorem \underline{Chapter 3.} Functions with nearly linear dependent small shifts and the Zygmund theorem §3.1. A spectral description of the class EP_{loc}^{n}(τ, κ) §3.2. Local behavior of functions from the class EP_{loc}^{n}(τ, κ) §3.3. Expanding the estimate §3.4. The Zygmund theorem on the unit circle §3.5. The Zygmund theorem on the real line \underline{Chapter 4.} Some remarks and supplements §4.1. Estimates of exponential polynomials with complex exponents §4.2. Functions with spectrum of finite Lebesgue measure §4.3. The Mandelbrojt theorem Bibliography Introduction

Exponential polynomials constitute a classical object of study in analysis. However, while their “global” behavior (i.e., the behavior in a neighborhood of infinity) was perfectly investigated long ago, many “local” questions (i.e., questions about their behavior near a finite point) have remained open until the present time. As an example we can name the problem of finding sharp estimates for the maximum of the modulus of an exponential polynomial on an interval I in terms of its maximum on a subset E of I having positive Lebesgue measure. This problem is solved in this article in the case where the measure of the set E is small compared with the length of I.

No less classical is another subject of this paper, the so-called uncertainty principle claiming that a function f and its Fourier transform f̂ cannot be small simultaneously (in an appropriate sense).

We shall show that one of the versions of this principle, namely, the Morgan theorem on the “strong annihilation” of two intervals on the real line (this terminology is borrowed from Jöricke and Havin’s book “The uncertainty principle in harmonic analysis”), which is usually proved by analytic continuation technique, has quite “elementary” nature, and, moreover, elementary methods of proof work in a much more general case, where analytic continuation technique cannot be applied (namely, when the intervals are replaced by two sets of finite measure). A quantitative version of Zygmund’s uniqueness theorem completes the article.

The following theorems may be regarded as main results of the paper.

Theorem I (the Turan lemma). Let p(t) = ∑_{k=1}^{n} c_{k}e^{λ_{k}t} (c_{k} ∈ ℂ, λ_{k} ∈ ℂ) be an exponential polynomial of order n. Let I ⊂ ℝ be an interval, and E a measurable subset of I of positive measure.

Then

sup_{t∈I} |p(t)| ⩽ e^{|I| max | Re λ_{k}|}\{\frac{Aμ(I)}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|.

(Here and below A denotes an absolute constant).

Theorem II (the Amrein–Berthier theorem). For every two sets E, Σ ⊂ ℝ of finite Lebesgue measure and every function f ∈ L^{2}(ℝ), the following inequality holds:

∥f∥_{L^{2}(ℝ)}^{2} ⩽ Ae^{Aμ(E)μ(Σ)}(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2}).

Theorem III (the Zygmund theorem). Let a set Λ = {m_{k}}_{k=−∞}^{∞} ⊂ ℤ be lacunary in the sense of Zygmund, i.e.,

sup_{r∈ℤ∖{0}} card{ (k′, k′′) : m_{k′} − m_{k′′} = r } =^{def} R(Λ) < +∞.

Then for every function f ∈ L^{2}(𝕋) with spectrum spec f lying in Λ and for every subset E of the unit circumference 𝕋 having positive measure, the following inequality holds:

∥f∥_{L^{2}(𝕋)}^{2} ⩽ exp\{\frac{B(ε, R(Λ))}{μ(E)^{2+ε}}\} ∫_{E} |f|^{2} dμ.

Here ε > 0 can be taken arbitrary small, and B(ε, R(Λ)) depends only on ε and R(Λ).

Corollary IV (The “logarithm summability” theorem). Under the above assumptions on Λ, for every function f ∈ L^{2}(𝕋) such that spec f ⊂ Λ and f is not identically zero, and every ε > 0 we have

∫_{𝕋} log^{\frac{1}{2} −ε}(1 + \frac{1}{|f|}) dμ < +∞.

This selection is not indisputable and characterizes the paper only in the sense that the reader whose interest is not roused by the above results is unlikely to read the rest of the paper, and vice versa.

How is this paper organized?

Chapter 1, devoted to estimating exponential polynomials with purely imaginary exponents, is in the core of the article. Chapters 2 and 3, devoted to the Amrein–Berthier theorem and the Zygmund theorem, respectively, are almost independent. If the reader prefers to read Chapter 3 first, from Chapter 2 he will only need the lattice averaging lemma (Lemma 2.1). Chapter 4 contains commentaries, some supplements to the results of Chapters 1–3, and some open problems. I have tried to make the exposition as selfcontained as possible. So, some readers may judge that the paper is overloaded with well-known classical results. I was fully aware of the latter, but, nevertheless, pursued the goal of making the topics in question attractive for people without (or with only a slight) knowledge of the subject. Moreover, I always thought that to skip a portion of a text might be much easier for the reader than to restore a skipped proof on the basis of the reference list.

Acknowledgement. First of all, I would like to thank V. P. Havin who acquainted me with this topic and actually forced me to publish the results. Also, it is due to him that chapter 3 can be read without great efforts. Besides, I would like to thank A. B. Aleksandrov, A. V. Malyshev, N. K. Nikolski, A. N. Podkorytov, S. E. Rukshin, N. A. Shirokov, and others who influenced my mathematical education.

Notation list.

ℤ is the set of integers;

ℕ = { z ∈ ℤ : z > 0 };

ℝ is the real line;

ℂ is the complex plane;

ℂ_{+} = { z ∈ ℂ : Im z > 0 } is the upper half-plane;

𝔻 = { z ∈ ℂ : |z| < 1 } is the unit disk;

𝕋 = ∂𝔻 is the unit circumference;

μ is the Lebesgue measure on ℝ or on 𝕋 normalized by μ([0, 1]) = 1 and μ(𝕋) = 1, respectively. This normalization ensures that the standard mapping from [0, 1] onto 𝕋 defined by z = e^{2πit} preserves the measure;

|I| is the length of an interval I ⊂ ℝ or an arc I ⊂ 𝕋. In the first case |I| = μ(I), while in the second |I| = 2πμ(I).

The Lebesgue and Hardy spaces L^{p} and H^{p}, respectively, have their usual sense.

We denote by f̂ the Fourier transform of a function f ∈ L^{2}(ℝ) understood in the sense of the Plancherel theorem, i.e., as a limit in L^{2}(ℝ) of the functions

f̂_{n}(λ) =^{def} ∫_{−n}^{n} f(x)e^{−2πiλx} dx

I prefer to write 2π in the exponent, so, for the reader who is accustomed to the more widespread (though less convenient) definition containing the factor 1/\sqrt{2π} before the integral, some numerical constants will need correction. The inverse Fourier transform is denoted by f̌.

We shall not distinguish between 1-periodic functions defined on the real line and functions defined on 𝕋. In either case we denote by f̂_{k} the k-th Fourier coefficient of a function f:

f̂_{k} = ∫_{0}^{1} f(t)e^{−2πikt} dt = ∫_{𝕋} f(z)z^{−k} dμ(z).

In Chapter 3, sometimes, we shall write f̂(k) instead of f̂_{k}.

supp f = { f ≠ 0 } is the support of a function f. If f ∈ L^{p}(ℝ), the support is defined up to a set of measure zero;

spec f = supp f̂ is the spectrum of a function f;

P{A} is the probability of a random event A;

Eξ is the mathematical expectation of a random variable ξ.

Notation connected with exponential polynomials. By an exponential polynomial we understand a finite sum

p(t) = ∑_{k=1}^{n} c_{k}e^{λ_{k}t} (c_{k} ∈ ℂ, λ_{k} ∈ ℂ).

The number of nonvanishing terms in this sum is called the order of an exponential polynomial p(t) and is denoted by ord p. The numbers c_{k} are the coefficients of p, and the numbers λ_{k} are its exponents. The set of all exponents of an exponential polynomial p is called its spectrum and is denoted by spec p. If all the exponents are purely imaginary, we shall sometimes change this notation and define the spectrum of p by spec p = \{ \frac{λ_{k}}{2πi} : k = 1, …, n \}; this set is the spectrum of the polynomial p treated as a distribution on the real line ℝ. The choice of definition will always be either inessential or clear from the context.

Some authors define an exponential polynomial as a more general expression p(t) = ∑_{k=1}^{n} q_{k}(t)e^{λ_{k}t}, where q_{k}(t) are algebraic polynomials of degrees deg q_{k} = d_{k}. It is natural to define the order of such an exponential polynomial as the sum ord p = ∑_{k=1}^{n}(d_{k} + 1). An obvious approximation argument shows that all the results of Chapter 1 remain valid in this more general setting.

\underline{C} hapter 1. Estimates of exponential polynomials

Our main purpose in this chapter is to prove the inequality

(1.1)  sup_{t∈I} |p(t)| ⩽ \{\frac{Aμ(I)}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|,

where p(t) = ∑_{k=1}^{n} c_{k}e^{iλ_{k}t} is an exponential polynomial with purely imaginary exponents, I ⊂ ℝ is an interval, E ⊂ I is a measurable set of positive Lebesgue measure, and A is an absolute constant.

§1.1. Historical remarks

In this section we are going to show what could have been well known by now if the fate would not have decided otherwise. For the first time estimate (1.1) in the special case where E is also an interval was derived by Turan from certain inequalities for sums of powers of n complex numbers (see [11]). Namely, Turan proved the following lemma.

Lemma. Let z_{1}, …, z_{n} be complex numbers, |z_{j}| ⩾ 1, j = 1, …, n. Let

b_{1}, …, b_{n} ∈ ℂ, S_{m} =^{def} ∑_{k=1}^{n} b_{k}z_{k}^{m} (m ∈ ℤ).

Then

(1.2) |S_{0}| ⩽ n\{\frac{2e(m + n − 1)}{n}\}^{n−1} max_{k=m+1}^{m+n} |S_{k}| ⩽ \{\frac{4e(m + n − 1)}{n}\}^{n−1} max_{k=m+1}^{m+n} |S_{k}|

for all m ∈ ℤ_{+}.

Turan’s proof was purely algebraic and was based on a construction of a polynomial q(z) = 1 + ∑_{k=1}^{n} γ_{k}z^{m+k} such that q(z_{j}) = 0 for each j = 1, …, n and ∑_{k=1}^{n} |γ_{k}| ⩽ n\{\frac{2e(m+n−1)}{n}\}^{n−1}. Obviously, in this case S_{0} = − ∑_{k=1}^{n} γ_{k}S_{m+k}, whence (1.2) follows immediately. The polynomial q(z) is exhibited explicitly, as a product q(z) = ∏_{k=1}^{n}(1 − \frac{z}{z_{k}} )σ_{m}(z), where σ_{m}(z) is the m-th partial sum of the series ∏_{k=1}^{n}(1− \frac{z}{z_{k}} )^{−1} = ∑_{k=0}^{∞} β_{k}z^{k}. The fact that the coefficients at the powers z, z^{2}, …, z^{m} vanish follows from a simple observation that the s-th coefficient in the expansion of the product 1 ≡ ∏_{j=1}^{n}(1− \frac{z}{z_{j}} ) ∑_{k=0}^{∞} β_{k}z^{k} depends only on β_{s−n}, …, β_{s}. The terms of the series ∑_{k=0}^{∞} β_{k}z^{k} are majorized by those of (1 − z)^{−n} = ∑_{k=0}^{∞} \frac{(k+n−1)!}{k! (n−1)!}z^{k}. Thus, all coefficients of σ_{m}(z) do not exceed

\frac{(m+n−1)!}{m! (n−1)!} ⩽ \{\frac{e(m+n−1)}{n}\}^{n−1}. The coefficients of the polynomial ∏_{j=1}^{n}(1 − \frac{z}{z_{j}} ) are not greater than \binom{n}{k}. Carefully examining everything contributing to γ_{k}, we get the estimates

|γ_{k}| ⩽ \{\frac{e(m + n − 1)}{n}\}^{n−1} ∑_{s=k}^{n} \binom{n}{s}

and

∑_{k=1}^{n} |γ_{k}| = \frac{1}{2} ∑_{k=1}^{n}(|γ_{k}| + |γ_{n+1−k}|) ⩽ 2^{n−1}n\{\frac{e(m + n − 1)}{n}\}^{n−1},

which prove (1.2).

It follows immediately from (1.2) that

|p(t_{0})| ⩽ \{\frac{4e(m + n − 1)}{n}\}^{n−1} max_{k=m+1}^{m+n} |p(t_{k})|

if the points t_{k} = t_{0} + kδ form an arithmetic progression (it suffices to put z_{j} = e^{iλ_{j}δ}, b_{j} = c_{j}e^{iλ_{j}t_{0}}); now inequality (1.1) for the case where E is an interval can be derived in an almost immediate way (with the constant A = 4e).

Turan himself said no word about the case of an arbitrary measurable set E. But using the same idea in its simplest form (with m = 0, which corresponds to the polynomial q(z) = ∏_{k=1}^{n}(1 − \frac{z}{z_{k}} ) and to the estimate |S_{0}| ⩽ 2^{n} max_{k=1}^{n} |S_{k}|), one can prove that

(1.3)  max_{t∈I} |p(t)| ⩽ 2^{n}\{\frac{μ(I)}{μ(E)}\}^{2n^{2}} max_{t∈E} |p(t)|.

Indeed, the inequality |p(t_{0})| ⩽ 2^{n} max_{t∈E} |p(t)| holds if t_{0} is the first term of an arithmetic progression t_{k} = t_{0} + kδ (k = 0, 1, …, n) with all other terms belonging to E. It is quite natural to think that such a progression exists for the points t_{0} ∈ I near which E is “sufficiently dense”. We shall see that all points t_{0} for which there is an open interval J satisfying t_{0} ∈ J ⊂ I and

(∗)  μ(E ∩ J) > (1 − \frac{1}{2n})μ(J)

will fit.

Indeed, such a point t_{0} splits the interval J into two subintervals J_{−} and J_{+}. At least for one of them (*) is also valid. Assume, for certainty, that μ(E ∩ J_{+}) > (1 − \frac{1}{2n})μ(J_{+}). Applying the lattice averaging lemma (Lemma 2.1 of Chapter 2), we see that the average number of points t_{k} = t_{0} + kδ (k ∈ ℕ) belonging to J_{+} ∖ E as δ runs over the interval (\frac{μ(J_{+})}{2n}, \frac{μ(J_{+})}{n}) is less than 1. Hence, there exists a positive value δ < \frac{μ(J_{+})}{n} such that none of the points t_{1}, …, t_{n} belongs to J_{+} ∖ E. But all these points lie in J_{+} and, consequently, in E.

If the reader deems the lattice averaging lemma little-known or nontrivial, he may restrict himself to the density estimate μ(E ∩ J) > (1 − \frac{1}{n^{2}} )μ(J), which simplifies the proof: one needs only to estimate the measure of the set of all δ ∈ [0, \frac{μ(J_{+})}{n}] such that t_{k} ∈ J_{+} ∖ E for each k separately. In this case the exponent 2n^{2} should be replaced by n^{3}.

Now let E_{1} = ⋃\{ J : J ⊂ I is open, μ(E ∩ J) > (1 − \frac{1}{2n})μ(J) \}. Clearly, the open set E_{1} contains all density points of E, so E ⊂ E_{1} up to a set of measure zero. If E_{1} ≠ I, then for each constituent interval Q of E_{1} the inequality μ(E ∩ Q) ⩽ (1 − \frac{1}{2n})μ(Q) holds (otherwise E_{1} would include some interval strictly larger than Q). Thus, the set E_{1} has the following two properties:

sup_{t∈E_{1}} |p(t)| ⩽ 2^{n} sup_{t∈E} |p(t)|, (A_{1})

μ(E_{1}) ⩾ (1 − \frac{1}{2n})^{−1}μ(E) ⩾ e^{1/(2n)}μ(E) or E_{1} = I. (B_{1})

Iterating this procedure (constructing E_{2} starting with E_{1} in the same way and so on), we obtain a sequence of sets E_{1}, E_{2}, … such that

sup_{t∈E_{1}} |p(t)| ⩽ 2^{nk} sup_{t∈E} |p(t)|, (A_{k})

μ(E_{k}) ⩾ e^{k/(2n)}μ(e) or E_{k} = I. (B_{k})

If k > 2n log \frac{μ(I)}{μ(E)}, then the first case in (B_{k}) cannot occur; therefore, E_{[2n log \frac{μ(I)}{μ(E)} +1]} = I,

whence

sup_{t∈I} |p(t)| ⩽ 2^{(2n log \frac{μ(I)}{μ(E)} +1)n} sup_{t∈E} |p(t)| ⩽ 2^{n}\{\frac{μ(I)}{μ(E)}\}^{2n^{2}} sup_{t∈E} |p(t)|.

As far as the leading idea of this reasoning (estimating along arithmetic progressions) was suggested by Turan and all the remaining part of the proof follows from this main idea in quite a natural way and is nothing but mathematical routine, one should regard inequality (1.3) to be due to Turan and date it back to 1953.

Often it is desirable to have an upper estimate of the sum of absolute values of coefficients ∑_{k=1}^{n} |c_{k}| rather than of the maximum max_{t∈I} |p(t)|. In this case the “separation condition” λ_{j+1} − λ_{j} ⩾ Δ > 0 (j = 1, …, n − 1) is usually imposed on the exponents λ_{1} < λ_{2} < ⋯ < λ_{n} ∈ ℝ. A desired estimate can be derived from (1.1) (or (1.3)) by using the well-known Salem inequality: if μ(I) ⩾ 4π/Δ, then

\frac{1}{n}(∑_{k=1}^{n} |c_{k}|)^{2} ⩽ ∑_{k=1}^{n} |c_{k}|^{2} ⩽ \frac{4}{|I|} ∫_{I} |p(t)|^{2} dt ⩽ 4 max_{t∈I} |p(t)|^{2}.

If the interval I is too small to apply the Salem inequality, we can take a wider interval I′ ⊃ I of length 4π/Δ. So, we get

(1.1′)  ∑_{k=1}^{n} |c_{k}| ⩽ \{ \begin{matrix}\{\frac{Aμ(I)}{μ(E)}\}^{n−1} max_{t∈E} |p(t)| & if Δμ(I) ⩾ 4π, \\ \{\frac{4πA}{Δμ(E)}\}^{n−1} max_{t∈E} |p(t)| & if Δμ(I) < 4π, \end{matrix}

where A is an absolute constant (slightly greater than that in (1.1)).

Concerning the state of affairs in the reality, I do not know any publication containing an estimate essentially better than

sup_{t∈I} |p(t)| ⩽ \{\frac{2μ(I)}{μ(E)}\}^{2^{n}} sup_{t∈E} |p(t)|.

§1.2. Two useful lemmas

This section contains the proofs of two propositions frequently used in the rest of the paper, namely, a weak type estimate of the logarithmic derivative of a polynomial on the unit circumference and on the real line, and the Langer lemma on the distribution of zeros of exponential polynomials. The first of them is a version of the Kolmogorov inequality for the harmonic conjugate function and is aimed merely at calculating certain constants. The second one has been included because it is relatively little-known. Here are the precise statements.

Lemma 1.2. If P (z) is an algebraic polynomial of degree n, then

μ\{ x ∈ ℝ : |\frac{d}{dx} log P (x)| > y \} ⩽ \frac{8n}{y}

and

μ\{ z ∈ 𝕋 : |\frac{d}{dz} log P (z)| > y \} ⩽ \frac{8n}{πy} .

Remark. It is worth noting that \frac{d}{dz} log P (z) = \frac{\frac{d}{dz} P (z)}{P (z)}, and, therefore, one can regard the inequalities of Lemma 1.2 as certain versions of the Bernstein estimate for the derivative of an algebraic polynomial P (z).

Lemma 1.3. Let p(z) = ∑_{k=1}^{n} c_{k}e^{iλ_{k}z} (0 = λ_{1} < λ_{2} < ⋯ < λ_{n} = λ) be an exponential polynomial not vanishing identically. Then the number of complex zeros of p(z) in an open vertical strip x_{0} < Re z < x_{0} + Δ of width Δ does not exceed (n − 1) + \frac{λΔ}{2π}.

Proof of Lemma 1.2. First we shall prove the inequality for the real line. Let z_{1}, …, z_{n_{1}}; ζ_{1}, …, ζ_{n_{2}} (n_{1} + n_{2} = n) be complex zeros of the polynomial P enumerated in such a way that Im z_{j} ⩽ 0 (j = 1, …, n_{1}), Im ζ_{j} > 0 (j = 1, …, n_{2}). We have

\frac{d}{dz} log P (z) = ∑_{j=1}^{n_{1}} \frac{1}{z − z_{j}} + ∑_{j=1}^{n_{2}} \frac{1}{z − ζ_{j}} = ∑_{1}(z) + ∑_{2}(z).

The function ∑_{1}(z) is analytic in the upper half-plane ℂ_{+}, and

Im ∑_{1}(z) = ∑_{j=1}^{n_{1}} \frac{Im z_{j} − Im z}{|z − z_{j}|^{2}} < 0

for all z ∈ ℂ_{+}.

Let h(ξ) be the harmonic measure of the set ℝ ∖ [−y, y] with respect to the upper halfplane and a point ξ ∈ ℂ_{+}. Repeating an argument of Koosis, we put u(z) =^{def} h(−∑_{1}(z)). The function u(z) is harmonic in ℂ_{+}, 0 ⩽ u(z) ⩽ 1, u(it) ⟶_{t→+∞} 0, and u(z) ⩾ 1/2 if

|∑_{1}(z)| ⩾ y (the latter fact follows from the geometric description of the harmonic measure as a ratio to π of the angle at which a subset of ℝ is seen from the point ξ).

Moreover, we have

lim_{t→+∞} πtu(it) = ∫_{ℝ} u(x) dx ⩾ \frac{1}{2}μ\{ x ∈ ℝ : |∑_{1}(x)| > y \}.

On the other hand, an easy computation shows that

lim_{t→+∞} πtu(it) = lim_{t→+∞} πth(\frac{in_{1}}{t} + O(\frac{1}{t^{2}} )) = \frac{2n_{1}}{y} .

Hence

μ\{ x ∈ ℝ : |∑_{1}(x)| > y \} ⩽ \frac{4n_{1}}{y} .

Similarly,

μ\{ x ∈ ℝ : |∑_{2}(x)| > y \} ⩽ \frac{4n_{2}}{y} .

Combining these estimates, we obtain

μ\{ x ∈ ℝ : |∑(x)| > y \} ⩽ μ\{ x ∈ ℝ : |∑_{1}| > \frac{n_{1}}{n} y \} + μ\{ x ∈ ℝ : |∑_{2}| > \frac{n_{2}}{n} y \} ⩽ \frac{8n}{y},

as desired.

Now we pass to the case of the circumference. As above, we split the zeros of P (z) into two groups: z_{1}, …, z_{n_{1}} ∈ 𝔻 and ζ_{1}, …, ζ_{n_{2}} ∈ ℂ ∖ 𝔻. Then

\frac{d}{dz} log P (z) = \frac{1}{z}( ∑_{j=1}^{n_{1}} \frac{z}{z − z_{j}} + ∑_{j=1}^{n_{2}} \frac{z}{z − ζ_{j}} ) = \frac{1}{z}(∑_{1}(z) + ∑_{2}(z)).

The factor 1/z can be disregarded since its absolute value is equal to 1. The estimate for ∑_{1}(z) is essentially the same as above: having established the inequality

Re ∑_{1}(z) = ∑_{j=1}^{n_{1}} \frac{|z|^{2} − Re z\overline{z}_{j}}{|z − z_{j}|^{2}} ⩾ 0

for z ∈ ℂ ∖ 𝔻, we consider the function u(z) =^{def} h(i∑_{1}(z)), which is harmonic outside the unit disk, and derive the estimate

u(∞) = \frac{2 arctg \frac{n_{1}}{y}}{π} = ∫_{𝕋} u(z) dμ(z) ⩾ \frac{1}{2}μ\{ z ∈ 𝕋 : |∑_{1}(z)| > y \},

which implies

μ\{|∑_{1}(z)| > y\} ⩽ \frac{4}{π} arctg \frac{n_{1}}{y} ⩽ \frac{4}{π} \frac{n_{1}}{y} .

The function Re ∑_{2}(z) may change sign. Therefore, we use another inequality

Re ∑_{2}(z) = n_{2} − ∑_{j=1}^{n_{2}} \frac{|ζ_{j}|^{2} − Re z\overline{ζ}_{j}}{|z − ζ_{j}|^{2}} ⩽ n_{2} (z ∈ 𝔻).

This time we choose the function h (see above) to be harmonic in ℂ_{+} − in_{2}. In order to obtain the estimate μ\{|∑_{2}| > y\} ⩽ \frac{4}{π} \frac{n_{1}}{y}, we can restrict ourselves to values y > n_{2}. Let h(ξ) be the harmonic measure of (ℝ − in_{2}) ∖ I (where I is the interval cut off from the line ℝ − in_{2} by the circle centered at 0 and of radius y) with respect to the half-plane ℂ_{+} − in_{2} and the point ξ ∈ ℂ_{+} − in_{2}. Drawing a picture, one can easily check that the function u(z) =^{def} h(−i∑_{2}(z)) is harmonic in 𝔻, u(z) ⩾ ^{\frac{π−arccos \frac{n_{2}}{y}}{π}} if |∑_{2}(z)| > y, and

u(0) = h(0) = ^{\frac{2 arcsin \frac{n_{2}}{y}}{π}} ; therefore,

μ\{ |∑_{2}| > y \} ⩽ \frac{2 arcsin \frac{n_{2}}{y}}{π − arccos \frac{n_{2}}{y}} = \frac{2 arcsin \frac{n_{2}}{y}}{π + arcsin \frac{n_{2}}{y}} .

Now, to get the desired estimate it suffices to verify that \frac{2θ}{π/2+θ} ⩽ \frac{4}{π} sin θ for each θ ∈ [0, \frac{π}{2} ]. The last inequality is equivalent to \frac{sin θ}{θ} + \frac{2}{π} sin θ ⩾ 1. Taking into account that sin θ ⩾ θ − \frac{1}{6}θ^{3} for every θ ⩾ 0 and sin θ ⩾ \frac{2}{π} θ for every θ ∈ [0, \frac{π}{2} ], we have

\frac{sin θ}{θ} + \frac{2}{π} sin θ ⩾ 1 − \frac{1}{6}θ^{2} + \frac{4}{π^{2}} θ = 1 + θ(\frac{4}{π^{2}} − \frac{1}{6}θ) ⩾ 1 + θ(\frac{4}{π^{2}} − \frac{π}{12}),

and it remains to notice that π^{3} ⩽ 48.

As above, the estimate of ∑(z) results from the estimates of ∑_{1}(z) and ∑_{2}(z). Lemma 1.2 is proved.

Proof of lemma 1.3. Without loss of generality we assume that the coefficients c_{1} and c_{2} do not vanish and the boundary of the strip x_{0} < Re z < x_{0} + Δ is free of zeros of the exponential polynomial p(z). Following the reasoning of Langer, we make use of the argument principle to estimate the number of zeros of p(z) in the rectangle Q = { z : x_{0} < Re z < x_{0} + Δ, | Im z| ⩽ y }, as y → +∞.

On the upper edge of Q we have p(z) = c_{1} + O(e^{−λ_{2}y}). Therefore, the argument increment along this edge tends to 0 as y → +∞. Similarly, the representation p(z) = c_{n}e^{iλz}(1+O(e^{−(λ−λ_{n−1})y})), which is valid on the lower edge of Q, implies that the argument increment along the lower edge tends to λΔ as y → +∞.

We show that the argument increment along any vertical segment { z = x + it : t ∈ [α, β] } free of zeros of p(z) does not exceed π(n − 1). Let ξ =^{def} e^{i arg p(x_{0}+iα)}. The function

q(t) =^{def} Im \overline{ξ}p(x_{0} + it) = ∑_{k=1}^{n} a_{k}e^{−λ_{k}t} (a_{k} = Im \overline{ξ}c_{k}e^{iλ_{k}x_{0}} ∈ ℝ) is a real exponential

polynomial. The definition of ξ ensures q(α) = 0. If q ≡ 0, then all values p(x_{0} + it) (t ∈ [α, β]) lie on the ray { ξy : y > 0 }, and therefore Δ_{[α,β]} arg p(x_{0} + it) = 0. Otherwise, real zeros of q(t) split the segment [α, β] into at most n − 1 intervals I_{j} (it is well known that a real exponential polynomial of order n has at most n − 1 real zeros). Within each of the intervals I_{j}, the values p(x_{0} + it) lie in one of the two half-planes generated by the line { ξy : y ∈ ℝ }, whence |Δ_{I_{j}} arg p(x_{0} + it)| ⩽ π. Adding these inequalities, we obtain |Δ_{[α,β]} arg p(x_{0} + it)| ⩽ π(n − 1), which proves the desired statement. In particular, the argument increment along each of the lateral edges of Q does not exceed π(n − 1). So the total argument increment of p(z) along the boundary of Q traced counter-clockwise can be estimated from above by a quantity tending to 2π(\frac{Δλ}{2π} + (n − 1)) as y → +∞, whence Lemma 1.3 clearly follows.

§1.3. The proof of estimate (1.1) in a special case

Here we shall prove inequality (1.1) for the case of a 1-periodic exponential polynomial p(t) = ∑_{k=1}^{n} c_{k}e^{2πim_{k}t}, where c_{k} ∈ ℂ, m_{1} < ⋯ < m_{n} ∈ ℤ, and for the segment I = [0, 1]. This case is singled out for two reasons: firstly, here the main idea of the proof is not obscured by inessential technical details and may be seen clearer, and, secondly, the constant A in (1.1) can be taken in this case much smaller than in the general case. Substituting z = e^{2πit}, we come to the following statement.

Theorem 1.4 (the Turan lemma for polynomials on the unit circumference). Let p(z) = ∑_{k=1}^{n} c_{k}z^{m_{k}} (c_{k} ∈ ℂ, m_{1} < ⋯ < m_{n} ∈ ℤ) be a trigonometric polynomial on the unit circumference 𝕋, and let E be a measurable subset of 𝕋. Then

∥p∥_{W} =^{def} ∑_{k=1}^{n} |c_{k}| ⩽ \{\frac{16e}{π} \frac{1}{μ(E)}\}^{n−1} sup_{z∈E} |p(z)| ⩽ \{\frac{14}{μ(E)}\}^{n−1} sup_{z∈E} |p(z)|.

(Here W stands in honor of Wiener, as usual).

Proof. We shall construct by induction a sequence of polynomials p_{n}, p_{n−1}, …, p_{1} such that

1) p_{n} = p;

2) ord p_{k} = k (k = 1, …, n);

3) ∥p_{k−1}∥_{W} ⩾ \frac{π}{16}∥p_{k}∥_{W};

4) the ratio φ_{k} =^{def} |\frac{p_{k−1}}{p_{k}} | admits the weak type estimate μ\{ z ∈ 𝕋 : φ_{k}(z) > t \} ⩽ 1/t for all t > 0.

The construction is as follows. Let p_{n} = p. The polynomial p_{k}(z) = ∑_{s=1}^{k} d_{s}z^{r_{s}} (r_{1} < ⋯ < r_{k} ∈ ℤ) being chosen, we introduce two polynomials

\underline{q}(z) =^{def} \frac{d}{dz} (z^{−r_{1}}p_{k}(z)) and \overline{q}(z) =^{def} \frac{d}{dz} (z^{−r_{k}}p_{k}(z)).

Obviously, ord \underline{q} = ord \overline{q} = k − 1. We have

∥\underline{q}∥_{W} = ∑_{s=1}^{k} |d_{s}|(r_{s} − r_{1}), ∥\overline{q}∥_{W} = ∑_{s=1}^{k} |d_{s}|(r_{k} − r_{s}),

whence

∥\underline{q}∥_{W} + ∥\overline{q}∥_{W} = (r_{k} − r_{1}) ∑_{s=1}^{k} |d_{s}| = r∥p_{k}∥_{W} ,

where r =^{def} r_{k} − r_{1}. So, at least one of the norms ∥\underline{q}∥_{W} and ∥\overline{q}∥_{W} is at least \frac{r}{2}∥p_{k}∥_{W}. We assume, for certainty, that ∥\overline{q}∥_{W} ⩾ \frac{r}{2}∥p_{k}∥_{W} (the other case is similar). Put p_{k−1}(z) = \frac{π}{8r} \overline{q}(z). Conditions 2) and 3) are satisfied, and we must check condition 4) only. We observe that z^{−r_{k}}p_{k}(z) = g(1/z), where g is an algebraic polynomial of degree r. But then \overline{q}(z) = −\frac{1}{z^{2}} g′(\frac{1}{z}), and

μ\{ z ∈ 𝕋 : φ_{k}(z) > t \} = μ\{ z ∈ 𝕋 : |\frac{g′(1/z)}{g(1/z)}| > \frac{8r}{π} t \} ⩽ \frac{1}{t}

LOCAL ESTIMATES OF EXPONENTIAL POLYNOMIALS 13

because of Lemma 1.2 and the simple observation that the substitution z → 1/z preserves Lebesgue measure on the unit circumference. Now we estimate the measure of the set of all points z ∈ 𝕋 for which |p_{1}(z)| is essentially greater than |p_{n}(z)| = |p(z)|. We have

|\frac{p_{1}(z)}{p_{n}(z)}| = ∏_{k=2}^{n} φ_{k}(z) ⩽ exp\{∑_{k=2}^{n} ψ_{k}(z)\},

where ψ_{k}(z) =^{def} log_{+} φ_{k}(z). The weak type estimate of φ_{k} gives the inequality μ{ψ_{k} > t} ⩽ e^{−t} for all t > 0. Let α > 0. We decompose ψ_{k}(z) into the sum of η_{k}(z) =^{def} min(ψ_{k}(z), α) and κ_{k}(z) =^{def} ψ_{k}(z) − η_{k}(z). Then ∑_{k=2}^{n} η_{k}(z) ⩽ α(n − 1) for all z ∈ 𝕋, and

∫_{𝕋} κ_{k}(z) dμ(z) = ∫_{α}^{+∞} μ{ψ_{k} > t} dt ⩽ ∫_{α}^{+∞} e^{−t} dt = e^{−α}.

Therefore,

∫_{𝕋}(∑_{k=2}^{n} κ_{k}(z)) dμ(z) ⩽ e^{−α}(n − 1)

and

μ{ z ∈ 𝕋 : ∑_{k=2}^{n} ψ_{k}(z) > (α + 1)(n − 1) } ⩽ μ{ z ∈ 𝕋 : ∑_{k=2}^{n} κ_{k}(z) > n − 1 } < e^{−α}.

Let α = log \frac{1}{μ(E)}; then e^{−α} = μ(E) and there exists a point z_{0} ∈ E for which ∑_{k=2}^{n} ψ_{k}(z_{0}) ⩽ (α + 1)(n − 1). Now we have

(\frac{π}{16})^{n−1}∥p∥_{W} ⩽ ∥p_{1}∥_{W} =^{(ord p_{1}=1!)} |p_{1}(z_{0})|

⩽ exp\{(1 + log \frac{1}{μ(E)})(n − 1)\}|p(z_{0})| = \{\frac{e}{μ(E)}\}^{n−1}|p(z_{0})|

⩽ \{\frac{e}{μ(E)}\}^{n−1} sup_{z∈E} |p(z)|,

and the theorem is proved. §1.4. The proof of the Turan lemma in the general case Theorem 1.5 (the Turan lemma). Let p(t) = ∑_{k=1}^{n} c_{k}e^{iλ_{k}t}, where c_{k} ∈ ℂ and λ_{1} < ⋯ < λ_{n} ∈ ℝ. If E is a measurable subset of the segment I = [−\frac{1}{2}, \frac{1}{2}], then

sup_{t∈I} |p(t)| ⩽ \{\frac{316}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|.

(It is clear that the restriction I = [−\frac{1}{2}, \frac{1}{2}] is not essential. The case of an arbitrary interval I can easily be reduced to this special case by a suitable linear substitution).

Proof of Theorem 1.5. The development of the proof depends on the interrelation of λ =^{def} λ_{n} − λ_{1} and n − 1.

Case 1: λ ⩽ n − 1. We shall see that in this case the exponential polynomial p(t) behaves on the interval [−\frac{1}{2}, \frac{1}{2}] like an algebraic polynomial of degree n − 1 and satisfies the estimate

sup_{t∈[− \frac{1}{2} , \frac{1}{2} ]} |p(t)| ⩽ \{\frac{154}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|.

If n = 1, the statement is obvious. Let n > 1. Without loss of generality, we assume that 0 = λ_{1} < ⋯ < λ_{n} = λ ⩽ n − 1. By virtue of the Langer lemma, complex zeros of the exponential polynomial p(z) are well separated, namely, each vertical strip of width Δ contains at most \frac{Δλ}{2π} + (n − 1) ⩽ (1 + \frac{Δ}{2π} )(n − 1) zeros. Let us enumerate z_{j} in the order of increase of | Re z_{j}|. For every j ∈ ℕ, the inequality | Re z_{j}| ⩾ π \frac{j−(n−1)}{(n−1)} holds (otherwise

the zeros z_{1}, …, z_{j} would lie in a vertical strip of width Δ < 2π \frac{j−(n−1)}{(n−1)}, but this strip can contain at most (1 + \frac{Δ}{2π} )(n − 1) < j zeros). Now we write the Hadamard factorization of p(z):

p(z) = ce^{az} ∏_{j=1}^{2(n−1)}(z − z_{j}) ∏_{j>2(n−1)}(1 − \frac{z}{z_{j}})e^{z/z_{j}} =^{def} ce^{az}Q(z)R(z).

We shall examine the behavior of each of the above three factors separately. The canonical product R(z). First of all, we notice that | Re z_{j}| ⩾ π if j > 2(n − 1). We have (since | Re z| < π):

|\frac{d}{dz} log R(z)| = | ∑_{j>2(n−1)}(\frac{1}{z_{j}} + \frac{1}{z − z_{j}} )| ⩽ |z| ∑_{j>2(n−1)} \frac{1}{|z_{j}||z − z_{j}|}

⩽ |z| ∑_{j>2(n−1)} \frac{1}{| Re z_{j}|(| Re z_{j}| − | Re z|)}.

whence it follows, since z ∈ [−\frac{1}{2}, \frac{1}{2}], that

|\frac{d}{dz} log R(z)| ⩽ |z| ∑_{j>2(n−1)} \frac{1}{| Re z_{j}|(| Re z_{j}| − 1/2)} ⩽ 2|z| ∑_{j=1}^{∞} \frac{1}{| Re z_{2(n−1)+j}|^{2}}

⩽ 2|z| ∑_{j=1}^{∞} \frac{1}{(π + \frac{πj}{n−1})^{2}} ⩽ \frac{2(n − 1)}{π}|z| ∫_{π}^{∞} \frac{dt}{t^{2}}

= \frac{2|z|}{π^{2}}(n − 1).

If z is purely imaginary, then

|\frac{d}{dz} log R(z)| ⩽ |z| ∑_{j>2(n−1)} \frac{1}{| Re z_{j}|^{2}} ⩽ \frac{|z|}{π^{2}}(n − 1).

Now,

∫_{−1/2}^{1/2}|\frac{d}{dz} log R(z)| dz ⩽ \frac{2(n − 1)}{π^{2}} ∫_{−1/2}^{1/2} |z| dz = \frac{n − 1}{2π^{2}},

and, therefore,

max_{z∈[− \frac{1}{2} , \frac{1}{2} ]} |R(z)| ⩽ exp\{\frac{n − 1}{2π^{2}}\} min_{z∈[− \frac{1}{2} , \frac{1}{2} ]} |R(z)|.

The factor ce^{az}.

The simplest way to estimate | Re a| is to consider the argument increment of p(z) along a segment [−iω, iω] (ω > 0). It follows from the proof of the Langer lemma that |Δ_{[−iω,iω]} arg p| ⩽ π(n − 1). The argument increment brought in by each of the zeros of Q(z) does not exceed π. So, we have

|Δ_{[−iω,iω]} arg Q| ⩽ 2π(n − 1);
|Δ_{[−iω,iω]} arg R| ⩽ ∫_{−ω}^{ω}|\frac{d}{dz} log R(it)| dt ⩽ \frac{n − 1}{π^{2}} ∫_{−ω}^{ω} |t| dt = \frac{n − 1}{π^{2}}ω^{2};
Δ_{[−iω,iω]} arg(ce^{az}) = 2ω Re a.

The identity

Δ_{[−iω,iω]} arg p = 2ω Re a + Δ_{[−iω,iω]} arg Q + Δ_{[−iω,iω]} arg R

implies

| Re a| ⩽ min_{ω>0}(\frac{3π}{2ω} + \frac{ω}{2π^{2}} )(n − 1) = \sqrt{\frac{3}{π}} (n − 1).

Thus, the absolute values of all the factors except Q(z) change only slightly on the interval [−\frac{1}{2}, \frac{1}{2}]. More precisely,

max_{z∈[− \frac{1}{2} , \frac{1}{2} ]}|\frac{p(z)}{Q(z)}| ⩽ exp\{(\sqrt{\frac{3}{π}} + \frac{1}{2π^{2}} )(n − 1)\} min_{z∈[− \frac{1}{2} , \frac{1}{2} ]}|\frac{p(z)}{Q(z)}|

⩽ 3^{n−1} min_{z∈[− \frac{1}{2} , \frac{1}{2} ]}|\frac{p(z)}{Q(z)}|.

It remains to examine The behavior of the polynomial Q(z). Let 0 < h < 1/8. We shall carry out the Cartan lemma construction. Let n_{1} be the

maximal integer for which there exists a disk D_{1} of radius \frac{n_{1}}{n−1}h containing at least n_{1} zeros of the polynomial Q (it is clear that D_{1} contains exactly n_{1} zeros of Q because otherwise n_{1} could be enlarged). Let n_{2} be the maximal integer for which there exists a disk D_{2} of radius \frac{n_{2}}{n−1}h containing at least n_{2} zeros of Q among those not lying in D_{1}, and so on, till all the zeros of Q are covered. Putting D′_{k} = 2D_{k} (i.e., the disk centered at the same point and of double radius), we obtain the sequence of integers n_{1} ⩾ ⋯ ⩾ n_{s} with the sum n_{1} + ⋯ + n_{s} = 2(n − 1) and the corresponding sequence of disks D′_{1}, …, D′_{s} with the sum of radii equal to 4h. We fix a point z ∈ [−\frac{1}{2}, \frac{1}{2}] ∖ ⋃_{k=1}^{s} D′_{k} and enumerate the zeros of Q in the order of increase of |z − z_{j}|. Following Cartan, we shall show that |z − z_{j}| ⩾ \frac{j}{n−1}h. Indeed, if this is not the case, then the disk D centered at z and of radius \frac{j}{n−1}h contains at least j zeros of Q. Choose an m ∈ \{1, …, s\} such that n_{1} ⩾ ⋯ ⩾ n_{m} ⩾ j > n_{m+1} ⩾ ⋯ ⩾ n_{s}. For every z ∉ ⋃_{k=1}^{s} D′_{k} the distance between z and the center of D_{k} is at least \frac{2n_{k}}{n−1}h ⩾ \frac{n_{k}}{n−1}h + \frac{j}{n−1}h if k ⩽ m. Hence, D does not intersect any of the disks D_{1}, …, D_{m}. But if this were true, the disk D (or a disk with a larger number of zeros) would have been taken instead of D_{m+1} at the (m + 1)-th step. This contradiction proves the claim.

Besides, the Langer lemma implies the inequality |z − z_{j}| ⩾ π \frac{j−(n−1)}{(n−1)} (otherwise the

zeros z_{1}, …, z_{j} would lie in a disk of radius strictly less than π \frac{j−(n−1)}{(n−1)}, and, consequently,

in the strip of width Δ < 2π \frac{j−(n−1)}{(n−1)}). Thus, we have

\frac{|Q(z)|}{max\{ |Q(t)| : t ∈ [−\frac{1}{2}, \frac{1}{2}] \}} ⩾ ∏_{j=1}^{2(n−1)} \frac{|z − z_{j}|}{max\{ |t − z_{j}| : t ∈ [−\frac{1}{2}, \frac{1}{2}] \}} ⩾ ∏_{j=1}^{2(n−1)} \frac{|z − z_{j}|}{1 + |z − z_{j}|}

= ∏_{j=1}^{n−1} \frac{|z − z_{j}|}{1 + |z − z_{j}|} × ∏_{j=1}^{n−1} \frac{|z − z_{n−1+j}|}{1 + |z − z_{n−1+j}|} ⩾ ∏_{j=1}^{n−1} \frac{\frac{j}{n−1}h}{1 + \frac{j}{n−1}h} × ∏_{j=1}^{n−1} \frac{\frac{πj}{n−1}}{1 + \frac{πj}{n−1}}

⩾ (8h)^{n−1} × ∏_{j=1}^{n−1} \frac{\frac{j}{n−1} \frac{1}{8}}{1 + \frac{j}{n−1} \frac{1}{8}} × ∏_{j=1}^{n−1} \frac{\frac{3j}{n−1}}{1 + \frac{3j}{n−1}} .

But for each θ > 0 we have

∏_{j=1}^{n−1} \frac{\frac{j}{n−1}θ}{1 + \frac{j}{n−1}θ} ⩾ exp\{(n − 1) ∫_{0}^{1} log \frac{θt}{1 + θt} dt\} = \{\frac{θ}{(1 + θ)^{1+1/θ}} \}^{n−1},

whence it follows that

\frac{|Q(z)|}{max\{ |Q(t)| : t ∈ [−\frac{1}{2}, \frac{1}{2}] \}} ⩾ (8h)^{n−1}\{8 × (\frac{9}{8})^{9} × \frac{4\sqrt[3]{4}}{3}\}^{−(n−1)} ⩾ \{\frac{8h}{32\sqrt[3]{4}}\}^{n−1}.

The measure of the exceptional set [−\frac{1}{2}, \frac{1}{2}]∩(⋃_{k=1}^{s} D′_{k}) is at most 8h. Combining all these

estimates, we find (h =^{def} μ(E)/8):

sup_{t∈I} |p(t)| ⩽ \{\frac{96\sqrt[3]{4}}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)| ⩽ \{\frac{154}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|.

Case 2: λ > n − 1. We shall reduce this case to Case 1 in the same way as in §3. To do this, we need a weak type estimate of the logarithmic derivative of an exponential polynomial.

Lemma 1.6. Let g(t) = ∑_{k=1}^{n} c_{k}e^{iλ_{k}t} (c_{k} ∈ ℂ, 0 = λ_{1} < ⋯ < λ_{n} = λ). If λ ⩾ n − 1, then μ\{ t ∈ [−\frac{1}{2}, \frac{1}{2}] : |\frac{d}{dt} log g(t)| > y \} ⩽ 29λ/y for all y > 0.

Assume for the moment that Lemma 1.6 is proved; we can easily finish the proof of Theorem 1.2 by constructing a sequence of exponential polynomials p_{n}, p_{n−1}, …, p_{s} (s ⩾ 1) such that

1) p_{n} = p;

2) ord p_{k} = k (k = s, …, n);

3) ∥p_{k−1}∥_{∞} ⩾ \frac{1}{58}∥p_{k}∥_{∞} (k = s + 1, …, n);

4) the ratio φ_{k} =^{def} |\frac{p_{k−1}}{p_{k}} | satisfies the weak type estimate μ\{ x ∈ [−\frac{1}{2}, \frac{1}{2}] : φ_{k}(x) > t \} ⩽ \frac{1}{t} for t > 0;

5) the difference between the greatest and the smallest exponent of p_{s} does not exceed s − 1 (i.e., p_{s} meets the condition of Case 1 investigated above).

The construction is almost the same as in §3. The difference is that, firstly, we make use of the identity \underline{q}(t) − \overline{q}(t) = i(ρ_{k} − ρ_{1})p_{k}(t), where

p_{k}(t) =^{def} ∑_{m=1}^{k} d_{k}e^{iρ_{m}t}  (ρ_{1} < ⋯ < ρ_{k} ∈ ℝ),
\underline{q}(t) =^{def} e^{iρ_{1}t} \frac{d}{dt}(e^{−iρ_{1}t}p_{k}(t)),
\overline{q}(t) =^{def} e^{iρ_{k}t} \frac{d}{dt}(e^{−iρ_{k}t}p_{k}(t)),

to estimate the sum of norms ∥\underline{q}∥_{∞}+∥\overline{q}∥_{∞} from below, and, secondly, we stop the sequence at the polynomial p_{s} satisfying the condition of Case 1, i.e., at that very moment when we cannot apply Lemma 1.6 to estimate φ_{s} once more. The same reasoning as in §3 shows that |\frac{p_{s}(t)}{p_{n}(t)} | ⩽ \{\frac{2e}{μ(E)}\}^{n−s} outside an exceptional set E′ of measure μ(E′) ⩽ μ(E)/2. Thus,

\begin{aligned}(\frac{1}{58})^{n−s}∥p∥_{∞} & ⩽ ∥p_{s}∥_{∞} ⩽ \{\frac{154}{μ(E ∖ E′)}\}^{s−1} sup_{t∈E∖E′} |p_{s}(t)| \\ & ⩽ \{\frac{308}{μ(E)}\}^{s−1}\{\frac{2e}{μ(E)}\}^{n−s} sup_{t∈E} |p(t)|. \end{aligned}

Now Theorem 1.5 easily follows if we take into account the inequality 116e < 316.

It remains to prove Lemma 1.6. We proceed like we did in Case 1, but now the number λ will play the main role instead of n − 1. Let z_{j} be the complex zeros of g(z) enumerated in the order of increase of | Re z_{j}|. The Langer lemma yields | Re z_{j}| ⩾ π \frac{j−(n−1)}{(n−1)} ⩾ \frac{π}{λ}(j − (n − 1)). We again write the Hadamard factorization

g(z) = ce^{az} ∏_{j⩽2λ}(z − z_{j}) ∏_{j>2λ}(1 − \frac{z}{z_{j}})e^{z/z_{j}} =^{def} ce^{az}Q(z)R(z).

The estimate for |\frac{d}{dz} log R(z)|. As before, we notice that | Re z_{j}| ⩾ π if j > 2λ. Let | Re z| < π/2, then

|\frac{d}{dz} log R(z)| ⩽ |z| ∑_{j>2λ} \frac{1}{| Re z_{j}|(| Re z_{j}| − π/2)} ⩽ 2|z| ∑_{j>2λ} \frac{1}{| Re z_{j}|^{2}}

⩽ 2|z| ∑_{j>2λ} \frac{λ^{2}}{π^{2}} \frac{1}{(j − (n − 1))^{2}} ⩽ 2\frac{λ^{2}}{π^{2}} |z| ∑_{j>2λ} ∫_{j−(n−1)−1/2}^{j−(n−1)+1/2} \frac{dt}{t^{2}}.

But if j > 2λ > 2(n − 1), then j ⩾ 2n − 1, and j − (n − 1) − 1/2 ⩾ 1/2 ⩾ λ. Therefore,

∑_{j>2λ} ∫_{j−(n−1)−1/2}^{j−(n−1)+1/2} \frac{dt}{t^{2}} ⩽ ∫_{λ}^{∞} \frac{dt}{t^{2}} = \frac{1}{λ},

and |\frac{d}{dz} log R(z)| ⩽ \frac{2|z|λ}{π^{2}} if | Re z| < π/2. In particular, |\frac{d}{dz} log R(z)| ⩽ \frac{λ}{π^{2}} on the interval [−\frac{1}{2}, \frac{1}{2}].

The estimate for |a|. One may proceed like it was done above to obtain the estimate for Re a, i.e., to consider the argument increment of g(z) along the segment [−iω \frac{\overline{a}}{|a|}, iω \frac{\overline{a}}{|a|}], where 0 < ω < π (if we let ω → π, the estimate for R(z) fails) but for diversity we go another way and use … the Turan lemma! Consider an exponential polynomial g̃(t) =^{def} e^{λt}g(−\frac{\overline{a}}{|a|}t) on the interval t ∈ [−\frac{3}{2}, \frac{3}{2}]; its remarkable property is that the real parts of exponents in its terms are nonnegative. The reasoning of the first half of §1 ensures the estimate

sup_{t∈[− \frac{3}{2} ,− \frac{1}{2} ]} |g̃(t)| ⩽ sup_{t∈[− \frac{3}{2} , \frac{3}{2} ]} |g̃(t)| ⩽ (12e)^{n−1} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]} |g̃(t)| ⩽ (12e)^{λ} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]} |g̃(t)|.

The function Q(−\frac{\overline{a}}{|a|}t) is an algebraic polynomial of degree at most 2λ, consequently, it is a limit of exponential polynomials of order at most 2λ + 1 with purely imaginary exponents. Applying the Turan lemma again, we obtain the inequalities

sup_{t∈[− \frac{3}{2} ,− \frac{1}{2} ]}|Q(−\frac{\overline{a}}{|a|}t)| ⩾ (12e)^{−2λ} sup_{t∈[− \frac{3}{2} , \frac{3}{2} ]}|Q(−\frac{\overline{a}}{|a|}t)|

⩾ (12e)^{−2λ} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]}|Q(−\frac{\overline{a}}{|a|}t)|

and

inf_{t∈[− \frac{3}{2} ,− \frac{1}{2} ]}|R(−\frac{\overline{a}}{|a|}t)| ⩾ exp\{− ∫_{−3/2}^{3/2}|\frac{d}{dt} log R(−\frac{\overline{a}}{|a|}t)| dt\} sup_{t∈[− \frac{3}{2} , \frac{3}{2} ]}|R(−\frac{\overline{a}}{|a|}t)|

⩾ exp\{− ∫_{−3/2}^{3/2} \frac{2|t|λ}{π^{2}} dt\} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]}|R(−\frac{\overline{a}}{|a|}t)|

⩾ e^{−λ/2} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]}|R(−\frac{\overline{a}}{|a|}t)|.

If |a| > λ, then inf_{t∈[− \frac{3}{2} ,− \frac{1}{2} ]} |ce^{(λ−|a|)t}| ⩾ e^{|a|−λ} sup_{t∈[ \frac{1}{2} , \frac{3}{2} ]} |ce^{(λ−|a|)t}|. Since g̃(t) = ce^{(λ−|a|)t}Q(−\frac{\overline{a}}{|a|}t)R(−\frac{\overline{a}}{|a|}t), these inequalities may hold simultaneously only if |a| ⩽ λ(3 log(12e) + 1/2 + 1) ⩽ \frac{25}{2}λ.

The polynomial Q(z). By Lemma 1.2, the polynomial Q(z) satisfies the weak type estimate μ\{ t ∈ [−\frac{1}{2}, \frac{1}{2}] : |\frac{d}{dt} log Q(t)| > y \} ⩽ \frac{16λ}{y} on the segment [−\frac{1}{2}, \frac{1}{2}].

If we combine all the above estimates and make use of the inequality |\frac{d}{dt} log g(t)| ⩽ |a| + |\frac{d}{dt} log R(t)| + |\frac{d}{dt} log Q(t)|, we obtain

μ\{ t ∈ [−\frac{1}{2}, \frac{1}{2}] : |\frac{d}{dt} log g(t)| > y \}

⩽ μ\{ t ∈ [−\frac{1}{2}, \frac{1}{2}] : |\frac{d}{dt} log Q(t)| > y − 13λ \}

⩽ \frac{16λ}{y − 13λ} ⩽ \frac{29λ}{y}

for y ⩾ 29λ. But if y < 29λ, then the corresponding estimate becomes trivial because \frac{29λ}{y} ⩾ 1 = μ([−\frac{1}{2}, \frac{1}{2}]). Lemma 1.6 is proved.

\underline{C} hapter 2. Random periodization technique and the Morgan theorem

In this chapter we are going to present a very simple method (borrowed from geometry of numbers and, in the main, going back to Siegel), which will allow us to reduce the study of a function f ∈ L^{2}(ℝ) with spectrum of finite measure to the study of some family of trigonometric polynomials with uniformly bounded orders on the unit circumference 𝕋. In this way the following inequality of “uncertainty principle type” will be derived from the Turan lemma:

(2.1)  ∥f∥_{L^{2}(ℝ)}^{2} ⩽ Ae^{Aμ(E)μ(Σ)}(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2});

here E and Σ are arbitrary sets of finite Lebesgue measure (and A is an absolute constant). This inequality essentially improves the famous Amrein–Berthier theorem. An immediate consequence of (2.1) is the uniqueness Theorem 2.3, resembling the Beurling theorem on the divergence of the double integral

∫∫_{ℝ^{2}} |f(x)||f̂(y)|e^{2π|x||y|} dx dy

for every function f ∈ L^{2}(ℝ) not vanishing identically, and, in a sense, generalizing the latter. In Theorems 2.4 and 2.6 we try to treat the case where we have some information about “smallness” of |f| and |f̂| not on the whole real line ℝ but only on one of its halflines, say, ℝ_{−} (though we do not succeed very much). We postpone the discussion of the corresponding classical results to §3.

§2.1. Lattice averaging lemma and random periodization operators

The following simple lemma will be very useful for us.

Lemma 2.1 (the lattice averaging lemma). Let φ: ℝ → ℝ_{+} be a positive summable function, and let ε > 0 be fixed. Then

∫_{1}^{2} ∑_{k∈ℤ∖\{0\}} φ(kεv) dv ⩽ \frac{1}{ε} ∫_{ℝ} φ(t) dt

and

∫_{1}^{2} ∑_{k∈ℤ∖\{0\}} φ(\frac{k}{εv} ) dv ⩽ 4ε ∫_{ℝ} φ(t) dt.

Proof. We have:

\begin{matrix}∫_{1}^{2} ∑_{k>0} φ(kεv) dv = ∑_{k>0} ∫_{1}^{2} φ(kεv) dv =^{(t=kεv)} ∑_{k>0} \frac{1}{εk} ∫_{εk}^{2εk} φ(t) dt \\ = \frac{1}{ε} ∑ ∫_{\begin{matrix}εk<t<2εk \\ k>0 \end{matrix}} \frac{φ(t)}{k} dt = \frac{1}{ε} ∫_{0}^{+∞} φ(t)( ∑_{\frac{t}{2ε} <k< \frac{t}{ε}} \frac{1}{k} ) dt ⩽ \frac{1}{ε} ∫_{0}^{+∞} φ(t) dt, \end{matrix}

since ∑_{\frac{t}{2ε} <k< \frac{t}{ε}} \frac{1}{k} ⩽ 1 for t > 0.

Similarly,

∫_{1}^{2} ∑_{k>0} φ(kεv) dv ⩽ \frac{1}{ε} ∫_{−∞}^{0} φ(t) dt.

Summing these inequalities, we obtain the first statement of the lemma. Further,

\begin{aligned}∫_{1}^{2} ∑_{k>0} φ(\frac{k}{εv} ) dv & = ∑_{k>0} ∫_{1}^{2} φ(\frac{k}{εv} ) dv = ∑_{k>0} \frac{k}{ε} ∫_{k/(2ε)}^{k/ε} \frac{φ(t)}{t^{2}} dt = \frac{1}{ε} ∑ ∫_{\begin{matrix}\frac{k}{2ε} <t< \frac{k}{ε} \\ k>0 \end{matrix}} \frac{φ(t)k}{t^{2}} dt \\ & = \frac{1}{ε} ∫_{0}^{+∞} φ(t)( ∑_{εt<k<2εt} k)\frac{dt}{t^{2}} ⩽ 4ε ∫_{0}^{+∞} φ(t) dt, \end{aligned}

since ∑_{εt<k<2εt} k ⩽ 4ε^{2}t^{2} for t > 0. The rest is clear.

Now let E ⊂ ℝ be a measurable set of finite measure. We consider an arbitrary function f ∈ L^{2}(ℝ) supported on E and fix a positive number ε. We define the random periodization g of the function f by

g(t) = g(ε, v|t) =^{def} \frac{1}{\sqrt{εv}} ∑_{k∈ℤ} f(\frac{k + t}{εv}).

Here v is a random variable equidistributed on the interval (1,2). The series in the definition of g converges in L_{loc}^{1}(ℝ); for every v this series represents a 1-periodic function. An easy computation shows that the Fourier coefficients of g are ĝ_{m} = \sqrt{εv}f̂(mεv) (m ∈ ℤ). ¿From the lattice averaging lemma we shall derive several simple but useful properties of the random periodization.

Proposition 2.2.

(A) μ{ t ∈ (0, 1) : g(t) ≠ 0 } ⩽ 2εμ(E); (B) E∥g∥_{L^{2}(0,1)}^{2} ⩽ 2ε|f̂(0)|^{2} + 2∥f∥_{L^{2}(ℝ)}^{2} ⩽ 2(εμ(E) + 1)∥f∥_{L^{2}(ℝ)}^{2}.

Let Σ ⊂ ℝ be measurable, 0 ∈ Σ. We consider a random lattice Λ = Λ(ε, v) =^{def} { sεv : s ∈ ℤ } and denote 𝔐 = { s ∈ ℤ : sεv ∈ Σ }.

(C) E(card 𝔐 − 1) ⩽ \frac{μ(Σ)}{ε}; (D) E ∑_{m∈ℤ∖𝔐} |ĝ_{m}|^{2} ⩽ 2 ∫_{ℝ∖Σ} |f̂|^{2}.

Proof. (A): The measure of the set of all points t ∈ (0, 1) for which the summand f(\frac{k+t}{εv}) in the series defining g does not vanish is equal to μ(εvE ∩ (k, k + 1)). Therefore,

μ{ t ∈ (0, 1) : g(t) ≠ 0 } ⩽ ∑_{k∈ℤ} μ(εvE ∩ (k, k + 1)) = μ(εvE) ⩽ 2εμ(E).

(B):

E∥g∥_{L^{2}(0,1)}^{2} = E ∑_{k∈ℤ} |ĝ_{k}|^{2} = E|ĝ_{0}|^{2} + E ∑_{k∈ℤ∖{0}} |ĝ_{k}|^{2}.

But |ĝ_{0}|^{2} = εv|f̂(0)|^{2} ⩽ 2ε|f̂(0)|^{2}, and

\begin{aligned}E ∑_{k∈ℤ∖\{0\}} |ĝ_{k}|^{2} & = ∫_{1}^{2}( ∑_{k∈ℤ∖\{0\}} εv|f̂(kεv)|^{2}) dv ⩽ 2ε ∫_{1}^{2}( ∑_{k∈ℤ∖\{0\}} |f̂(kεv)|^{2}) dv \\ & ⩽ 2 ∫_{ℝ} |f̂|^{2} = 2∥f∥_{L^{2}(ℝ)}^{2}. \end{aligned}

It remains to notice that

|f̂(0)|^{2} = |∫_{E} f|^{2} ⩽ μ(E) ∫_{E} |f|^{2} = μ(E)∥f∥_{L^{2}(ℝ)}^{2}.

(C): Since card 𝔐 = 1 + ∑_{k∈ℤ∖{0}} χ_{Σ}(kεv), we have

E(card 𝔐 − 1) = ∫_{1}^{2} ∑_{k∈ℤ∖\{0\}} χ_{Σ}(kεv) dv ⩽ \frac{1}{ε} ∫_{ℝ} χ_{Σ} = \frac{μ(Σ)}{ε}.

(D):

E ∑_{m∈ℤ∖𝔐}|ĝ_{m}|^{2} = ∫_{1}^{2}( ∑_{k∈ℤ∖{0}} εv|f̂(kεv)|^{2}χ_{ℝ∖Σ}(kεv)) dv

= 2ε ∫_{1}^{2}( ∑_{k∈ℤ∖{0}}(|f̂(kεv)|^{2}χ_{ℝ∖Σ})(kεv)) dv

⩽ 2 ∫_{ℝ} |f̂|^{2}χ_{ℝ∖Σ} = 2 ∫_{ℝ∖Σ} |f̂|^{2}. (By L.A.L.)

§2.2. Strong annihilation of sets of finite measure

Let E and Σ be two measurable subsets of ℝ. Borrowing the terminology from Jöricke and Havin, we say that E and Σ annihilate if for every function f ∈ L^{2}(ℝ) the conditions supp f ⊂ E, spec f ⊂ Σ imply that f vanishes identically. We say that E and Σ strongly annihilate if there exists a constant C > 0 such that the inequality

(∗)  ∥f∥_{L^{2}(ℝ)}^{2} ⩽ C(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2})

holds for every function f ∈ L^{2}(ℝ). The strong annihilation condition can be written in a form which is less symmetric but more convenient to verify: E and Σ strongly annihilate if and only if

(∗∗)  ∫_{Σ} |f̂|^{2} ⩽ C′ ∫_{ℝ∖Σ} |f̂|^{2}

for every f ∈ L^{2}(ℝ) supported on E.

There is a relationship between the best possible constants C and C′:

C′ = ctg^{2} α, C = \frac{1}{2 sin^{2} \frac{α}{2}} = \frac{1}{1 − cos α},

where α is the angle between the subspaces L^{2}(E) =^{def} { f ∈ L^{2}(ℝ) : supp f ⊂ E } and L^{2}(Σ̂) =^{def} { f ∈ L^{2}(ℝ) : spec f ⊂ Σ } of the Hilbert space L^{2}(ℝ). The proof of this statement is a simple exercise in geometry. Denoting by P_{E} and PΣ̂ the orthogonal projection onto L^{2}(E) and L^{2}(Σ̂), respectively, we have:

cos α = sup{ |(f, g)| : f ∈ L^{2}(E), g ∈ L^{2}(Σ̂), ∥f∥_{L^{2}(ℝ)}^{2} = ∥g∥_{L^{2}(ℝ)}^{2} = 1 }
= sup{ |(PΣ̂f, g)| : … }
= sup{ ∥PΣ̂f∥_{L^{2}(ℝ)} : f ∈ L^{2}(E), ∥f∥_{L^{2}(ℝ)}^{2} = 1 },

and

C′ = sup\{ \frac{∫_{Σ} |f̂|^{2}}{∫_{ℝ∖Σ} |f̂|^{2}} : f ∈ L^{2}(E), ∥f∥_{L^{2}(ℝ)}^{2} = 1 \}

= sup\{ \frac{∥PΣ̂f∥_{L^{2}(ℝ)}^{2}}{1 − ∥PΣ̂f∥_{L^{2}(ℝ)}^{2}} : f ∈ L^{2}(E), ∥f∥_{L^{2}(ℝ)}^{2} = 1 \}

= \frac{cos^{2} α}{1 − cos^{2} α} = ctg^{2} α.

The computation of the constant C is slightly more complicated. Denote by β and γ the angles between f and the subspaces L^{2}(E) and L^{2}(Σ̂), respectively. It is clear that 0 < β, γ < \frac{π}{2} , β + γ ⩾ α. Since

∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2} = ∥f − P_{E}f∥_{L^{2}(ℝ)}^{2} + ∥f − PΣ̂f∥_{L^{2}(ℝ)}^{2}

= (sin^{2} β + sin^{2} γ)∥f∥_{L^{2}(ℝ)}^{2} ⩾ 2 sin^{2} \frac{α}{2}∥f∥_{L^{2}(ℝ)}^{2},

we have C ⩽ \frac{1}{2 sin^{2} \frac{α}{2}} . To verify the reverse inequality, it suffices to exhibit a function f for which the angles β and γ are close to \frac{α}{2}. This can be done as follows. One can choose g ∈ L^{2}(E) and h ∈ L^{2}(Σ̂) so that ∥g∥_{L^{2}(ℝ)} = ∥h∥_{L^{2}(ℝ)} = 1 and Re(h, g) ≈ cos α, and then

put f =^{def} \frac{1}{2}(g + h).

It should be noted that, proceeding in the same way, one can describe the image of the unit ball of L^{2}(ℝ) under the mapping

L^{2}(ℝ) ∋ f ⟶ (∫_{ℝ∖E} |f|^{2}, ∫_{ℝ∖Σ} |f̂|^{2}) ∈ ℝ_{+}^{2}

provided that each of the subspaces L^{2}(E) and L^{2}(Σ̂) contains a vector making an angle arbitrarily close to \frac{π}{2} with the other subspace (this condition is certainly satisfied if both E and Σ have zero density at infinity, i.e,. if lim_{A→+∞} \frac{μ(E∩[−A,A])}{A} = lim_{A→+∞} \frac{μ(Σ∩[−A,A])}{A} = 0; the corresponding vectors can be chosen among those of the form fe^{iλt} and τ_{λ}g, where f ∈ L^{2}(E), g ∈ L^{2}(Σ̂) and λ is a suitable number from a sufficiently large interval centered at 0). This image turns out to be the square [0, 1]^{2} with the upper-right angle cut off along the curve arccos \sqrt{x}+arccos \sqrt{y} = α (this simple observation is due to Slepian and Pollac).

Excluding α from the formulas for C and C′, we get

C = C′ + 1 + \sqrt{C′(C′ + 1)} ⩽ 2C′ + \frac{3}{2}.

Now we state the main theorem of this section.

Theorem 2.2. For every two sets E and Σ of finite Lebesgue measure and every function f ∈ L^{2}(ℝ), the following inequality holds:

∥f∥_{L^{2}(ℝ)}^{2} ⩽ 130e^{66μ(E)μ(Σ)}(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2}).

Proof. As it was shown above, it suffices to prove that

∫_{Σ} |f̂|^{2} ⩽ 64e^{66μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2}

for every function f ∈ L^{2}(E). We set ε = \frac{1}{4μ(E)} and introduce the random periodization g of the function f. By (A),

μ\{ t ∈ (0, 1) : g(t) = 0 \} =^{def} μ(F) ⩾ 1 − 2εμ(E) = \frac{1}{2}.

We decompose g into a sum p + q, where

p(t) =^{def} ∑_{m:mεv∈Σ∪{0}} ĝ_{m}e^{2πimt} =^{def} ∑_{m∈𝔐} ĝ_{m}e^{2πimt}

(the second relation defines 𝔐) and q(t) =^{def} ∑_{m∈ℤ∖𝔐} ĝ_{m}e^{2πimt}. We have

E∥q∥_{L^{2}(0,1)}^{2} = E ∑_{m∈ℤ∖𝔐}|ĝ_{m}|^{2} ⩽^{(D)} 2 ∫_{ℝ∖Σ} |f̂|^{2},

whence

P\{∥q∥_{L^{2}(0,1)}^{2} > 4 ∫_{ℝ∖Σ} |f̂|^{2}\} < \frac{1}{2}.

Next,

E(ord p − 1) = E(card 𝔐 − 1) ⩽^{(C)} \frac{μ(Σ)}{ε} = 4μ(E)μ(Σ).

Consequently,

P\{ord p > 1 + 8μ(E)μ(Σ)\} < \frac{1}{2}.

We see that,with positive probability, the following 4 events take place simultaneously: (1) μ(F) ⩾ \frac{1}{2};

(2) ∥q∥_{L^{2}(0,1)}^{2} ⩽ 4 ∫_{ℝ∖Σ} |f̂|^{2};

(3) ord p ⩽ 1 + 8μ(E)μ(Σ);

(4) ε|f̂(0)|^{2} = \frac{1}{4μ(E)}|f̂(0)|^{2} ⩽ |p̂_{0}|^{2} = |ĝ_{0}|^{2} (indeed, (1) and (4) always hold, while each of (2) and (3) does not hold with probability

less than \frac{1}{2}). Since g|_{F} ≡ 0, we have p|_{F} ≡ −q|_{F} and ∫_{F} |p|^{2} = ∫_{F} |q|^{2}. Hence μ\{ t ∈ F :

|p(t)|^{2} ⩾ 16 ∫_{ℝ∖Σ} |f̂|^{2} \} ⩽ \frac{1}{4}, and, since μ(F) ⩾ \frac{1}{2}, we get

μ\{ t ∈ (0, 1) : |p(t)| ⩽ 4(∫_{ℝ∖Σ} |f̂|^{2})^{1/2} \} ⩾ \frac{1}{4}.

Now a special case of the Turan lemma (Theorem 1.4) implies

\begin{aligned}\frac{1}{4μ(E)}|f̂(0)|^{2} & ⩽ |p̂_{0}|^{2} ⩽ (∑_{k} |p̂_{k}|)^{2} ⩽ \{(\frac{14}{1/4})^{ord p−1}4(∫_{ℝ∖Σ} |f̂|^{2})^{1/2}\}^{2} \\ & ⩽ 16 × 56^{16μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2}, \end{aligned}

whence

|f̂(0)|^{2} ⩽ 64μ(E)e^{16 log 56μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2}.

If we take the function f_{1}(x) =^{def} f(x)e^{−2πixy} instead of f(x) and the set Σ−y instead of Σ, we arrive at the same estimate for |f̂(y)| and, since y is arbitrary, for sup_{ℝ} |f̂|. Integrating this estimate over Σ, we get the inequality

∫_{Σ} |f̂|^{2} ⩽ 64μ(E)μ(Σ)e^{16 log 56μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2} ⩽ 64e^{66μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2},

which proves the theorem.

Of course, the constant factor in the exponent is estimated very roughly. Analyzing the proof thoroughly one may observe that our method provides a possibility to estimate the constants C and C′ from above by A_{δ}e^{(w+δ)μ(E)μ(Σ)}, where δ > 0 is an arbitrary small positive number, A_{δ} is a constant depending on δ only, and w =^{def} inf_{α∈(0,1)}(\frac{2}{α} log \frac{16e}{π} \frac{1}{1−α}) < 10.6, which is nevertheless very far from the conjectured inequality C ⩽ A_{δ}e^{( \frac{π}{2} +δ)μ(E)μ(Σ)} (the latter cannot be essentially improved, which can be shown by the example f = e^{−πx^{2}}, E = Σ = [−a, a], a → +∞).

Theorem 2.2 is substantial only if μ(E)μ(Σ) ⩾ 1. If μ(E)μ(Σ) < 1, then much simpler ideas work: for every function f ∈ L^{2}(ℝ) supported on E, we have sup |f̂| ⩽ ∥f∥_{L^{1}(ℝ)} ⩽ \sqrt{μ(E)}∥f∥_{L^{2}(ℝ)}, whence ∫_{Σ} |f̂|^{2} ⩽ μ(E)μ(Σ)∥f∥_{L^{2}(ℝ)}^{2}. Consequently, ∫_{ℝ∖Σ} |f̂|^{2} ⩾ (1 −

μ(E)μ(Σ))∥f∥_{L^{2}(ℝ)}^{2} and ∫_{Σ} |f̂|^{2} ⩽ \frac{μ(E)μ(Σ)}{1−μ(E)μ(Σ)} ∫_{ℝ∖Σ} |f̂|^{2}. Recalling the relationship between C and C′, we get

∥f∥_{L^{2}(ℝ)} ⩽ \frac{1}{\sqrt{1 − μ(E)μ(Σ)}}(∫_{ℝ∖E} |f|^{2} + ∫_{ℝ∖Σ} |f̂|^{2})

for every f ∈ L^{2}(ℝ).

Any nontrivial estimate of the constant C implies some restrictions on the simultaneous distribution of the absolute values of the functions f and f̂. We say that a decreasing function Φ: (0, +∞) → [0, +∞] majorizes |f| in distribution if

μ{ t ∈ ℝ : |f(t)| > y } ⩽ μ{ t ∈ (0, +∞) : Φ(t) > y }

for all y > 0. This condition is equivalent to the inequality

μ{ t ∈ ℝ : |f(t)| > Φ(y) } ⩽ y

for all y > 0.

Let Φ and Ψ be majorize |f| and |hf| in distributions. Theorem 2.2 immediately implies the following statement.

Corollary 2.3. For every a, b > 0,

∫_{a}^{+∞} Φ^{2} + ∫_{b}^{+∞} Ψ^{2} ⩾ \frac{1}{130}e^{−66ab}∥f∥_{L^{2}(ℝ)}^{2}.

In particular,

\underline{lim}_{a,b→+∞} e^{66ab}(∫_{a}^{+∞} Φ^{2} + ∫_{b}^{+∞} Ψ^{2}) = 0

implies f ≡ 0.

It follows that the majorants Φ and Ψ cannot decrease as e^{−Bx^{p}} and e^{−B′x^{p′}}, respectively, 1/p + 1/p′ = 1 and B^{1/p}B′^{1/p′} > 33. The last statement can be regarded as an interesting complement to the Morgan theorem discussed in the next section.

§2.3. The Morgan theorem

The following theorem was proved by Morgan [9] in 1934.

Theorem. Let p ∈ [2, +∞), p′ = \frac{p}{p−1} ∈ (1, 2], f ∈ S.

(A) If f(x) = O(e^{−B|x|^{p}}), f̂(x) = O(e^{−B′|x|^{p′}}) as x → ±∞, and B^{1/p}B′^{1/p′} > b_{1}(p) = \frac{2π}{p^{1/p}p′^{1/p′}} | cos \frac{πp′}{2} |^{1/p′}, then f ≡ 0.

(B) If f(x) = O(e^{−B|x|^{p}}), f̂(x) = O(e^{−B′|x|^{p′}}) as x → −∞, and B^{1/p}B′^{1/p′} > b_{2}(p) = \frac{2π}{p^{1/p}p′^{1/p′}} sin \frac{π}{p′}, then f ≡ 0.

Constants b_{1}(p) and b_{2}(p) are the best possible. The conclusion of (A) remains valid if f rapidly decreases only on the left semiaxis while in case (B) an example of a nontrivial function f on which the value b_{2}(p) is “attained” can be chosen so that f(−x) = \overline{f(x)}; the latter means that it does not matter whether the decrease of f is “one-sided” or “twosided”, and the situation is determined by the character of decrease of f̂ (the functions f and f̂ do not have the same rights because p and p′ do not).

We make several remarks concerning the Morgan theorem before proving it.

1) The statement of the theorem is invariant with respect to the linear change of variable x → αx (α > 0). Substituting \overline{f} in place of f, we can see that it does not matter on which particular semiaxis f̂ rapidly decreases.

2) The condition f ∈ S may be replaced by the condition of belonging to an arbitrary other reasonable function class between S and S′. To see this, it suffices to convolve f with a boundedly supported infinitely differentiable function and then to do the same with f̂.

3) Moreover, the convolution allows us to derive (B) from (A) provided we do not aim at making the constant b_{2}(p) best possible. To do this, one can consider the function g =^{def} (f\overline{f̃}) ∗ (\overline{f}f̃) (under the assumptions of (B)), where f̃(x) =^{def} f(−x). Easy estimates show that g decreases on each of the two semiaxis as e^{−B( \frac{|x|}{2} )^{p}}, while ĝ decreases as e^{−B′( \frac{|x|}{2} )^{p′}}, and, therefore, one can take b_{2}(p) = 4b_{1}(p).

4) The Paley–Wiener theorem implies that the one-sided condition f(x) = O(e^{− \frac{(2π+δ)|x|^{p}}{p}}), x → −∞, is equivalent to the analytic continuability of the function f̂ into the upper half-

plane \{Im z ⩾ 0\} with the growth estimate f̂(z) = O(exp\{(2π − δ)\frac{| Im z|^{p′}}{p′} \}), and the

two-sided condition f(x) = O(e^{− \frac{(2π+δ)|x|^{p}}{p}}), x → ±∞, means that f̂ can be extended to an entire function with the same growth estimate (provided f ∈ S, of course). Consequently (the details are left to the reader), the Morgan theorem is equivalent to the following statement.

Let q ∈ (1, 2], and let F (z) be a function analytic in the upper half-plane, continuous up to its boundary and satisfying the conditions F |_{ℝ} ∈ S and F (z) = O(e^{(1−ε)| Im z|^{q}}) with some ε ∈ (0, 1).

(A) If F (x) = O(e^{−B|x|^{q}}) as x → ±∞, x ∈ ℝ, and B > | cos \frac{πq}{2} |, then F ≡ 0.

(B) If F (x) = O(e^{−B|x|^{q}}) as x → −∞, x ∈ ℝ, and B > (sin \frac{π}{2} )^{q}, then F ≡ 0.

The constants | cos \frac{πq}{2} | and (sin \frac{π}{2} )^{q} cannot be improved even for entire functions satisfying the same growth condition.

Proof of the Morgan theorem. (A) We set G(z) = e^{−( \frac{z}{i})^{q}}F (z) (choosing the branch of the q-th power to be positive on (0, +∞)). G(z) is bounded in a small angle bisected by the imaginary axis. Therefore, G ∈ H^{∞}(ℂ_{+}) by the Fragmen–Lindelöff principle (the latter can be applied since q ⩽ 2). But log |G(x)| behaves on the real line like −(B + cos \frac{πq}{2} )|x|^{q}, whence ∫_{ℝ} log |G(x)|\frac{dx}{1+x^{2}} = −∞ provided B > | cos \frac{πq}{2} |. This implies G ≡ 0 and F ≡ 0.

(B) We set G(z) = e^{(z sin \frac{π}{q})^{q}}F (z). This function is bounded on (0, +∞) and in a small angle bisected by the ray Γ\frac{π}{q} (Γ_{α} =^{def} \{ e^{iα}t : t ⩾ 0 \}). Therefore, of the Fragmen– Lindelöff principle, G is bounded inside the angle of angular measure \frac{π}{q} bounded by the rays [0, +∞) = Γ_{0} and Γ\frac{π}{q}. But on these rays G(z) behaves like e^{−δ|z|^{q}}, and, therefore, the integral of log |G| over the boundary of the angle with respect to the harmonic measure diverges, whence G ≡ 0 and F ≡ 0.

Now we shall show how to construct examples of symmetric (with respect to ℝ) entire functions (i.e., those satisfying F (\overline{z}) = \overline{F (z)}) with almost optimal correlation between the rate of growth and decrease. If q = 2, then F (z) = e^{−z^{2}} is an example for whatever we could imagine. Let q ∈ (1, 2). Consider the canonical product P (z) = ∏_{n=1}^{∞}(1+\frac{z}{n^{1/q}}) exp(−\frac{z}{n^{1/q}}). The function log |F (z)| can be estimated from above by the integral

Re ∫_{0}^{+∞}(log(1 + \frac{z}{t^{1/q}} ) − \frac{z}{t^{1/q}} ) dt = Re(z^{q} ∫_{Γ_{−q arg z}}(log(1 + \frac{1}{s^{1/q}} ) − \frac{1}{s^{1/q}} ) ds)
= −γ Re z^{q} = −γ|z|^{q} cos(q arg z),

up to a summand of order o(|z|^{q}) depending only on |z| and not depending on arg z ∈ [−π, π].

(A) The absolute value of the function F (z) = P (z)P (−z) does not exceed exp{−γ(cos qθ+ cos q(π − θ))R^{q} + o(R^{q})\} = exp\{−2γ cos \frac{πq}{2} cos q(θ − \frac{π}{2} )R^{q} + o(R^{q})\} on the rays Re^{±iθ} (θ ∈ [0, π]). The ratio \frac{cos q(θ− \frac{π}{2} )}{(sin θ)^{q}} attains its maximum when θ = \frac{π}{2} ; therefore, |F (z)| ⩽ exp\{2γ| cos \frac{πq}{2} || Im z|^{q} + o(| Im z|^{q})\} (we can replace o(|z|^{q}) by o(| Im z|^{q}) because cos q(θ − \frac{π}{2} ) is negative if θ is near to 0 or π, whence it follows that F is a rapidly decreasing function in a small angle bisected by the real axis). On the other hand, |F (x)| ⩽ exp\{−2γ cos^{2} \frac{πq}{2} |x|^{q} + o(|x|^{q})\} as x → ±∞, x ∈ ℝ.

In case (B), we want the absolute value of the function F (z) to be dominated by exp\{−β cos((\frac{π}{q} − \frac{π}{2} ) + qθ)R^{q} + o(R^{q})\} with some β > 0, on the rays Re^{±iθ}. Since the

ratio \frac{− cos( \frac{π}{q} − \frac{π}{2} +qθ)}{(sin θ)^{q}} attains its maximum at the point θ = \frac{π}{q}, we have

|F (z)| ⩽ exp\{β \frac{cos(\frac{π}{q} − \frac{π}{2} )}{(sin \frac{π}{q})^{q}}| Im z|^{q} + o(| Im z|^{q})\},

while |F (x)| ⩽ exp\{−β cos(\frac{π}{q} − \frac{π}{2} )x^{q} + o(x^{q})\} as x → +∞.

We define F by F (z) = P (z)P (−α^{1/q}z) with a suitable α ∈ (0, +∞). Then

log |F (Re^{±iθ})| ⩽ −γ(cos qθ + α cos q(π − θ))R^{q} + o(R^{q})
= −β(α) cos(qθ + φ(α))R^{q} + o(R^{q}).

The phase shift φ(α) runs over the interval (0, 2π − πq) as α runs from 0 to +∞. But 0 < \frac{π}{q} − \frac{π}{2} < 2π − πq for q ∈ (1, 2), so the desired value of α really exists.

The condition F |_{ℝ} ∈ S follows in both cases from the fast decrease of F in the strip −1 < Im z < 1 (in case (B) it should be noted that cos(\frac{π}{q} − \frac{π}{2} + πq) > 0).

The proof of the Beurling theorem stated at the beginning of this chapter is similar to the above reasoning and is also based on the analytic continuation technique, so we omit it. A detailed proof can be found in [8].

Comparing the statements of the Morgan theorem and Theorem 2.3, it is quite natural to ask the following question: What happens if the condition of fast distribution decrease is one-sided (i.e., it holds only on one of the semiaxis)? All I know about that is presented in §4 and §5. §4 is devoted to the case where f rapidly decreases on the whole real line ℝ, while f̂ only on ℝ_{−}. In this case the following uniqueness theorem, rather close to Theorem 2.3, can be proved.

Theorem 2.4. Let f ∈ L^{2}(ℝ). If Φ majorizes the distribution of |f| on the whole real line ℝ and Ψ majorizes the distribution of |f̂| on the left semiaxis ℝ_{−}, then the condition

\underline{lim}_{a,b→+∞} e^{97ab log ab}(∫_{a}^{+∞} Φ^{2} + ∫_{b}^{+∞} Ψ^{2}) = 0

implies f ≡ 0.

The constant 97 in the exponent can be replaced by 1+ε, but this is of no value because even the logarithmic factor might have been superfluous.

Nevertheless, Theorem 2.4 allows us to catch the best possible rate of decrease for majorants but on a rougher level (with order in place of type): this theorem implies that the estimates Φ(x) = O(e^{−δx^{p}}), Ψ(x) = O(e^{−δx^{q}}) (δ > 0, \frac{1}{p} + \frac{1}{q} < 1) are incompatible unless f ≡ 0.

If the condition of fast decrease is one-sided for both f and f̂, we know much less: in §5 it will be shown only that the intersections of ℝ_{−} with the supports of f and f̂ cannot have finite measure simultaneously.

§2.4. The Turan lemma for H^{2}(𝔻) and a semi-one-sided uniqueness theorem

We shall start with proving the following lemma, which is also interesting in itself.

Lemma 2.5. Let f(z) = c_{0}z^{m_{0}} +c_{1}z^{m_{1}} +⋯+c_{n}z^{m_{n}} +⋯ ∈ H^{2}(𝔻) (0 ⩽ m_{0} < m_{1} < …), ∥f∥_{H^{2}(𝔻)} ⩽ 1. Let G(f) =^{def} exp ∫_{𝕋} log |f| dμ be the geometric mean of |f| over the unit circumference 𝕋. If G(f) ⩽ G ⩽ 1, then |c_{n}| ⩽ e^{n}(1 + log \frac{1}{G})^{n}G.

Proof. We prove the statement by induction on n. Without loss of generality, we can assume m_{0} = 0 (otherwise we consider z^{−m_{0}}f(z) instead of f(z)). Thus, the base of induction is reduced to the inequality |f(0)| ⩽ G(f) ⩽ G, which is immediate because log |f| is subharmonic. Now we assume that the estimate for the coefficient |c_{n−1}| has been proved.

How can the estimate for |c_{n}| be derived from this? The most natural way is to try to construct a function g(z) with the following properties:

1) ∥g∥_{H^{2}(𝔻)} ⩽ ∥f∥_{H^{2}(𝔻)}; 2) the (n − 1)-th nonvanishing coefficient of g(z) is not too far from c_{n};

3) G(g) is only slightly greater than G(f).

It turns out that one can put g = (1 − r^{2})(f′)_{r} with a suitable r ∈ (0, 1) (for a function h analytic in 𝔻, we denote h′(z) =^{def} \frac{d}{dz} h(z), h_{r}(z) =^{def} h(rz), z ∈ 𝔻).

In formal computations, it is convenient to use the mapping Φ_{r}: f → (1 − r^{2})(f′)_{r} defined on the set of all analytic functions in 𝔻. If f admits a reasonable extension to 𝕋, the Cauchy integral formula yields

Φ_{r}f(z) = ∫_{𝕋} \frac{(1 − f^{2})ζ}{(ζ − rz)^{2}} f(ζ) dμ(ζ) = 2 ∫_{𝕋} \frac{(1 − f^{2})ζ}{(ζ − rz)^{2}} Re f(ζ) dμ(ζ)

for z ∈ \overline{𝔻}.

Thus, Φ_{r} is a linear integral operator with the kernel k(z, ζ) = \frac{(1−r^{2})ζ}{(ζ−rz)^{2}} . Besides,

the function |k(z, ζ)| = \frac{(1−r^{2})ζ}{|ζ−rz|^{2}} (z, ζ ∈ 𝕋) is precisely the Poisson kernel. Therefore, ∥Φ_{r}∥_{L^{p}(𝕋)→L^{p}(𝕋)} ⩽ 1, p ∈ [1, +∞], and, in particular, ∥Φ_{r}f∥_{H^{2}} ⩽ ∥f∥_{H^{2}}. Similarly, ∥Φ_{r}f∥_{H^{p}} ⩽ 2∥ Re f∥_{L^{p}(𝕋)}, p ∈ [1, +∞]. Let us look at the action of the operator Φ_{r} on the coefficients of an analytic function f(z). If f(z) = a_{0} + a_{1}z + ⋯ + a_{k}z^{k} + …, then (Φ_{r}f)(z) = (1 − r^{2})a_{1} + 2(1 − r^{2})ra_{2}z + ⋯ + (k(1 − r^{2})r^{k−1}a_{k})z^{k−1} + . . .. Let r^{2} = \frac{k−1}{k+1},

then the factor at the coefficient a_{k} is equal to k(1 − r^{2})r^{k−1} = \frac{2k}{k+1}(\frac{k−1}{k+1})^{\frac{k−1}{2}} ⩾ \frac{2}{e}. Thus, if r is chosen suitably, then the absolute value of the (n − 1)-th nonvanishing coefficient of the function g = Φ_{r}f is at most \frac{2}{e}|c_{n}|.

Finally, we shall estimate G(Φ_{r}f) provided that f ∈ H^{2}, ∥f∥_{H^{2}} ⩽ 1, and G(f) ⩽ G ⩽ 1. We make use of a suitable factorization of f(z). Namely, let e^{φ(z)} be an outer function with modulus not exceeding |f(z)| on 𝕋 and such that the geometric mean of e^{φ(z)} over 𝕋 equals g and |e^{φ(z)}| = |f(z)| whenever |f(z)| ⩾ 1 (z ∈ 𝕋). For example, we can take φ(z) = φ_{1}(z) + iφ̃_{1}(z), where φ_{1}(z) is the harmonic extension of log max(|f(z)|, β) into the unit disk and φ̃_{1}(z) is its harmonic conjugate. If G(f) ⩽ G ⩽ 1, then the number β ∈ [0, 1] can actually be chosen so that G(e^{φ}) = exp ∫_{𝕋} φ_{1} dμ = G. The function f(z) is representable as a product f(z) = e^{φ(z)}h(z) with h(z) ∈ H^{∞}(𝔻), ∥h∥_{∞} ⩽ 1. Since

\begin{aligned}Φ_{r}f & = (1 − r^{2})(f′)_{r} = (1 − r^{2})((h′)_{r} + (φ′)_{r}h_{r})e^{φ_{r}} \\ & = (Φ_{r}h + (Φ_{r}φ)h_{r})e^{φ_{r}} =^{def} ψe^{φ_{r}}, \end{aligned}

we have G(Φ_{r}f) = G(ψ)G(e^{φ_{r}}). But G(e^{φ_{r}}) = exp ∫_{𝕋} φ_{1}(rz) dμ(z) = exp ∫_{𝕋} φ_{1} dμ = G(e^{φ}) = G, and G(ψ) ⩽ ∥ψ∥_{L^{1}(𝕋)} ⩽ 1 + ∥Φ_{r}φ∥_{L^{1}(𝕋)} ⩽ 1 + 2∥ Re φ∥_{L^{1}(𝕋)} = 1 + 2 ∫_{𝕋} |φ_{1}| dμ.

Moreover, ∫_{𝕋} φ_{1}^{+} dμ = ∫_{𝕋} log^{+} |f| dμ ⩽ ∫_{𝕋} \frac{|f|^{2}}{2e} dμ ⩽ \frac{1}{2e}, and ∫_{𝕋} φ_{1} dμ = log G. Therefore, ∫_{𝕋} |φ_{1}| dμ = − ∫_{𝕋} φ_{1} dμ + 2 ∫_{𝕋} φ_{1}^{+} dμ ⩽ log \frac{1}{G} + \frac{1}{e}, and we obtain the inequality G(Φ_{r}f) < 2(1 + log \frac{1}{G})G.

Besides, we have the trivial estimate G(Φ_{r}f) ⩽ ∥Φ_{r}f∥_{H^{2}} ⩽ ∥f∥_{H^{2}} ⩽ 1, whence G(Φ_{r}f) ⩽ min(1, 2(1 + log \frac{1}{G})G) =^{def} G′. Applying the induction assumption to the function g constructed above, we obtain \frac{2}{e}|c_{n}| ⩽ e^{n−1}(1 + log \frac{1}{G′})^{n−1}G′; since G ⩽ G′ ⩽ 2(1 + log \frac{1}{G})G, we have |c_{n}| ⩽ e^{n}(1 + log \frac{1}{G})^{n}G, which proves the lemma.

Lemma 2.5 implies the following simple corollary.

Corollary. If G ⩽ \frac{1}{e}, then

|c_{n}| ⩽ (2e)^{n}(log \frac{1}{G})^{n}G ⩽ (4e)^{n}n!\frac{1}{n!}(\frac{1}{2} log \frac{1}{G})^{n}G ⩽ (4en)^{n}\sqrt{G}.

This estimate is obviously valid also if G ∈ [\frac{1}{e}, +∞]. Compared with Lemma 2.5, the only advantage of this estimate is that now the right-hand side is monotone increasing as a function of G.

Later we shall encounter a function f which satisfies the restrictions ∥f∥_{H^{2}} ⩽ 1, ∫_{F} |f|^{2} ⩽ ρ^{2} < 1, μ(F) = ν ⩾ \frac{1}{2}. In this case the Jensen inequality implies

log G(f) = ∫_{F} log |f| + ∫_{𝕋≠F} log |f| ⩽ (log x ⩽ \frac{x^{2}}{2} )

⩽ \frac{1}{2} ∫_{𝕋} |f|^{2} + \frac{ν}{2} log(\frac{1}{ν} ∫_{F} |f|^{2}) ⩽ \frac{1}{2} + \frac{ν}{2} log(\frac{1}{ν} ρ^{2}),

and, therefore, G(f) ⩽ e^{1/2}(\frac{1}{ν})^{ν/2}ρ^{ν} ⩽ eρ^{ν} ⩽ eρ^{1/2}, which gives |c_{n}| ⩽ (4en)^{n}eρ^{1/4}. Now we derive a semi-one-sided uniqueness theorem from Lemma 2.5. Let f ∈ L^{2}(ℝ), ∥f∥_{L^{2}(ℝ)} = 1. We assume that Φ majorizes the distribution of |f| on

the whole real line ℝ and Ψ majorizes the distribution of |f̂| on the semiaxis ℝ_{−}. Having fixed two positive numbers a, b > 0, we choose some sets E ⊂ ℝ and Σ ⊂ ℝ_{−} so that

μ(E) = a, μ(Σ) = b,

∫_{ℝ∖E} |f|^{2} ⩽ ∫_{a}^{+∞} Φ^{2} =^{def} Φ_{a}^{2}, ∫_{ℝ∖Σ} |f̂|^{2} ⩽ ∫_{b}^{+∞} Ψ^{2} =^{def} Ψ_{b}^{2}.

We put f_{E} = fχ_{E},ε = \frac{1}{4a} and introduce the random periodization g of the function f_{E}. As in §2, we have μ\{ t ∈ (0, 1) : g(t) = 0 \} ⩾ \frac{1}{2}. Besides,

∫_{ℝ_{−}∖Σ} |f̂_{E}|^{2} ⩽ 2(∫_{ℝ_{−}∖Σ} |f̂|^{2} + ∥f − f_{E}∥_{L^{2}(ℝ)}^{2}) ⩽ 2(Φ_{a}^{2} + Ψ_{b}^{2}).

To consider the Fourier coefficients ĝ_{m} with negative indices, we split these indices, as before, into two sets: 𝔐_{1} =^{def} { m ∈ ℤ : m < 0, mεv ∈ Σ } and 𝔐_{2} =^{def} ℤ_{−} ∖ 𝔐_{1}. Properties (B)–(D) of random periodization imply that E card 𝔐_{1} ⩽ \frac{b}{ε} = 4ab, and, therefore, P\{card 𝔐_{1} ⩾ 12ab\} < \frac{1}{3}. Besides,

E ∑_{m∈𝔐_{2}} |ĝ_{m}|^{2} ⩽ 2 ∫_{ℝ_{−}∖Σ} |f̂_{E}|^{2} ⩽ 4(Φ_{a}^{2} + Ψ_{b}^{2}),

whence

P\{ ∑_{m∈𝔐_{2}} |ĝ_{m}|^{2} > 12(Φ_{a}^{2} + Ψ_{b}^{2})\} < \frac{1}{3}.

Finally,

E∥g∥_{L^{2}(0,1)}^{2} ⩽ 2(εa + 1)∥f_{E}∥_{L^{2}(ℝ)}^{2} ⩽ 3 ⟹ P\{∥g∥_{L^{2}(0,1)} > 3\} < \frac{1}{3}.

Thus, with a positive probability the following 4 events take place simultaneously: (1) card 𝔐_{1} ⩽ 12ab;

(2) ∑_{m∈𝔐_{2}} |ĝ_{m}|^{2} ⩽ 12(Φ_{a}^{2} + Ψ_{b}^{2});

(3) ∥g∥_{L^{2}(0,1)} ⩽ 3;

(4) μ\{ t ∈ (0, 1) : g(t) = 0 \} =^{def} μ(F) ⩾ \frac{1}{2}.

Putting p(t) = ∑_{ℤ∖𝔐_{2}} ĝ_{m}e^{2πimt}, we see that the function p(z) has at most 12ab nonvanishing Fourier coefficients with negative indices, if any. Besides,

∫_{F} |p|^{2} dμ = ∫_{F} |g − p|^{2} dμ ⩽ ∫_{𝕋} |g − p|^{2} dμ = ∑_{m∈𝔐_{2}} |ĝ_{m}|^{2} < 12(Φ_{a}^{2} + Ψ_{b}^{2}),

and ∥p∥_{L^{2}(0,1)} ⩽ ∥g∥_{L^{2}(0,1)} ⩽ 3. Applying the estimate from the final remark after Lemma 2.4 with ρ^{2} = \frac{3}{4}(Φ_{a}^{2} + Ψ_{b}^{2}) to the function \frac{1}{3}p (this is not perfectly well because there we dealt with functions from H^{2}(𝔻), but the reduction is trivial), we conclude that

\frac{1}{\sqrt{4a}}|f̂_{E}(0)| ⩽ |p̂_{0}| = |ĝ_{0}| ⩽ 3e(4e × 12ab)^{12ab}\{\frac{3}{4}(Φ_{a}^{2} + Ψ_{b}^{2})\}^{1/8},

whence |f̂_{E}(0)| ⩽ 20\sqrt{a}(144ab)^{12ab}(Φ_{a}^{2} + Ψ_{b}^{2})^{1/8} =^{def} Q_{0}(a, b). Let λ > 0. The same reasoning applied to the function f(t)e^{2πiλt} (with the same set E, and the set Σ shifted to the right by λ) leads to the inequality |f̂_{E}(−λ)| ⩽ Q_{0}(a, b).

Now the absolute value of the Fourier image of the function f(t)e^{−iλt} has the distribution which cannot be majorized by Ψ on the left semi-axis because the interval [0, λ] carrying an unknown distribution has been shifted there. But, surely, the distribution is majorized by the function

Ψ_{(λ)}(x) =^{def} \{ \begin{matrix}+∞, & 0 ⩽ x ⩽ λ, \\ Ψ(x − λ), & x > λ, \end{matrix}

consequently,

|f̂_{E}(λ)| ⩽ 20\sqrt{a}(144a(b + λ))^{12a(b+λ)}(Φ_{a}^{2} + Ψ_{b}^{2})^{1/8} =^{def} Q_{λ}(a, b).

(We have replaced b by b + λ and used the obvious identity (Ψ_{(λ)})_{b+λ}^{2} = Ψ_{b}^{2}).

Assuming that Φ_{a}^{2} + Ψ_{b}^{2} < e^{−97ab log ab} as a, b → +∞ along certain sets, we have Q_{λ}(a, b) → 0 on the same set of pairs (a, b) for every λ ⩾ 0. Then f̂_{E}(y) → 0 for every y ∈ ℝ, while f̂_{E} → f̂ in the sense of L^{2}(ℝ). Thus, f̂ ≡ 0 and f ≡ 0. The theorem is proved.

§2.5. The one-sided Amrein–Berthier theorem

We start with a simple approximation lemma.

Lemma 2.6. Let I ⊂ ℝ be an interval, F ∈ L^{2}(I), and let e ⊂ I be a measurable subset of I. Suppose that for each ρ ∈ (0, \frac{1}{2}) we can find a function Φ_{ρ} ∈ H^{2}(ℂ_{+}) such that

1) ∫_{I∖e} |Φ_{ρ} − F |^{2} dμ ⩽ Cρ^{2};

2) ∥Φ_{ρ}∥_{H^{2}(ℂ_{+})} ⩽ Cρ^{−1/6}.

Then there exists a connected region Ω ⊂ ℂ_{+} with Lipschitz boundary and a function Φ ∈ H^{2}(Ω) such that

1) μ(I ∖ ∂Ω) ⩽ Aμ(e) (A is an absolute constant);

2) Φ = F a.e. on ∂Ω ∩ I.

Proof. Let Γ(x) denote the set { z ∈ ℂ_{+} : Im z ⩾ | Re z − x| }, i.e., the right angle with vertex x ∈ ℝ bisected by a line parallel to the imaginary axis. Let u be the Poisson integral of the function χ_{e}. In other words, u(z) is the harmonic measure of the set e with respect to the region ℂ_{+} and the point z ∈ ℂ_{+}, i.e., u(z) = ω_{ℂ_{+}}(e, z). By the well-known weak type estimate of the nontangential maximal function, the measure of the open set E = \{ x ∈ ℝ : sup_{Γ(x)} u > \frac{1}{4} \} does not exceed Aμ(e). We put Ω =^{def} D_{+} ∩ (⋃_{x∈I∖E} Γ(x)), where D_{+} is the open half-disk in ℂ_{+} whose diameter is the interval I. The set Ω is the half-disk D_{+} from which we cut off all isosceles right triangles based on the constituent intervals of E. Having drawn the corresponding picture, one can easily verify that Ω is a connected and simply connected region with Lipschitz boundary, and ∂Ω ∩ I = I ∖ E. For z ∈ Ω we have

ω_{ℂ_{+}}(I ∖ e, z) = ω_{ℂ_{+}}(I, z) − ω_{ℂ_{+}}(e, z) ⩾ \frac{1}{2} − u(z) ⩾ \frac{1}{4}.

Now we estimate the norm ∥Φ_{ρ} − Φ_{ρ/2}∥_{H^{2}(Ω)}. (As usual, we define the H^{2}(Ω)-norm of

a function φ ∈ H^{2}(Ω) by ∥φ∥_{H^{2}(Ω)} =^{def} inf{ u(ζ_{0}) : Δu = 0 in Ω, |φ|^{2} ⩽ u in Ω }, where ζ_{0}

is a fixed point from Ω. Another form of this definition is ∥φ∥_{H^{2}(Ω)} =^{def} ∥φ ∘ ψ∥_{H^{2}(𝔻)}, where ψ is a conformal mapping of the unit disk 𝔻 onto Ω with ψ(0) = ζ_{0}.) It is well known that ∥φ∥_{H^{2}(Ω)}^{2} = lim_{n→∞} ∫_{∂Ω_{n}} |φ(ζ)|^{2} dω_{Ω_{n}}(ζ, ζ_{0}), where Ω_{n} ↑ Ω are simply connected regions with smooth boundaries containing ζ_{0} and relatively compact in Ω. The function φ = Φ_{ρ}−Φ_{ρ/2} belongs to H^{2}(ℂ_{+}), and ∥φ∥_{H^{2}(ℂ_{+})} ⩽ ∥Φ_{ρ}∥_{H^{2}(ℂ_{+})}+∥Φ_{ρ/2}∥_{H^{2}(ℂ_{+})} ⩽ 3Cρ^{1/6}. Since the function log |φ| is subharmonic in ℂ_{+}, it admits the following estimate at the points ζ ∈ ∂Ω_{n}:

log |φ(ζ)| ⩽ ∫_{ℝ} log |φ(x)|P (ζ, x) dx
= ∫_{e′} log |φ(x)|P (ζ, x) dx + ∫_{ℝ∖e′} log |φ(x)|P (ζ, x) dx

⩽ γ(ζ) log(∫_{e′} P (ζ, x)|φ(x)|^{2} dx)^{\frac{1}{2}}

+ (1 − γ(ζ)) log(∫_{ℝ∖e′} P (ζ, x)|φ(x)|^{2} dx)^{\frac{1}{2}} .

(Here P is the Poisson kernel for ℂ_{+}, e′ =^{def} I ∖ e, and γ(ζ) =^{def} ω_{ℂ_{+}}(I ∖ e, ζ)). Replacing in the last sum ∫_{ℝ∖e′} by ∫_{ℝ}, exponentiating both sides, and taking into account that γ(ζ) ⩾ \frac{1}{4}, we obtain

|φ(ζ)|^{2} ⩽ (∫_{e′} P (ζ, x)|φ(x)|^{2} dx)^{\frac{1}{4}} (∫_{ℝ} P (ζ, x)|φ(x)|^{2} dx)^{\frac{3}{4}} .

Therefore,

∫_{∂Ω_{n}} |φ(ζ)|^{2} dω_{Ω_{n}}(ζ, ζ_{0})

⩽ ∫_{∂Ω_{n}}(∫_{e′} P (ζ, x)|φ(x)|^{2} dx)^{\frac{1}{4}} (∫_{ℝ} P (ζ, x)|φ(x)|^{2} dx)^{\frac{3}{4}} dω_{Ω_{n}}(ζ, ζ_{0})

⩽ (∫_{∂Ω_{n}}(∫_{e′} P (ζ, x)|φ(x)|^{2} dx) dω_{Ω_{n}}(ζ, ζ_{0}))^{\frac{1}{4}}

× (∫_{∂Ω_{n}}(∫_{ℝ} P (ζ, x)|φ(x)|^{2} dx) dω_{Ω_{n}}(ζ, ζ_{0}))^{\frac{3}{4}}

= (∫_{e′} P (ζ_{0}, x)|φ(x)|^{2} dx)^{\frac{1}{4}} (∫_{ℝ} P (ζ_{0}, x)|φ(x)|^{2} dx)^{\frac{3}{4}}

since P (ζ, x) is harmonic with respect to ζ ∈ Ω_{n}. Consequently,

∥φ∥_{H^{2}(Ω)}^{2} ⩽ (∫_{e′} P (ζ_{0}, x)|φ(x)|^{2} dx)^{\frac{1}{4}} (∫_{ℝ} P (ζ_{0}, x)|φ(x)|^{2} dx)^{\frac{3}{4}}

⩽ max_{x∈ℝ} P (ζ_{0}, x) × (∫_{e′} |φ(x)|^{2} dx)^{\frac{1}{4}} (∫_{ℝ} |φ(x)|^{2} dx)^{\frac{3}{4}}

⩽ C′ρ^{\frac{1}{2}} ρ^{− \frac{1}{3}} = C′ρ^{\frac{1}{6}} ,

i.e., ∥Φ_{ρ} − Φ_{ρ/2}∥_{H^{2}(Ω)} ⩽ C′ρ^{\frac{1}{12}} . This estimate immediately implies that the sequence

{Φ_{2^{−k}}}_{k=1}^{∞} is fundamental in H^{2}(Ω), whence there exists a limit lim_{H^{2}(Ω)} Φ_{2^{−k}} =^{def} Φ. Since every density point of e belongs to E, we have Φ_{2^{−k}} → F on ∂Ω ∩ I = I ∖ E a.e. with respect to the Lebesgue measure on ℝ or, what is the same, with respect to the arc length on ∂Ω. But ∂Ω is rectifiable, and, therefore, Φ_{2^{−k}} → F a.e. with respect to the harmonic measure ω_{Ω}(·, ζ_{0}). So, Φ = F a.e. on ∂Ω ∩ I with respect to both of these measures (in the sense of nontangential boundary values). Thus, Lemma 2.6 is proved.

Now we take f ∈ L^{2}(ℝ) and suppose that the sets spec_{−} f =^{def} spec f∩ℝ_{−} and supp_{−} f =^{def} supp f ∩ ℝ_{−} have finite measure; we want to show that f ≡ 0. If μ(supp_{−} f) = 0, then f̂ is the boundary function of some function from H^{2}(ℂ_{−}), and there is nothing to prove. Let μ(supp_{−} f) > 0. Replacing, if needed, f(x) by f(Bx), where B > 0 is large enough, we can assume that

a) μ(supp_{−} f) ⩽ \frac{1}{2};

b) μ(supp_{−} f ∩ (−∞, −1)) ⩽ \frac{δ}{2}μ(supp_{−} f ∩ (−1, 0)), where δ is a small positive number to be chosen later. Replacing f(x) by e^{iλx}f(x) with sufficiently large λ > 0, we can also assume that

c) μ(spec_{−} f) ⩽ \frac{1}{7}.

Let Σ =^{def} spec_{−} f. The set Σ_{1} =^{def} ⋃_{k∈ℤ}(Σ + k) is a 1-periodic set containing Σ. We set

σ =^{def} μ(Σ_{1} ∩ (0, 1)), then σ ⩽ μ(spec_{−} f) ⩽ \frac{1}{7}. Let ρ ⩽ \frac{1}{2}. There exists a trigonometric polynomial q(ξ) = q_{ρ}(ξ) = ∑_{k=0}^{n} q_{k}e^{−2πikξ} with the following properties:

1) q_{0} = 1;

2) max_{ξ∈ℝ} |q(ξ)| ⩽ ρ^{−1/6};

3) ∫_{ℝ_{−}} |f̂(ξ)|^{2}|q(ξ)|^{2} dξ ⩽ 2∥f∥_{L^{2}(ℝ)}ρ^{2}.

Indeed, consider an outer function Q(z) such that

|Q(e^{2πit})| = \{ \begin{matrix}ρ, & t ∈ Σ_{1}, \\ ρ^{− \frac{σ}{1−σ}} , & t ∉ Σ_{1}. \end{matrix}

¿From this definition it follows that Q(0) = 1, sup_{|z|⩽1} |Q(z)| ⩽ ρ^{−1/6}, and

∫_{ℝ_{−}} |f̂(ξ)|^{2}|Q(e^{2πiξ})|^{2} dξ ⩽ ρ^{2} ∫_{Σ} |f̂(ξ)|^{2} dξ ⩽ ∥f∥_{L^{2}(ℝ)}ρ^{2}.

Now it is clear that we can take q(ξ) = \overline{Q_{n}(e^{2πiξ})}, where Q_{n} is the Fejer polynomial for the function Q of a sufficiently large degree n.

Let Φ_{ρ} =^{def} (qf̂ξ_{ℝ_{+}})^{∨} = (qf̂)^{∨} − (qf̂χ_{Σ})^{∨}. We have (qf̂)^{∨}(x) = ∑_{k=0}^{n} q̂_{k}f(x − k) = f(x)

for every x ∈ (−1, 0) ∖ ⋃_{k=1}^{∞}(supp_{−} f + k). If e =^{def} (−1, 0) ∩ (⋃_{k=1}^{∞}(supp_{−} f + k)), then, obviously, μ(e) ⩽ μ(supp_{−} f ∩ (−∞, −1)). Moreover,

∥qf̂χ_{Σ}∥_{L^{2}(ℝ)}^{2} = ∫_{ℝ_{−}} |f̂(ξ)|^{2}|q(ξ)|^{2} dξ ⩽ 2∥f∥_{L^{2}(ℝ)}ρ^{2}.

Applying Lemma 2.6 to I = (−1, 0), F = f, and the set e defined above, we get a region Ω and a function Φ ∈ H^{2}(Ω) such that Φ = f, a.e. on ∂Ω ∩ I and μ(I ∖ ∂Ω) ⩽ Aμ(e). Now from a) and b) it follows that if δA < 1,then ∂Ω intersects each of the sets (−1, 0) ∩ supp f and (−1, 0) ∖ supp f by a set of positive measure, and, therefore, Φ vanishes on a subset of positive length in ∂Ω, not being zero identically. This contradiction completes the proof.

\underline{C} hapter 3. Functions with nearly linear dependent small shifts and the Zygmund theorem

Our main purpose in this chapter is to prove the quantitative version of the Zygmund theorem. However, we start with a study of some auxiliary class of functions that behave “in the small” like exponential polynomials with pure imaginary exponents.

Definition. Let v ∈ ℕ, τ, κ > 0. We shall say that f ∈ EP_{loc}^{n}(τ, κ) if f ∈ L^{2}(ℝ) and for every t ∈ (0, τ) the functions f_{kt}(·) =^{def} f(· + kt) (k = 0, …, n) are nearly linear dependent, which means, more precisely, that there exist numbers a_{0}(t), …, a_{n}(t) ∈ ℂ such that ∑_{k=0}^{n} |a_{k}(t)|^{2} = 1 and ∥∑_{k=0}^{n} a_{k}(t)f_{kt}∥_{L^{2}(ℝ)} ⩽ κ. Since any n+1 shifts on an exponential polynomial of order n are linear dependent, the above property really makes the function f somewhat similar to an exponential polynomial. This relationship is rather close in the following sense:

1) Every exponential polynomial p of order n with spectrum λ_{1}, …, λ_{n} ∈ ℝ satisfies the equation p̂(·) ∏_{j=1}^{n}(· − λ_{j}) = 0; similarly, every function f ∈ EP_{loc}^{n}(τ, κ) satisfies the inequality

∥f̂(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} ⩽ {20(n + 1)}^{2n}κ

with suitable λ_{1}, …, λ_{n} ∈ ℝ (here θ_{τ} (x) =^{def} min(1, τ|x|)). It is quite natural to call the set of these numbers λ_{1}, …, λ_{n} the essential spectrum of the function f (the latter is not uniquely determined, of course, but the arbitrariness in its choice is very like that in the choice of n points of spectrum for an exponential polynomial whose true order is strictly less than n).

The proof of the above statement is given in §3.1.

2) On intervals of length comparable with τ a function f ∈ EP_{loc}^{n}(τ, κ) behaves like an exponential polynomial of order n. More precisely, in §3.2 we show that for each essential spectrum λ_{1}, …, λ_{n} of the function f there exists an “error function” Φ ∈ L^{2}(ℝ) with the properties that its L^{2}-norm does not exceed {200(n + 1)}^{2n}κ and for every interval J ⊂ ℝ of length |J| ⩽ Mτ (M > 1) one can find an exponential polynomial p^{(J)} with spectrum λ_{1}, …, λ_{n} satisfying the inequality |f(x) − p^{(J)}(x)| ⩽ M^{n}Φ(x) for all x ∈ J.

If we think n to be fixed and let the number κ tend to zero, the described properties are characteristic. Indeed, if we define a_{k}(t) by ∏_{j=1}^{n}(z − e^{2πiλ_{j}}) = ∑_{k=0}^{n} a_{k}(t)z^{k}, then the inequality ∥f̂(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j}))∥_{L^{2}(ℝ)} ⩽ κ implies

\begin{aligned}∥∑_{k=0}^{n} a_{k}(t)f_{kt}∥_{L^{2}(ℝ)} & = ∥f̂(·) ∏_{j=1}^{n}(e^{2πit·} − e^{2πitλ_{j}})∥_{L^{2}(ℝ)} \\ & ⩽ (2π)^{n}∥f̂(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} ⩽ (2π)^{n}κ \end{aligned}

since |e^{2πitλ} − e^{2πitλ_{j}} | ⩽ 2πθ_{τ} (λ − λ_{j}) for every t ∈ (0, τ), λ ∈ ℝ. But, if a_{k}(t) are those chosen above, then it is evident that ∑_{k=0}^{n} |a_{k}(t)|^{2} ⩾ |a_{n}(t)|^{2} = 1, so f ∈ EP_{loc}^{n}(τ, (2π)^{n}κ).

On the other hand, if a function f can be approximated on every interval of length nτ by an exponential polynomial with spectrum λ_{1}, …, λ_{n}, and if the error of the approximation is majorized by a function Φ, ∥Φ∥_{L^{2}(ℝ)} ⩽ κ, then with the same choice of a_{k}(t), for each x ∈ ℝ we have

\begin{aligned}|∑_{k=0}^{n} a_{k}(t)f_{kt}(x)| & = |∑_{k=0}^{n} a_{k}(t)f(x + kt)| = |∑_{k=0}^{n} a_{k}(t)(f(x + kt) − p(x + kt))| \\ & ⩽ ∑_{k=0}^{n} |a_{k}(t)|Φ(x + kt) ⩽ (∑_{k=0}^{n} |a_{k}(t)|^{2})^{\frac{1}{2}} (∑_{k=0}^{n} Φ(x + kt)^{2})^{\frac{1}{2}} . \end{aligned}

Here p stands for an exponential polynomial approximating f on the interval [x, x + nτ], and, clearly, the latter contains all the points of the form x + kt (k = 0, …, n,t ∈ (0, τ)). Squaring the two sides of the last inequality and integrating over ℝ, we obtain the estimate

∥∑_{k=0}^{n} a_{k}(t)f_{kt}∥_{L^{2}(ℝ)} ⩽ \sqrt{n + 1}(∑_{k=0}^{n} |a_{k}(t)|^{2})^{\frac{1}{2}} ∥Φ∥_{L^{2}(ℝ)} ⩽ \sqrt{n + 1}(∑_{k=0}^{n} |a_{k}(t)|^{2})^{\frac{1}{2}} κ,

which coincides (up to normalization) with the condition f ∈ EP_{loc}^{n}(τ, \sqrt{n + 1}κ).

In §3.3 we try to find an analogue of the Turan lemma for functions of class EP_{loc}^{n}(τ, κ). It will be shown that, if f ∈ EP_{loc}^{n}(τ, κ) is small (say, in L^{2}-sense) outside a set E of finite measure, f cannot be too large on a certain part of E, the measure of this part depending on how well E can be approximated by a union of intervals of length comparable with τ, or, in other words, how much E changes under a shift by nτ. In §3.4 and §3.5 we apply the technique developed in §§3.1–3.3 to prove quantitative versions of the Zygmund theorem for the unit circumference and the real line, respectively. It should be noted that in §§3.1–3.3 we deal with functions defined on the real line because this allows us to avoid digressions to unessential technical details and analysis of various cases. Therefore, §3.5 should, perhaps, precede §3.4. I placed the case of circumference first merely because theorems of this kind for Fourier series are more usual than those for Fourier integrals. The changes that should be made in the arguments of §§3.1–3.3 to deal with the case of circumference are pointed out in §3.4.

§3.1. A spectral description of the class EP_{loc}^{n}(τ, κ)

The main result of this section has already been stated in the introduction to this chapter. It remains only to prove it. Our reasoning will be based on a simple observation: because of the identity

∥∑_{k=0}^{n} a_{k}(t)f_{kt}∥_{L^{2}(ℝ)} = ∥f̂(·)q_{t}(e^{2πit·})∥_{L^{2}(ℝ)},

where q_{t}(z) =^{def} ∑_{k=0}^{n} a_{k}(t)z^{k}, z ∈ ℂ, the norm of the linear combination ∑_{k=0}^{n} a_{k}(t)f_{kt} can be small only if large values of f̂ are multiplied by small values of q_{t}.

First we show that f̂ is essentially concentrated on n intervals of length about \frac{1}{n^{2}τ}. More precisely, the following statement is true.

Lemma 3.1. Let δ ⩽ \frac{1}{8n(n+1)}. If f ∈ EP_{loc}^{n}(τ, κ), then there exist n intervals I_{1}, …, I_{n}, each of length \frac{2δ}{τ}, such that

∫_{ℝ∖∪_{1}^{n}I_{k}} |f̂|^{2} ⩽ 4(n + 1)(\frac{4e}{δ})^{2n}κ^{2}.

Proof. The shift operator being continuous in L^{2}(ℝ), we can assume the functions a_{k}(t) in the definition of the class EP_{loc}^{n}(τ, κ) to be piecewise constant and, thus, measurable. Averaging the inequality ∫_{ℝ} |f̂(λ)|^{2}|q_{t}(e^{2πitλ})|^{2} dλ ⩽ κ^{2} over the interval (0, τ), we obtain

∫_{ℝ} |f̂(λ)|^{2}\{∫_{0}^{τ} |q_{t}(e^{2πitλ})|^{2} \frac{dt}{τ}\} dλ =^{def} ∫_{ℝ} |f̂(λ)|^{2}ρ^{2}(λ) dλ ⩽ κ^{2}.

Let us show that the function ρ^{2}(λ) cannot be very small simultaneously at n + 1 points ξ_{1} < ⋯ < ξ_{n+1} satisfying ξ_{j+1} − ξ_{j} ⩾ \frac{2δ}{τ} for j = 1, …, n. Indeed, for every t ∈ (\frac{τ}{2} , τ),

the set S_{t} =^{def} \{ ξ ∈ ℝ : |q_{t}(e^{2πitξ})| < (\frac{δ}{4e})^{n} \} is a union of at most n \frac{1}{t}-periodic systems of intervals of the form \{ J + \frac{k}{t} : k ∈ ℤ \}, where J is an interval of length \frac{δ}{t} < \frac{2δ}{τ} or less. To

see this, one can argue as follows. Consider the set U =^{def} \{ z ∈ 𝕋 : |q_{t}(z)| < (\frac{δ}{4e})^{n} \}. Its boundary is determined by the equation |q_{t}(z)|^{2} = (\frac{δ}{4e})^{2n} or, which is the same, by

(∑_{k=0}^{n} a_{k}(t)z^{k})(∑_{k=0}^{n} \overline{a}_{k}(t)z^{−k}) = (\frac{δ}{4e})^{2n}

(we have used the identity \overline{z} = 1/z for z ∈ 𝕋). The left-hand side is a rational function of degree at most 2n. Hence the number of solutions of this equation does not exceed 2n, and, therefore, U consists of at most n arcs. The length of each of them can be estimated from above by 2πδ, in accordance with the Turan lemma (which is applied here in its simplest form: q_{t} is a polynomial of degree n) if we take into account that

sup_{z∈𝕋} |q_{t}(z)| ⩾ (∫_{𝕋} |q_{t}|^{2} dμ)^{1/2} = (∑_{k=0}^{n} |a_{k}(t)|^{2})^{1/2} = 1.

On the real line each of these arcs becomes a system of intervals of the above kind.

If

∑_{j=1}^{n+1}|q_{t}(e^{2πitξ_{j}})|^{2} < (\frac{δ}{4e})^{2n}

for some t ∈ (0, τ), then all the points ξ_{j} belong to S_{t}. By virtue of the Dirichlet principle, two of them, say, ξ_{j′} and ξ_{j′′}(j′ < j′′), lie in the same periodic system of intervals constituting the set S_{t}. This means that there exists k ∈ ℤ such that |ξ_{j′′} −ξ_{j′} − \frac{k}{t} | ⩽ \frac{δ}{t} < \frac{2δ}{τ}. Since ξ_{j′′} − ξ_{j′} > \frac{2δ}{τ}, we have k > 0. Let G be the union of \frac{n(n+1)}{2} intervals of length \frac{4δ}{τ} centered at the points ξ_{j′′} −ξ_{j′} (1 ⩽ j′ < j′′ ⩽ n). The above condition is equivalent to the following one: there exists k ∈ ℕ such that k/t ∈ G or, which is the same, ∑_{k=1}^{∞} χ_{G}(\frac{k}{t} ) ⩾ 1. The lattice averaging lemma (with φ = χ_{G} and ε = τ/2) implies

\begin{aligned}∫_{τ/2}^{τ} ∑_{k>0} χ_{G}(\frac{k}{t} ) dt & = \frac{τ}{2} ∫_{1}^{2} ∑_{k>0} χ_{G}(\frac{k}{(τ/2)v} ) dv \\ & ⩽ \frac{τ}{2} × 4\frac{τ}{2} ∫_{ℝ} χ_{G}(ξ) dξ ⩽ \frac{τ}{2} × 4\frac{τ}{2} × \frac{1}{4τ} = \frac{τ}{4} , \end{aligned}

since μ(G) ⩽ \frac{n(n+1)}{2} \frac{4δ}{τ} = \frac{2δn(n+1)}{τ} ⩽ \frac{1}{4τ} (we have assumed that δ ⩽ \frac{1}{8n(n+1)}). Thus,

the measure of the set of all t ∈ (τ/2, τ) satisfying ∑_{j=1}^{n+1} |q_{t}(e^{2πitξ_{j}})|^{2} < (\frac{δ}{4e})^{2n} does not exceed τ/4. Then we have

∑_{j=1}^{n+1} ρ^{2}(ξ_{j}) = ∫_{0}^{τ} ∑_{j=1}^{n+1}|q_{t}(e^{2πitξ_{j}})|^{2} \frac{dt}{τ} ⩾ \frac{1}{4}(\frac{δ}{4e})^{2n}.

Now let S =^{def} \{ λ ∈ ℝ : ρ^{2}(λ) < \frac{1}{4(n+1)}(\frac{δ}{4e})^{2n} \}. It is obvious that

∫_{ℝ∖S} |f̂(λ)|^{2} dλ ⩽ 4(n + 1)(\frac{4e}{δ})^{2n}κ^{2}.

On the other hand, we cannot choose n + 1 points ξ_{1} < ⋯ < ξ_{n+1} from S so that ξ_{j+1} − ξ_{j} ⩾ \frac{2δ}{τ} for each j = 1, …, n, which precisely means that S can be covered by n intervals of length \frac{2δ}{τ}. Lemma 3.1 is proved.

Let S̃ = ∪_{1}^{n}Ĩ_{j}, where Ĩ_{j} is a \frac{δ}{τ} -neighborhood of the interval I_{j}, i.e., the interval of length \frac{4δ}{τ} centered at the same point. By the same argument as above, it can easily be shown that there exists t_{0} ∈ (τ/2, τ) such that any arithmetic progression with difference t_{0} contains at most one point from S̃. Indeed, the existence of an arithmetic progression with difference 1/t containing 2 or more points from S̃ is equivalent to the existence of a number k ∈ ℤ ∖ {0} such that k/t ∈ G̃ =^{def} S̃− S̃. Taking into account that G̃ is symmetric with respect to the origin, we obtain ∑_{k∈ℤ∖{0}} χG̃(k/t) ⩾ 2 (together with k/t, the set G̃ must contain −k/t). But μ(G̃) ⩽ \frac{8δ}{τ} n^{2}, and now it follows from the lattice averaging lemma that

∫_{τ/2}^{τ} ∑_{k∈ℤ∖{0}} χG̃(k/t) dt ⩽ τ^{2}μ(G̃) ⩽ 8δn^{2}τ < τ,

because δ ⩽ \frac{1}{8n(n+1)} < \frac{1}{8n^{2}} . Thus, the measure of the set of “bad” t ∈ (τ/2, τ) is strictly less than τ/2, which proves the claim.

Now we show that for every λ ∈ ∪_{1}^{n}I_{j} the absolute value of the factor q_{t_{0}}(e^{2πit_{0}λ}) can be estimated from below by ∏_{j=1}^{n} θ_{τ} (λ − λ_{j}) if the numbers λ_{j} ∈ ℝ are suitably chosen. Let z_{1}, …, z_{n} be all complex zeros of the polynomial q_{t_{0}}(z). We have q_{t_{0}}(z) = α ∏_{j=1}^{n}(z − z_{j}).

Let ζ_{j} =^{def} \frac{z_{j}}{|z_{j}|}, p(z) =^{def} ∏_{j=1}^{n}(z − ζ_{j}). Since the ratio |\frac{z−ζ_{j}}{z−z_{j}} | attains its maximum \frac{2}{1+|z_{j}|} at the point z = −ζ_{j}, we have

|\frac{p(z)}{q_{t_{0}}(z)}| ⩽ \frac{1}{|α|} ∏_{j=1}^{n} \frac{2}{1 + |z_{j}|}

for every z ∈ 𝕋. But, as shown above, sup_{𝕋} |q_{t_{0}}| ⩾ 1. Therefore |α| ⩾ ∏_{j=1}^{n} \frac{1}{1+|z_{j}|}, whence

we obtain the inequality |p(z)| ⩽ 2^{n}|q_{t_{0}}(z)| (z ∈ 𝕋). Since ∥f̂(·)q_{t_{0}}(e^{2πit_{0}·})∥_{L^{2}(ℝ)} ⩽ κ, we have ∥f̂(·)p(e^{2πit_{0}·})∥_{L^{2}(ℝ)} ⩽ 2^{n}κ. The real zeros of the trigonometric polynomial p(e^{2πit_{0}·}) form n arithmetic progressions with difference 1/t_{0}. We denote by λ_{1}, …, λ_{m} the zeros lying in S̃. It should be noticed that m ⩽ n because each of the above progressions has at most one point in S̃. Each λ_{j} (j = 1, …, m) corresponds to the root e^{2πit_{0}λ_{j}} of the polynomial p(z). To the set λ_{1}, …, λ_{m} we add certain numbers λ_{m+1}, …, λ_{n}, so that the set {e^{2πit_{0}λ_{j}}}_{j=1}^{n} coincide with the set of all roots of p. Then

|p(e^{2πit_{0}λ})| = 2^{n} ∏_{j=1}^{n}|sin πt_{0}(λ − λ_{j})| (λ ∈ ℝ).

Since | sin πx| ⩾ 2|x| (x ∈ [−\frac{1}{2}, \frac{1}{2}]), the factor | sin πt_{0}(λ − λ_{j})| must be at least 2t_{0}d_{j}(λ), where d_{j}(λ) is the distance from λ to the nearest point of the form λ_{j} + k/t_{0}, k ∈ ℤ. But if λ ∈ ∪_{k=1}^{n}I_{k}, there can be at most one point of this kind whose distance from λ is less than δ/τ, namely, λ_{j} itself. Hence, for every λ ∈ ∪_{k=1}^{n}I_{k} we get

d_{j}(λ) ⩾ min(\frac{δ}{τ} , |λ − λ_{j}|) ⩾ \frac{δ}{τ} min(1, τ|λ − λ_{j}|) = \frac{δ}{τ} θ_{τ} (λ − λ_{j}),

because δ ⩽ \frac{1}{8n(n+1)} < 1. The inequality t_{0} ⩾ \frac{τ}{2} implies | sin πt_{0}(λ − λ_{j})| ⩾ 2t_{0}q(\frac{δ}{τ})θ_{τ} (λ − λ_{j}) ⩾ δθ_{τ} (λ − λ_{j}). Therefore, |p(e^{2πit_{0}λ})| ⩾ 2^{n}δ^{n} ∏_{j=1}^{n} θ_{τ} (λ − λ_{j}), and

∫_{⋃_{1}^{n} I_{k}} |f̂(λ)|^{2}|∏_{j=1}^{n} θ_{τ} (λ − λ_{j})|^{2} dλ ⩽ 2^{−2n}δ^{−2n}∥f̂(·)p(e^{2πit_{0}·})∥_{L^{2}(ℝ)}^{2} ⩽ δ^{−2n}κ.

Combining this inequality with the above estimate ∫_{ℝ∖⋃_{1}^{n} I_{k}} |f̂|^{2} ⩽ 4(n + 1)(\frac{4e}{δ})^{2n}κ^{2}, we get

∥f̂(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} ⩽ \{4(n + 1)(4e)^{2n} + 1\}^{\frac{1}{2}} δ^{−n}κ.

It remains to notice that if δ = \frac{1}{8n(n+1)}, then the right-hand side is less than \{20(n+1)\}^{2n}κ.

§3.2. Local behavior of functions from the class EP_{loc}^{n}(τ, κ)

We start with recalling a well-known lemma on differential equations.

Lemma 3.2. Let D = ∏_{j=1}^{n}(e^{2πiλ_{j}x} \frac{d}{dx}e^{−2πiλ_{j}x}) be a differential operator of order n ⩾ 1 (λ_{1}, …, λ_{n} ∈ ℝ). Let a function f satisfy the equation Df = g on an interval J = [a, b]. Then there exists an exponential polynomial p with spectrum λ_{1}, …, λ_{n} such that sup_{x∈J} |f(x) − p(x)| ⩽ |J|^{n} \frac{1}{|J|} ∫_{J} |g(t)| dt.

Proof. Since the space of exponential polynomials with spectrum λ_{1}, …, λ_{n} is precisely the space of solutions of the homogeneous equation corresponding to the operator D, it suffices to find a solution φ of the equation Dφ = g such that |φ(x)| ⩽ |J|^{n} \frac{1}{|J|} ∫_{J} |g(t)| dt for all x ∈ J. As usual, one can prove by induction on n that φ = ∏_{j=1}^{n}(e^{2πiλ_{j}x}J e^{−2πiλ_{j}x})g fits, where J is the integral operator defined by (J h)(x) = ∫_{a}^{x} h(t) dt, and the lemma is proved.

Now let f ∈ EP_{loc}^{n}(τ, κ). As shown in §1, there exist λ_{1}, …, λ_{n} ∈ ℝ such that

∥f̂(·) ∏_{j=1}^{n} θ_{τ} (·−λ_{j})∥_{L^{2}(ℝ)} ⩽ κ′ with κ′ =^{def} \{20(n+1)\}^{2n}κ. We put I_{k} =^{def} (λ_{k} − \frac{1}{τ} , λ_{k} + \frac{1}{τ}),

Ĩ_{k} =^{def} (λ_{k} − \frac{2}{τ} , λ_{k} + \frac{2}{τ}), E_{0} =^{def} ℝ ∖ (⋃_{k=1}^{n} I_{k}),and E_{k} =^{def} I_{k} ∖ (⋃_{j=1}^{k−1} I_{j}). The sets

E_{0}, E_{1}, …, E_{n} form a partition of the real line. Now, the function f can be decomposed into the sum f = ∑_{k=0}^{n}(f̂χ_{E_{k}})^{∨} =^{def} ∑_{k=0}^{n} f_{k}. The spectrum of the function f_{k} lies in the set E_{k} ⊂ I_{k} (k = 1, …, n). Moreover,

κ′^{2} ⩾ ∥f̂(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)}^{2} = ∑_{k=0}^{n} ∥f̂_{k}(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)}^{2}.

Consequently,

∥f̂_{k}(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} ⩽ κ′

for each k = 0, …, n. But ∏_{j=1}^{n} θ_{τ} (· − λ_{j}) ≡ 1 on E_{0}, therefore ∥f_{0}∥_{L^{2}(ℝ)}^{2} ⩽ κ′. Now

let 1 ⩽ k ⩽ n. On the interval I_{k} we may keep only the factors θ_{τ} (· − λ_{j}) with λ_{j} ∈ Ĩ_{k} because all the other factors are identically equal to 1. Let n_{k} denote the number of points λ_{j} lying in Ĩ_{k}. Obviously, 1 ⩽ n_{k} ⩽ n. We define a differential operator D_{k} of order n_{k} by D_{k} =^{def} ∏_{λ_{j}∈Ĩ_{k}}(e^{2πiλ_{j}x} \frac{d}{dx}e^{−2πiλ_{j}x}). The function f_{k} has a bounded spectrum,

therefore, it is differentiable infinitely many times on ℝ. Setting g_{k} =^{def} D_{k}f_{k}, we have ĝ_{k}(·) = (2πi)^{n_{k}} ∏_{λ_{j}∈Ĩ_{k}}(· − λ_{j})f̂_{k}(·). Since |λ − λ_{j}| ⩽ \frac{3}{τ} θ_{τ} (λ − λ_{j}) for every λ ∈ I_{k}, λ_{j} ∈ Ĩ_{k}, we see that

\begin{aligned}∥g_{k}∥_{L^{2}(ℝ)} & = ∥ĝ_{k}(·)∥_{L^{2}(ℝ)} ⩽ (\frac{6π}{τ} )^{n_{k}} ∥f̂_{k}(·) ∏_{λ_{j}∈Ĩ_{k}} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} \\ & = (\frac{6π}{τ} )^{n_{k}} ∥f̂_{k}(·) ∏_{j=1}^{n} θ_{τ} (· − λ_{j})∥_{L^{2}(ℝ)} ⩽ (\frac{6π}{τ} )^{n_{k}}κ′. \end{aligned}

Let J ⊂ ℝ be an arbitrary interval of length |J| ⩽ Mτ (M ⩾ 1). In accordance with Lemma 3.2, there exists an exponential polynomial p_{k}^{(J)} with spectrum consisting of the points λ_{j} ∈

Ĩ_{k}, such that |f_{k}(x) − p_{k}^{(J)}(x)| ⩽ |J|^{n_{k}} \frac{1}{|J|} ∫_{J} |g_{k}(t)| dt for all x ∈ J. Let g̃_{k} =^{def} τ^{n_{k}}g_{k}. Then

the right-hand side can be estimated from above by M^{n_{k}} \frac{1}{|J|} ∫_{J} |g̃_{k}(t)| dt ⩽ M^{n}𝔐g̃_{k}(x),

where 𝔐h(x) =^{def} sup_{J:x∈J} \frac{1}{|J|} ∫_{J} |h(t)| dt is the Hardy–Littlewood maximal function. The well-known estimate of the norm of the maximal function operator in L^{2}(ℝ) yields

∥𝔐g̃_{k}∥_{L^{2}(ℝ)} ⩽ 4∥g̃∥_{L^{2}(ℝ)} = 4τ^{n_{k}}∥g_{k}∥_{L^{2}(ℝ)} ⩽ 4(6π)^{n_{k}}κ′.

If p^{(J)} =^{def} ∑_{k=1}^{n} p_{k}^{(J)}, then p^{(J)} is an exponential polynomial with spectrum λ_{1}, …, λ_{n}, and for every x ∈ J we have

\begin{aligned}|f(x) − p^{(J)}(x)| & ⩽ |f_{0}(x)| + M^{n} ∑_{k=1}^{n} 𝔐g̃_{k}(x) \\ & ⩽ M^{n}(|f_{0}(x)| + ∑_{k=1}^{n} 𝔐g̃_{k}(x)) =^{def} M^{n}Φ(x). \end{aligned}

The norm of the “error function” Φ(x) satisfies the inequality

\begin{aligned}∥Φ∥_{L^{2}(ℝ)} & ⩽ κ′ + ∑_{k=1}^{n} 4(4π)^{n_{k}}κ′ ⩽ (4n + 1)(6π)^{n}κ′ \\ & = (4n + 1)(6π)^{n}\{20(n + 1)\}^{2n}κ ⩽ \{200(n + 1)\}^{2n}κ, \end{aligned}

and this proves the desired statement.

§3.3. Expanding the estimate

First of all, we shall reformulate the Turan lemma for the setting where the smallness of an exponential polynomial is understood in the L^{2}-sense instead of the L^{∞}-sense as it was in Chapter 1.

Lemma 3.3. Let p(x) be an exponential polynomial of order n with pure imaginary exponents, I ⊂ ℝ an interval, and e ⊂ I a set of positive Lebesgue measure. Then

(∫_{I} |p(x)|^{2} dx)^{\frac{1}{2}} ⩽ \{\frac{Aμ(I)}{μ(e)}\}^{n− \frac{1}{2}} (∫_{e} |p(x)|^{2} dx)^{\frac{1}{2}} ,

where A is an absolute constant.

Proof. By the Chebyshev inequality, |p(x)|^{2} ⩽ \frac{2}{μ(e)} ∫_{e} |p(t)|^{2} dt on a subset e′ ⊂ e of measure μ(e′) ⩾ \frac{1}{2}μ(e). An application of the “L^{∞}-Turan lemma” to the set e′ leads to the estimate

sup_{x∈I} |p(x)|^{2} ⩽ \{\frac{Aμ(I)}{μ(e′)}\}^{2(n−1)} \frac{2}{μ(e)} ∫_{e} |p(t)|^{2} dt.

Integrating this inequality over I and replacing μ(e′) by \frac{1}{2}μ(e), we obtain the statement of the lemma. It should be noted that this result was essentially proved in Chapter 2. Here we recall it for convenience of the references.

Now let f ∈ EP_{loc}^{n}(τ, κ), let λ_{1}, …, λ_{n} be an essential spectrum of f, and Φ an “error function”corresponding to this spectrum. Let E ⊂ ℝ be a measurable set of finite measure. Suppose that ∫_{ℝ∖E} |f|^{2} ⩽ κ^{2}. (The reader may regard κ̃ to be comparable with κ, though from the formal point of view this is inessential). Fixing γ ∈ (0, 1) and M > 1, we decompose the real line into intervals of length Mτ and take one of them (say, J). Two cases are possible:

1) μ(J ∖ E) ⩾ γμ(J);

2) μ(J ∖ E) < γμ(J).

In case 1 we call the interval J white and in case 2 we call it black. If we think γ to be very small, we can say that every black interval consists mostly of points of E, while every white one contains a noticeable piece of ℝ ∖ E.

Let J be white. Denote by p an exponential polynomial with spectrum λ_{1}, …, λ_{n} approximating f on J up to an error M^{n}Φ. The Minkowski inequality implies

(∫_{J∖E} |p|^{2})^{\frac{1}{2}} ⩽ (∫_{J∖E} |f|^{2})^{\frac{1}{2}} + M^{n}(∫_{J} Φ^{2})^{\frac{1}{2}} .

Applying Lemma 3.3, we obtain the inequality

\begin{aligned}(∫_{J} |p|^{2})^{\frac{1}{2}} & ⩽ \{\frac{Aμ(J)}{μ(J ∖ E)}\}^{n− \frac{1}{2}} \{(∫_{J∖E} |f|^{2})^{\frac{1}{2}} + M^{n}(∫_{J} Φ^{2})^{\frac{1}{2}} \} \\ & ⩽ (\frac{A}{γ})^{n− \frac{1}{2}} \{(∫_{J∖E} |f|^{2})^{\frac{1}{2}} + M^{n}(∫_{J} Φ^{2})^{\frac{1}{2}} \}; \end{aligned}

therefore,

\begin{aligned}(∫_{J} |f|^{2})^{\frac{1}{2}} & ⩽ (∫_{J} |p|^{2})^{\frac{1}{2}} + M^{n}(∫_{J} Φ^{2})^{\frac{1}{2}} \\ & ⩽ (\frac{A}{γ})^{n− \frac{1}{2}} \{(∫_{J∖E} |f|^{2})^{\frac{1}{2}} + 2M^{n}(∫_{J} Φ^{2})^{\frac{1}{2}} \}. \end{aligned}

Squaring and summing over all the white intervals J, we find

∫_{⋃\{ J:J is white \}} |f|^{2} ⩽ (\frac{A′}{γ} )^{2n−1}\{∫_{ℝ∖E} |f|^{2} + M^{2n} ∫_{ℝ} Φ^{2}\}

⩽ (\frac{A′′n^{2}M}{γ} )^{2n}(κ^{2} + κ^{2}),

(∗∗)

where A′ and A′′ are absolute constants. (We used the estimate ∥Φ∥_{L^{2}(ℝ)}^{2} ⩽ {200(n + 1)}^{4n}κ^{2} proved in §2 and the condition ∫_{ℝ∖E} |f|^{2} ⩽ κ^{2}). Let E_{w} and E_{b} be the parts of E lying in white and black intervals, respectively. The last inequality can be regarded as an expansion of the estimate of f from ℝ ∖ E to E_{w}. Its usefulness depends on the measure of E_{w}: the more this measure is, the more valuable the expansion is.

Later we shall need the following simple observation relating the measure μ(E_{w}) with another characteristic of E, namely, the measure of the set (E + nτ) ∖ E.

Lemma 3.4. If γ < 1/2 and |t| ⩽ nτ, then

μ(E ∖ (E + t)) ⩽ μ(E_{w}) + (2γ + \frac{2n}{M})μ(E).

Proof. Obviously, μ(E ∖ (E + t)) ⩽ μ(E_{w}) + μ(E_{b} ∖ (E + t)). Let an interval J be black. Then

\begin{aligned}μ((E_{b} ∩ J) ∖ (E + t)) & ⩽ μ(J ∖ (E + t)) ⩽ μ(J ∖ (J + t)) + μ((J + t) ∖ (E + t)) \\ & ⩽ |t| + μ(J ∖ E) ⩽ nτ + γμ(J) = (γ + \frac{n}{M} )μ(J). \end{aligned}

It remains to notice that the number of black intervals does not exceed \frac{μ(E)}{(1−γ)μ(J)} ⩽ 2\frac{μ(E)}{μ(J)}.

Corollary 3.5. Let μ(E ∖ (E + nτ)) = ν, f ∈ EP_{loc}^{n}(τ, κ), ∫_{ℝ∖E} |f|^{2} ⩽ κ^{2}. There exists a subset E′ ⊂ E of measure μ(E′) ⩾ \frac{ν}{2} such that

∫_{(ℝ∖E)∪E′} |f|^{2} ⩽ (\frac{A′′′n^{3}μ(E)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2}).

Proof. We set γ =^{def} \frac{ν}{8μ(E)}, M =^{def} 8n\frac{μ(E)}{ν}. It is immediate from the condition μ(E ∖ (E + nτ)) = ν that μ(E) ⩾ ν, and, therefore, M > 1. We show that as E′ one can take the set E_{w} corresponding to these values of parameters. Indeed, by Lemma 3.4, we have

\begin{aligned}ν & = μ(E ∖ (E + nτ)) ⩽ μ(E_{w}) + (2γ + \frac{2n}{M})μ(E) \\ & = μ(E_{w}) + (\frac{ν}{4μ(E)} + \frac{2nν}{8nμ(E)})μ(E) = μ(E_{w}) + \frac{ν}{2} , \end{aligned}

and, therefore, μ(E_{w}) ⩾ \frac{ν}{2} . On the other hand, (∗∗) implies that

\begin{aligned}∫_{(ℝ∖E)∪E_{w}} |f|^{2} & ⩽ ∫_{ℝ∖E} |f|^{2} + ∫_{⋃\{ J:J is white \}} |f|^{2} ⩽ κ^{2} + (\frac{A′′n^{2}M}{γ} )^{2n}(κ^{2} + κ^{2}) \\ & = κ^{2} + (\frac{64A′′n^{3}μ(E)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2}) = (\frac{A′′′n^{3}μ(E)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2}) \end{aligned}

if A′′′ is suitably chosen.

Though the statement of Corollary 3.5 looks rather cumbersome, it turns out to be most convenient for applications, to which we pass now.

§3.4. The Zygmund theorem on the unit circle

In 1948, Zygmund proved the following uniqueness theorem.

Theorem. Let a set Λ = {m_{k}}_{k=−∞}^{∞} ⊂ ℤ satisfy the condition

(3.1)  sup_{r≠0} card{ (k′, k′′) : m_{k′} − m_{k′′} = r } =^{def} R(Λ) < +∞.

If f ∈ L^{2}(𝕋) and spec f ⊂ Λ, then

(3.2)  ∥f∥_{L^{2}(𝕋)}^{2} ⩽ C(E, Λ) ∫_{E} |f|^{2} dμ,

where E ⊂ 𝕋 is an arbitrary set of positive measure and the constant C(E, Λ) depends on E and Λ but not on f.

The conditions on the spectrum Λ sufficient for (3.2) have been weakened by many authors. As far as I know, the following result due to Mikheev is the best one at present.

Theorem. (Mikheev, 1975). If a spectrum Λ satisfies the condition

lim_{ε→0} sup\{\frac{∫_{E} |f|^{2} dμ}{∥f∥_{L^{2}(𝕋)}^{2}} : E ⊂ 𝕋, μ(E) < ε, f ∈ L^{2}(𝕋), spec f ⊂ Λ \} = 0,

then (3.2) still holds.

In particular, (3.2) holds for an arbitrary S_{p}-system Λ (p > 2) because in this case the Hölder inequality yields

∫_{E} |f|^{2} dμ ⩽ μ(E)^{1− \frac{2}{p}} (∫_{E} |f|^{p} dμ)^{\frac{2}{p}} ⩽ C(Λ)ε^{1− \frac{2}{p}} ∥f∥_{L^{2}(𝕋)}^{2}.

We are going to improve Zygmund’s result in other direction, namely, we shall show that the constant C(E, Λ) can be chosen depending only on the measure of the set E and the bound R(Λ) defined in (3.1). Here is the precise statement.

Theorem 3.6 (the Zygmund theorem). If a spectrum Λ satisfies (3.1), then, for an arbitrarily small ε > 0, in inequality (3.2) we can take

C(E, Λ) = exp\{\frac{B(ε, R(Λ))}{μ(E)^{2+ε}}\},

where B(ε, R(Λ)) is a constant depending only on ε and R(Λ).

Corollary 3.7 (the “logarithm summability” theorem). Under the same assumptions on Λ, if f ∈ L^{2}(𝕋) and spec f ⊂ Λ, then for each ε > 0

∫_{𝕋} log^{\frac{1}{2} −ε}(1 + \frac{1}{|f|}) dμ < +∞

unless f ≡ 0.

Unfortunately, condition (3.1) plays the same crucial role in our proof as it did in original Zygmund’s proof. It remains unclear, for example, whether the estimate of the constant C(E, Λ) depending only on the measure of E is possible if Λ is a union of two spectra, each of them satisfying (3.1). As to the sharpness of Theorem 3.6, it is not so bad as one could think: from a result due to Mandelbrojt it follows that for each ε > 0 there exists a spectrum Λ satisfying (3.1) for which the constant C(E, Λ) cannot be taken less than exp\{\frac{B}{μ(E)^{1−ε}} \} even if we restrict ourselves to arcs E ⊂ 𝕋.

Proof of Theorem 3.6. First of all we shall show that all the results of §§3.1–3.3 remain valid for the functions of class L_{per}^{2}(ℝ), i.e., 1-periodic functions which are square-integrable on the period. The norm of a function f from this class is defined by

∥f∥_{L_{per}^{2}(ℝ)} =^{def} (∫_{0}^{1} |f(x)|^{2} dx)^{\frac{1}{2}} .

The arguments in §§3.1 and 3.2 will need no correction if we replace ∥·∥_{L^{2}(ℝ)} by ∥·∥_{L_{per}^{2}(ℝ)} everywhere (the estimate of the norm of the maximal function operator remains valid in the 1-periodic case) and if we integrate f̂ (understood now as f̂(λ) =^{def} ∫_{0}^{1} f(x)e^{−2πiλx} dx)

with respect to the measure μ̂ =^{def} ∑_{k=−∞}^{∞} δ(· − k) instead of Lebesgue measure. As usual, we denote by δ the Dirac measure, i.e., the unit mass concentrated at the origin. Now it does not matter whether τ is greater or less than 1 and whether the numbers λ_{j} are integers or not (though the multiplication by e^{2πiλx} does not preserve the class L_{per}^{2}, the operator D = e^{2πiλx} \frac{d}{dx}e^{−2πiλx} = \frac{d}{dx} − 2πiλ is still well-defined and the formula D̂f(·) = 2πif̂(·)(· − λ) remains true on the support of the measure μ̂). In §3.3 we need the following additional corrections:

a) we assume τ ⩽ 1;

b) we assume E to be a 1-periodic set such that μ_{per}(E) < 1 (we denote μ_{per}(E) =^{def} μ(E ∩ (0, 1))).

Now the reader is invited to find out where the Lebesgue measure μ should be replaced by μ_{per}, taking into account the final correction:

c) in the argument preceding Corollary 3.5 we impose on M an additional restriction, namely, we require that the number \frac{1}{Mτ} be an integer.

This condition ensures that the sets E_{w} and E_{b} are 1-periodic automatically.

Thus, Corollary 3.5 remains valid if the number M defined in its proof by M = 8n\frac{μ_{per}(E)}{ν} satisfies the inequality Mτ ⩽ 1, for in this case one can choose an M′ ∈ [M, 2M] such that \frac{1}{M′τ} is an integer; replacing M by M′ results only in doubling the constant A′′′. If Mτ > 1, then we take M′ = \frac{1}{τ}. Since μ_{per}(E ∖ (E + nτ)) = ν, we have, at any rate,

μ_{per}(E) ⩾ ν and μ_{per}(ℝ ∖ E) ⩾ ν. Let γ′ =^{def} μ_{per}(ℝ ∖ E). Now our decomposition of the real line into intervals of length M′τ is a decomposition into periods, each period being white. Inequality (∗∗) takes the form

∫_{0}^{1} |f|^{2} ⩽ (\frac{A′′n^{2}M′}{γ′} )^{2n}(κ^{2} + κ^{2}) ⩽ (\frac{A′′′n^{3}μ_{per}(E)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2})

since M′ ⩽ M = 8n\frac{μ_{per}(E)}{ν} and \frac{1}{γ′} = \frac{1}{μ_{per}(ℝ∖E)} ⩽ \frac{2μ_{per}(E)}{ν} (the latter inequality can be rewritten as μ_{per}(E)μ_{per}(ℝ ∖ E) ⩾ \frac{ν}{2} , which is evident because each of the factors in at least ν and their sum is equal to 1). Thus, we can simply take E′ = ℝ ∖ E.

Instead of 1-periodic functions defined on the real line one can consider functions defined on 𝕋. In this setting, the functions f_{kt} in the definition of the class EP_{loc}^{n}(τ, κ) should be understood as f_{kt}(z) =^{def} f(e^{2πikt}z).

In these terms, Corollary 3.5 can be reformulated as follows.

Corollary 3.5′. Let μ(Ẽ ∖ e^{2πinτ} Ẽ) = ν, f ∈ EP_{loc}^{n}(τ, κ), and ∫_{𝕋∖Ẽ} |f|^{2} ⩽ κ^{2}. There exists a subset Ẽ′ ⊂ Ẽ of measure μ(Ẽ′) ⩾ \frac{ν}{2} such that

∫_{(𝕋∖Ẽ)∪Ẽ′} |f|^{2} ⩽ (\frac{A′′′n^{3}μ(Ẽ)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2}).

(We have replaced E by E′ because a set denoted by E is already present in the statement of Theorem 3.6, and we are going to apply the corollary not to this set but to its complement).

Now, at last, we can proceed to the proof of Theorem 3.6. Essentially, we shall copy the classical proof due to Zygmund, though some technicalities will differ slightly.

Let f ∈ L^{2}(𝕋), spec f ⊂ Λ, E ⊂ 𝕋. Repeating the argument of Zygmund, we write f(z) = ∑_{k=−∞}^{∞} f̂(m_{k})z^{m_{k}}, and

∫_{E} |f|^{2} dμ = ∫_{𝕋} χ_{E}|f|^{2} dμ = ∫_{𝕋} χ_{E}(z)| ∑_{k=−∞}^{∞} f̂(m_{k})z^{m_{k}} |^{2} dμ

= μ(E) ∑_{k=−∞}^{∞} |f̂(m_{k})|^{2} + ∑_{k′≠k′′} χ̂_{E}(m_{k′} − m_{k′′})f̂(m_{k′′})\overline{f̂}(m_{k′})

= μ(E)∥f∥_{L^{2}(𝕋)}^{2} + ⟨Q_{E}f, f⟩,

where Q_{E} is an operator in L^{2}(𝕋). The matrix of Q_{E} in the basis {z^{j}}_{j∈ℤ} is (q_{js})_{j,s∈ℤ}, and all q_{js} vanish except those of the form q_{m_{k′′} m_{k′}} k′ ≠ k′′, which are equal to χ̂_{E}(m_{k′′} −m_{k′}).

Condition (3.1) imposed on Λ implies that

∑_{j,s∈ℤ} |q_{js}|^{2} = ∑_{k′≠k′′} |χ̂_{E}(m_{k′} − m_{k′′})|^{2}

= ∑_{r≠0} |χ̂_{E}(r)|^{2} card{ (k′, k′′) : m_{k′} − m_{k′′} = r }

⩽ R(Λ) ∑_{r≠0} |χ̂_{E}(r)|^{2} = R(Λ)μ(E)(1 − μ(E)).

Hence Q_{E} is a selfadjoint Hilbert–Schmidt operator (the selfadjointness of Q_{E} follows from the fact that the function χ_{E} is real-valued, which implies χ̂_{E}(−r) = \overline{χ̂}_{E}(r). We have

∥Q_{E}∥ ⩽ ( ∑_{j,s∈ℤ} |q_{js}|^{2})^{\frac{1}{2}} ⩽ \sqrt{R(Λ)μ(E)(1 − μ(E))}.

If the set E is large enough, more precisely, if \sqrt{R(Λ)μ(E)(1 − μ(E))} ⩽ \frac{μ(E)}{2}, then

\begin{aligned}∫_{E} |f|^{2} dμ & = μ(E)∥f∥_{L^{2}(𝕋)}^{2} + ⟨Q_{E}f, f⟩ \\ & ⩾ μ(E)∥f∥_{L^{2}(𝕋)}^{2} − ∥Q_{E}∥∥f∥_{L^{2}(𝕋)}^{2} ⩾ \frac{μ(E)}{2}∥f∥_{L^{2}(𝕋)}^{2}, \end{aligned}

whence

(∗)  ∥f∥_{L^{2}(𝕋)}^{2} ⩽ \frac{2}{μ(E)} ∫_{E} |f|^{2} dμ.

Solving the inequality \sqrt{Rt(1 − t)} ⩽ \frac{t}{2}, we find that (∗) holds if μ(E) ⩾ \frac{4R(Λ)}{4R(Λ)+1}. Now we

shall show that (∗) holds also in the case μ(E) < \frac{4R(Λ)}{4R(Λ)+1} provided that f is orthogonal to some subspace V_{E} of the space L^{2}(𝕋) having rather low dimension.

We denote by σ_{1}, σ_{2}, … the eigenvalues of the operator Q_{E} enumerated so that |σ_{1}| ⩾ |σ_{2}| ⩾ . . .. Let V_{n} be the subspace generated by the eigenvectors corresponding to the first n eigenvalues of Q_{E}. The norm of the restriction of Q_{E} to the orthogonal complement V_{n}^{⊥} of the space V_{n} is equal to |σ_{n+1}|. It is well known that ∑_{s=1}^{∞} |σ_{s}|^{2} = ∑_{j,s∈ℤ} |q_{js}|^{2} ⩽

R(Λ)μ(E)(1 − μ(E)). Hence |σ_{n+1}| ⩽ \frac{1}{n+1}R(Λ)μ(E)(1 − μ(E)) ⩽ \frac{μ(E)^{2}}{4} provided that

n = n(E) =^{def} [\frac{4R(Λ)(1−μ(E))}{μ(E)}]. Thus, putting V_{E} =^{def} V_{n(E)}, we see, as before, that (∗) holds for every f ∈ L^{2}(𝕋) orthogonal to V_{E} with spec f ⊂ Λ. The above argument for “large” sets E was based actually on the relation n(E) = 0, which holds if (and only if) μ(E) ⩾ \frac{4R(Λ)}{4R(Λ)+1}.

Now let E be an arbitrary set of positive measure; let μ(E) ⩽ \frac{4R(Λ)}{4R(Λ)+1}. Let f ∈ L^{2}(𝕋), spec f ⊂ Λ. The proof of the desired estimate for ∥f∥_{L^{2}(𝕋)}^{2} in terms of ∫_{E} |f|^{2} dμ goes as follows.

We put Ẽ =^{def} 𝕋 ∖ E and n =^{def} [\frac{8R(Λ)}{μ(E)}] (the reason for this choice of n will become clear a bit later). Let τ > 0 be so small that for every t ∈ (0, τ) the measure of the set E_{t} =^{def} ⋂_{k=0}^{n} e^{−2πikt}E is at least \frac{μ(E)}{2}. For each k = 0, …, n, we have

\begin{aligned}∫_{E_{t}} |f_{kt}(z)|^{2} dμ(z) & = ∫_{E_{t}} |f(e^{2πikt}z)|^{2} dμ(z) ⩽ ∫_{e^{−2πikt}E} |f(e^{2πikt}z)|^{2} dμ(z) \\ & = ∫_{E} |f|^{2} dμ =^{def} κ^{2}. \end{aligned}

Therefore, for every linear combination g =^{def} ∑_{0}^{n} a_{k}f_{kt} with ∑_{k=0}^{n} |a_{k}|^{2} = 1 the following inequality holds:

∫_{E_{t}} |g|^{2} dμ ⩽ (∑_{k=0}^{n} |a_{k}|^{2})(∑_{k=0}^{n} ∫_{E_{t}} |f_{kt}|^{2} dμ) ⩽ (n + 1)κ^{2}.

Now we observe that

n(E_{t}) = [\frac{4R(Λ)(1 − μ(E_{t}))}{μ(E_{t})}] ⩽ [\frac{4R(Λ)}{μ(E_{t})}] ⩽ [\frac{8R(Λ)}{μ(E)}] = n

since μ(E_{t}) ⩾ \frac{1}{2}μ(E). It follows that we can choose the coefficients a_{k} in such a way that g is orthogonal to V_{E_{t}}. Inequality (∗) applied to the function g and the set E_{t} yields

∥g∥_{L^{2}(𝕋)}^{2} ⩽ \frac{2}{μ(E_{t})} ∫_{E_{t}} |g|^{2} dμ ⩽ \frac{4}{μ(E)}(n + 1)κ^{2} =^{def} κ^{2}.

Since t ∈ (0, τ) is arbitrary, this means that f ∈ EP_{loc}^{n}(τ, κ). Thus, we can apply Corollary 3.5′ to estimate the integral of |f|^{2} over a certain wider set E_{1} ⊃ E by the integral ∫_{E} |f|^{2} dμ multiplied by some factor W > 0.

The efficiency of this estimate crucially depends on the number ν =^{def} μ(Ẽ ∖ e^{2πinτ} Ẽ) which, on one hand, stands (in the power 4n) in the denominator of the factor W and, on the other hand, serves as an estimate from below for the measure μ(E_{1} ∖ E), so the greater ν is, the better estimate we have (and on the wider set). Unfortunately, the above restriction μ(E_{t}) ⩾ \frac{1}{2}μ(E) for every t ∈ (0, τ) hinders us in making ν large: to ensure this condition we must demand μ(Ẽ ∖ e^{2πis}Ẽ) ⩽ \frac{μ(E)}{2n} for every s ∈ (0, nτ) (at least I cannot devise anything better), and this does not allow us to take ν > \frac{μ(E)}{2n}. (The sufficiency of the above condition follows from the chain of inequalities

μ(E_{t}) = μ( ⋂_{k=0}^{n} e^{−2πikt}E)

= μ(E ∖ ⋃_{k=1}^{n}(e^{−2πikt}Ẽ ∖ Ẽ)) ⩾ μ(E) − ∑_{k=1}^{n} μ(e^{−2πikt}Ẽ ∖ Ẽ)

= μ(E) − ∑_{k=1}^{n} μ(Ẽ ∖ e^{2πikt}Ẽ) ⩾ μ(E) − n\frac{μ(E)}{2n} = \frac{μ(E)}{2},

since kt ∈ (0, nτ) if t ∈ (0, τ) and k = 1, …, n.) One could get ν = \frac{μ(E)}{2n} taking τ = \frac{σ}{n}, where σ is the smallest number among all s ∈ [0, 1] satisfying μ(Ẽ ∖ e^{2πis}Ẽ) ⩾ \frac{μ(E)}{2n}. But why does such σ really exist? Let us show that we are lucky at least here. Since the (nonnegative) function ψ(s) =^{def} μ(Ẽ ∖ e^{2πis}Ẽ) is continuous and ψ(0) = 0, it suffices to verify that ∫_{0}^{1} ψ(s) ds ⩾ \frac{μ(E)}{2n}. It is easy to show that ∫_{0}^{1} ψ(s) ds = μ(E)μ(Ẽ); besides, the assumption μ(E) ⩽ \frac{4R(Λ)}{4R(Λ)+1} made at the very beginning of this part of the reasoning allows

us to write the inequality μ(Ẽ) ⩾ \frac{1}{4R(Λ)+1}. It remains to notice that the denominator is

less than 2n = 2[\frac{8R(Λ)}{μ(E)}] (R(Λ) is an integer!).

Now all the parameters in the statement of Corollary 3.5′ are determined, and we may write its conclusion: there exists a set E_{1} =^{def} E ∪ Ẽ′ such that

1) μ(E_{1} ∖ E) = μ(Ẽ′) ⩾ \frac{ν}{2} = \frac{μ(E)}{4n} = \frac{μ(E)}{4[\frac{8R(Λ)}{μ(E)}]} ⩾ \frac{μ(E)^{2}}{32R(Λ)};

2)  ∫_{E_{1}} |f|^{2} dμ ⩽(\frac{A′′′n^{3}μ(Ẽ)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ^{2}) ⩽^{(μ(Ẽ)⩽1)} (\frac{A′′′n^{3}}{ν^{2}} )^{2n}(κ^{2} + κ^{2})

=^{(ν= \frac{μ(E)}{2n})}(\frac{4A′′′n^{5}}{μ(E)^{2}} )^{2n}(\frac{4}{μ(E)}(n + 1) + 1)κ^{2}

⩽^{(n⩽ \frac{8R(Λ)}{μ(E)} )}(\frac{4A^{(IV)}R(Λ)^{5}}{μ(E)^{7}} )^{\frac{16R(Λ)}{μ(E)}} \{\frac{4}{μ(E)}(\frac{8R(Λ)}{μ(E)} + 1) + 1\}κ^{2}

⩽(\frac{BR(Λ)}{μ(E)})^{120 \frac{R(Λ)}{μ(E)}} κ^{2} = (\frac{BR(Λ)}{μ(E)})^{120 \frac{R(Λ)}{μ(E)}} ∫_{E} |f|^{2} dμ

(A^{(IV)} and B > 0 are suitable constants). Now it is clear how to complete the proof of the Zygmund theorem: one simply needs to iterate the above result until the measure of the current expansion of the set E becomes greater than \frac{4R(Λ)}{4R(Λ)+1}. To make this iteration process more elegant, it is convenient to denote by C(μ) the best possible value of the constant C > 0 such that the inequality ∥f∥_{L^{2}(𝕋)}^{2} ⩽ C ∫_{E} |f|^{2} dμ holds for every function f ∈ L^{2}(𝕋) with spec f ⊂ Λ and every set E ⊂ 𝕋 of measure μ(E) ⩾ μ (a priori we do not suppose C(μ) < +∞, though this is clear now). All we have proved by now can be written as follows:

I) C(μ) ⩽ \frac{2}{μ} for μ ∈ [\frac{4R(Λ)}{4R(Λ)+1}, 1];

II) C(μ) ⩽ (\frac{BR(Λ)}{μ})^{120 \frac{R(Λ)}{μ}} C(μ + \frac{μ^{2}}{32R(Λ)}) for μ < \frac{4R(Λ)}{4R(Λ)+1}.

To derive an explicit expression for the constant C(μ), we need the following simple observation.

Lemma (on solutions of difference equations). Let f: (0, 1) → [0, +∞) be a monotone decreasing function and μ_{∗} ∈ (0, 1). Let a function ψ: (0, 1) → [0, +∞], bounded on [μ_{∗}, 1), satisfy the difference inequality \frac{ψ(μ)−ψ(μ+d(μ))}{d(μ)} ⩽ f(μ), where μ ∈ (0, μ_{∗}) and the function d: (0, μ_{∗}) → (0, +∞) meets the next two natural restrictions:

1) if μ ∈ (0, μ_{∗}), then μ + d(μ) ∈ (0, 1);

2) if ν > 0, then inf_{μ∈[ν,μ_{∗})} d(μ) > 0.

If, in addition, for every μ ∈ (0, μ_{∗})

3) f(μ) ⩽ P f(μ + d(μ)) with some constant P > 0, then ψ(μ) ⩽ sup_{[μ_{∗},1)} ψ + P ∫_{μ}^{1} f(x) dx.

The proof of this lemma is shorter than its statement. Starting with a number μ_{0} = μ we construct an increasing sequence μ_{0} < μ_{1} < μ_{2} < … by setting μ_{k+1} =^{def} μ_{k} + d(μ_{k}). Properties 1) and 2) imply that this sequence will stop at some term μ_{s} ∈ [μ_{∗}, 1). Then

ψ(μ) = ψ(μ_{s}) + ∑_{k=0}^{s−1}{ψ(μ_{k}) − ψ(μ_{k+1})}.

It remains to notice that ψ(μ_{s}) ⩽ sup_{[μ_{∗},1)} ψ, and each of the summands of the last sum satisfy

\begin{aligned}ψ(μ_{k}) − ψ(μ_{k+1}) & = ψ(μ_{k}) − ψ(μ_{k} + d(μ_{k})) ⩽ d(μ_{k})f(μ_{k}) \\ & ⩽ P d(μ_{k})f(μ_{k+1}) ⩽ P ∫_{μ_{k}}^{μ_{k+1}} f(x) dx, \end{aligned}

because f is monotone decreasing.

Now we set μ_{∗} = \frac{4R(Λ)}{4R(Λ)+1} and ψ(μ) =^{def} log C(μ). By virtue of condition II), we have

ψ(μ) − ψ(μ + \frac{μ^{2}}{32R(Λ)}) ⩽ 120\frac{R(Λ)}{μ} log \frac{BR(Λ)}{μ} ⩽ \frac{B_{ε}}{μ^{1+ε}} ,

where ε > 0 is arbitrarily small and B_{ε} > 0 depends only on ε and R(Λ). Setting d(μ) =^{def} \frac{μ^{2}}{32R(Λ)} (conditions 1) and 2) are clearly satisfied), we find

\frac{ψ(μ) − ψ(μ + d(μ))}{d(μ)} ⩽ 32R(Λ)\frac{B_{ε}}{μ^{3+ε}} =^{def} f(μ).

The estimate f(μ) ⩽ P f(μ + d(μ)) holds, for example, with P = 2^{3+ε}; applying the last lemma, we conclude that ψ(μ) grows near the origin not faster than \frac{1}{μ^{2+ε}}, which proves the theorem.

§3.5. The Zygmund theorem on the real line

The most direct analogue of (3.1) on the real line is the condition

(3.1′)  sup_{t∈ℝ} μ(Σ ∩ (Σ + t)) =^{def} R(Σ) < +∞.

But it is easy to understand that (3.1′) implies that Σ is of finite measure (it suffices to take t = 0). This case has already been fully investigated in Chapter 2.

It is worth recalling, however, that Zygmund himself proved his theorem for Fourier series with the Hadamard gaps. From many points of view, a proper analogue of the Hadamard lacunarity condition on the real line is the condition

(3.3)  sup_{k∈ℤ} μ(Σ ∩ I_{k}) =^{def} D(Σ) < +∞,

where the intervals

I_{k} =^{def} \{ \begin{matrix}[−2^{|k|}, −2^{|k|−1}), & k < 0, \\ [−1, 1), & k = 0, \\ [2^{k−1}, 2^{k}), & k > 0 \end{matrix}

form a “diadic decomposition” of the real line ℝ. Such spectra Σ satisfy the following condition, which is somewhat weaker than (3.1′): there exists a partition Σ = ⋃_{j∈ℤ} Σ_{j} such that

(3.4)  sup_{j∈ℤ} μ(Σ_{j}) < +∞,  sup_{t∈ℝ} μ( ⋃_{j′≠j′′}(Σ_{j′} ∩ (Σ_{j′′} + t))) =^{def} R̃(Σ) < +∞.

That (3.3) implies (3.4) will be verified later on; now we shall show that from (3.4) it follows that the estimate

∥f∥_{L^{2}(ℝ)}^{2} ⩽ exp exp{Bμ(E)} ∫_{ℝ∖E} |f|^{2} dμ

is valid for every function f ∈ L^{2}(ℝ) with spec f ⊂ Σ and every set E ⊂ ℝ of finite Lebesgue measure. (The constant B depends only on upper bounds in (3.4).) In accordance with the partition Σ = ∪_{j∈ℤ}Σ_{j}, we decompose f into the sum f = ∑_{j∈ℤ}(f̂χ_{Σ_{j}})^{∨} =^{def} ∑_{j∈ℤ} f_{j}. We have

∫_{E} |f|^{2} dμ = ∫_{E}|∑_{j∈ℤ} f_{j}|^{2} dμ = ∑_{j∈ℤ} ∫_{E} |f_{j}|^{2} dμ + ∑_{j′≠j′′} ∫_{E} f_{j′} \overline{f}_{j′′} dμ.

In Chapter 2 it was shown that

∫_{E} |f_{j}|^{2} dμ ⩽ (1 − e^{−Aμ(E)μ(Σ_{j})})∥f_{j}∥_{L^{2}(ℝ)}^{2},

therefore, the first sum does not exceed

(1 − e^{−A′μ(E)}) ∑_{j∈ℤ} ∥f_{j}∥_{L^{2}(ℝ)}^{2} = (1 − e^{−A′μ(E)})∥f∥_{L^{2}(ℝ)}^{2},

where A′ =^{def} A sup_{j∈ℤ} μ(Σ_{j}). The second sum is equal to

∑_{j′≠j′′} ∫_{ℝ} χ_{E}(x)(∫_{Σ_{j′}} f̂(λ′)e^{2πiλ′x} dλ′)\overline{(∫_{Σ_{j′′}} f̂(λ′′)e^{2πiλ′′x} dλ′′)} dx

= ∑_{j′≠j′′} ∫∫∫_{\begin{matrix}x∈ℝ \\ λ′∈Σ_{j′} ,λ′′∈Σ_{j′′} \end{matrix}} χ_{E}(x)e^{2πi(λ′−λ′′)x}f̂(λ′)\overline{f̂(λ′′)} dλ′ dλ′′ dx

= ∑_{j′≠j′′} ∫∫_{\begin{matrix}λ′∈Σ_{j′} \\ λ′′∈Σ_{j′′} \end{matrix}} χ̂_{E}(λ′′ − λ′)f̂(λ′)\overline{f̂(λ′′)} dλ′ dλ′′ = ⟨Q_{E}f, f⟩,

where Q_{E} is an operator in L^{2}(ℝ) with the symbol

q(λ′′, λ′) = \{ \begin{matrix}χ̂_{E}(λ′′ − λ′), & λ′ ∈ Σ_{j′}, λ′′ ∈ Σ_{j′′}, j′ ≠ j′′, \\ 0 & otherwise. \end{matrix}

(If we denote P_{E}f =^{def} χ_{E}f, PΣ̂f =^{def} (f̂χ_{Σ})^{∨}, we may think the operator Q_{E} to be the difference between the operator PΣ̂P_{E}PΣ̂ and its “diagonal” part ∑_{j∈ℤ} PΣ̂_{j}P_{E}PΣ̂_{j}, like we did in §3.4). But

∫∫_{ℝ^{2}} |q(λ′′, λ′)|^{2} dλ′ dλ′′ = ∫_{ℝ}|χ̂_{E}(t)|^{2}μ(∪_{j′≠j′′}(Σ_{j′} ∩ (Σ_{j′′} + t))) dt

⩽ sup_{t∈ℝ} μ(∪_{j′≠j′′}(Σ_{j′} ∩ (Σ_{j′′} + t))) ∫_{ℝ}|χ̂_{E}(t)|^{2} = R̃(Σ)μ(E),

and again we see that Q_{E} is a selfadjoint Hilbert–Schmidt operator whose norm ∥Q_{E}∥ satisfies

∥Q_{E}∥ ⩽ (∫∫_{ℝ^{2}} |q(λ′′, λ′)|^{2} dλ′ dλ′′)^{1/2} ⩽ \sqrt{R̃(Σ)μ(E)}.

If μ(E) ⩽ μ_{∗} with μ_{∗} determined by the equation \sqrt{R̃(Σ)μ_{∗}} = \frac{1}{2}e^{−A′μ_{∗}}, then

(∗) ∫_{E} |f|^{2} dμ ⩽ (1 − \frac{1}{2}e^{−A′μ(E)})∥f∥_{L^{2}(ℝ)}^{2} and ∥f∥_{L^{2}(ℝ)}^{2} ⩽ 2e^{A′μ(E)} ∫_{ℝ∖E} |f|^{2} dμ.

If μ(E) > μ_{∗}, then we can prove, by the same way as in §4, that (∗) is valid for all the functions f ∈ L^{2}(ℝ) with spec f ⊂ Σ that are orthogonal to some subspace V_{E} ⊂ L^{2}(ℝ) of dimension dim V_{E} = n(E) =^{def} [4R̃(Σ)μ(E)e^{2A′μ(E)}].

For an arbitrary set E ⊂ ℝ of finite measure satisfying μ(E) > μ_{∗}, we put n =^{def} [8R̃(Σ)μ(E)e^{4A′μ(E)}]. It τ is so small that for every t ∈ (0, nτ) the measure of the set

E_{t} =^{def} ∪_{k=0}^{n}(E − kt) is less than 2μ(E) (this implies the estimate n(E_{t}) ⩽ n), then we again notice that the function f belongs to EP_{loc}^{n}(τ, κ) with κ^{2} = 2e^{A′μ(E)}(n + 1)κ^{2}, where κ̃ =^{def} ∫_{ℝ∖E} |f|^{2} dμ (f ∈ L^{2}(ℝ), spec f ⊂ Σ as above).

Let τ be the smallest positive number satisfying μ(E ∖(E +nτ)) = \frac{μ(E)}{n} =^{def} ν. Applying Corollary 3.5, we see that there exists a subset E′ ⊂ E such that μ(E′) ⩾ \frac{ν}{2} ⩾ \frac{1}{16R̃(Σ)}e^{−4A′μ(E)}, and

∫_{(ℝ∖E)∪E′} |f|^{2} ⩽ (\frac{A′′′n^{3}μ(E)^{2}}{ν^{2}} )^{2n}(κ^{2} + κ)

⩽ {A^{(IV)}R̃(Σ)^{5}μ(E)^{4}e^{20A′μ(E)}}^{8R̃(Σ)μ(E)e^{4A′μ(E)}}{2e^{A′μ(E)}(e^{4A′μ(E)} + 1)} ∫_{ℝ∖E} |f|^{2}

⩽ exp exp{Bμ(E)} ∫_{ℝ∖E} |f|^{2} dμ

if the constants A^{(IV)} and B are suitably chosen (we have used the fact that μ(E) ⩾ μ_{∗} > 0).

Denoting by C(μ) the best possible constant in the inequality

∥f∥_{L^{2}(ℝ)}^{2} ⩽ C(μ) ∫_{ℝ∖E} |f|^{2} dμ (μ > 0, μ(E) ⩽ μ, f ∈ L^{2}(ℝ), spec f ⊂ Σ),

we obtain the properties I) C(μ) ⩽ 2e^{A′μ} for 0 < μ < μ_{∗}; II) C(μ) ⩽ e^{e^{Bμ}}C(μ − \frac{1}{16R̃(Σ)}e^{−4A′μ}) for μ ⩾ μ_{∗},

whence it is not hard to derive the desired estimate

C(μ) ⩽ exp exp{B(Σ)μ}.

It remains to show that for every spectrum Σ sparse in the sense of (3.3), there exists a partition Σ = ∪_{j∈ℤ}Σ_{j} satisfying (3.4). Let k > 0. Well-known estimates of the one-sided maximal function on the real line imply that the half of the interval I_{k} that is nearest to the origin contains a point ξ_{k} such that

μ(Σ ∩ I_{k} ∩ [ξ_{k}, ξ_{k} + s]) ⩽ \frac{2μ(Σ ∩ I_{k})}{μ(I_{k})}s ⩽ 2^{2−k}D(Σ)s

for all s > 0. Similarly, if k < 0, then we define the point ξ_{k}, again from the half of I_{k} that is nearest to the origin, by the condition

μ(Σ ∩ I_{k} ∩ [ξ_{k} − s, ξ_{k}]) ⩽ \frac{2μ(Σ ∩ I_{k})}{μ(I_{k})}s ⩽ 2^{2−|k|}D(Σ)s (s > 0).

Let, finally, ξ_{0} = 0. We introduce

Σ_{j} =^{def} \{ \begin{matrix}[ξ_{j−1}, ξ_{j}) ∩ Σ, & j > 0, \\ [ξ_{j}, ξ_{j+1}) ∩ Σ, & j < 0. \end{matrix}

It is clear that μ(Σ_{j}) ⩽ 2D(Σ). Now, for t ∈ ℝ, let j_{0} be the smallest positive integer such that 2^{j_{0}} ⩾ |t|. Let us take an arbitrary interval [a, b] of length |t| centered at a point lying to the right from the origin. How can it happen that its right end lies in Σ_{j} while the left one lies in some Σ_{m} with m < j? Obviously, this occurs only if b ∈ Σ_{j} ∩ [ξ_{j−1}, ξ_{j−1} + |t|). Since b ⩾ \frac{|t|}{2}, we have j > 0. If j ⩾ j_{0} + 3, then the distance between the point ξ_{j−1} and the right end of the interval I_{j−1} is greater than 2^{j−3} ⩾ 2^{j_{0}} ⩾ |t|, and, therefore, the latter set coincides with Σ ∩ I_{j−1} ∩ [ξ_{j−1}, ξ_{j−1} + |t|). On the other hand, if 0 < j < j_{0} − 2, then all points from Σ_{j} lie to the left of the point \frac{|t|}{2} (in this case from the definition of j_{0} it follows that 2^{j−1} < |t| and, consequently, ξ_{j} < 2^{j} ⩽ 2^{j_{0}−2} < \frac{|t|}{2}). Therefore, the measure of the set of right ends (and, of course, the measure of the set of left ends, whose value is the same) of all intervals [a, b] with centers to the right of the origin and such that a and b lie in different Σ_{j}, does not exceed

\begin{matrix}∑_{j_{0}−1⩽j⩽j_{0}+2} μ(Σ_{j}) + ∑_{j=j_{0}+3}^{∞} μ(Σ ∩ I_{j−1} ∩ [ξ_{j−1}, ξ_{j−1} + |t|)) \\ ⩽ 4 × 2D(Σ) + ∑_{j=j_{0}+3}^{∞} 2^{3−j}D(Σ)|t| = 8D(Σ) + 2D(Σ)\frac{|t|}{2^{j_{0}}} < 10D(Σ). \end{matrix}

Considering similarly the intervals with centers to the left of the origin, we conclude that the second supremum in (3.4) does not exceed 20D(Σ). That’s all!

\underline{C} hapter 4. Some remarks and supplements

§4.1. Estimates of exponential polynomials with complex exponents

Let p(t) = ∑_{k=1}^{n} c_{k}e^{λ_{k}t} be an exponential polynomial of order n. The same reasoning as we used to prove the classical Turan lemma (see Chapter 1, §1) shows that, J ⊂ I ⊂ ℝ being two intervals,

sup_{t∈I} |p(t)| ⩽ e^{|I| max | Re λ_{k}|}\{\frac{4eμ(I)}{μ(J)}\}^{n−1} sup_{t∈J} |p(t)|

(essentially, this was already mentioned when we used the Turan lemma to estimate the constant a in the Hadamard factorization p(z) = ce^{az}Q(z)R(z)).

Now we are going to show how the same statement can be obtained for an arbitrary subset E ⊂ I of positive measure.

Proposition 4.1. Under the same conditions on p(t),

∑_{t∈I} |p(t)| ⩽ e^{|I| max | Re λ_{k}|}\{\frac{Aμ(I)}{μ(E)}\}^{n−1} sup_{t∈E} |p(t)|

(here E is a measurable subset of I, A is a universal constant).

Proof. We dwell on the main ideas only. Without loss of generality, I = [−\frac{1}{2}, \frac{1}{2}]. The main difficulty is that the Langer lemma, which provided us with quite sharp information on the local distribution of zeros of an exponential polynomial, cannot be applied now. The following simple observations based on combining the Turan lemma for two intervals with the Jensen inequality (or Schwarz’s lemma, if the reader wishes) partly compensate for this lack.

Lemma 4.2. Let p(t) = ∑_{k=1}^{n} c_{k}e^{λ_{k}t} (c_{k} ∈ ℂ, λ_{k} ∈ ℂ) be an exponential polynomial of order n, λ =^{def} max |λ_{k}|. Then the number of zeros of p(z) inside each disk of radius r > 0 does not exceed 4(n − 1) + 7λr.

Proof. Let D_{r} be a disk of radius r centered at z_{0}, and D_{R} a larger disk of radius R > r centered at the same point. We put m =^{def} max_{z∈D_{r}} |p(z)|, M =^{def} max_{z∈D_{R}} |p(z)|.

Considering the restrictions of p(z) to all the intervals

I_{ζ} =^{def} { z_{0} + tζ : t ∈ [−r, R] } ⊃ J_{ζ} =^{def} { z_{0} + tζ : t ∈ [−r, r] } (ζ ∈ 𝕋)

and applying the Turan lemma, we get

M ⩽ \{\frac{4e(R + r)}{2r}\}^{n−1}e^{λ(R+r)}m.

On the other hand, for each zero ξ of p(z) lying in D_{r}, the maximum the absolute value of the corresponding Blaschke factor B_{ξ}(z) =^{def} \frac{R(z−ξ)}{R^{2}−\overline{(ξ−z_{0})}(z−z_{0})} in the disk D_{r} is equal to

\frac{2rR}{R^{2}+r^{2}} . Hence m ⩽ (\frac{2rR}{R^{2}+r^{2}} )^{N}M, where N is the number of zeros of p(z) lying in D_{r}.

Comparing these two inequalities, we obtain

N ⩽ \{log(\frac{2rR}{R^{2} + r^{2}} )\}^{−1}\{(n − 1) log \frac{4e(R + r)}{2r} + λ(R + r)\}.

Putting R = 6r we arrive at N ⩽ \frac{1}{log \frac{37}{12}} \{(n−1) log(14e)+7λr\}, and the statement follows. The following estimate of the Bernstein lemma type can easily by derived from Lemma 4.2 (This is an analogue of Lemma 1.6 used before).

Lemma 1.6′. Let g(t) =^{def} ∑_{k=1}^{n} c_{k}e^{iλ_{k}t}, λ_{k} ∈ ℂ, λ =^{def} max |λ_{k}| > n − 1. Then

μ\{ t ∈ [−\frac{1}{2}, \frac{1}{2}] : |\frac{d}{dt} log g(t)| > y\} ⩽ \frac{Aλ}{y}

for every y > 0, where A is a universal constant.

The proof of Lemma 1.6′ almost entirely follows the lines of the proof of Lemma 1.6 and is based on the factorization

g(z) = ce^{az} ∏_{j⩽11λ}(z − z_{j}) ∏_{j⩽11λ}(1 − \frac{z}{z_{j}} )e^{z/z_{j}},

where the zeros z_{j} of p(z) are enumerated so that the numbers |z_{j}| are monotone nondecreasing. Lemma 4.2 implies the inequality |z_{j}| ⩾ \frac{j−4(n−1)}{7λ} ⩾ \frac{j−4λ}{7λ} > 1 (valid for j > 11λ), which can fully replace the inequality |z_{j}| ⩾ π \frac{j−λ}{λ} used before.

Lemma 1.6′ allows us to reduce the proof of the Turan lemma to the case λ = diam spec p =^{def} max_{k′,k′′} |λ_{k′} − λ_{k′′} | ⩽ n − 1 by constructing by induction a sequence of exponential polynomials p_{n}, p_{n−1}, …, p_{s} such that

1) p_{n} = p;

2) ord p_{k} = k (k = s, …, n);

3) spec p_{k−1} ⊂ spec p_{k} and ∥p_{k−1}∥_{∞} ⩾ δ∥p_{k}∥_{∞} (k = s + 1, …, n; δ > 0 is an absolute constant);

4) the ratio φ_{k} =^{def} |\frac{p_{k−1}}{p_{k}} | satisfies the weak type estimate

μ\{ x ∈ [−\frac{1}{2}, \frac{1}{2}] : φ_{k}(x) > t \} ⩽ \frac{1}{t}

for every t > 0; 5) diam spec p_{s} ⩽ s − 1. Unlike the case of pure imaginary exponents, now we must look after the spectrum

spec p_{k}. Fortunately, that leads to no additional trouble since it can easily be seen that the spectra of the auxiliary polynomials \overline{q} and \underline{q} are obtained from the spectrum of p_{k}(z) merely by excluding one point.

Now let diam spec p ⩽ n − 1. Then p(z) = e^{λ_{1}z}p̃(z), where λ̃_{k} =^{def} λ_{k} − λ_{1}, |λ̃_{k}| ⩽ n − 1, and it suffices to show that

sup ; t∈[− \frac{1}{2} , \frac{1}{2}] |p̃(t)| ⩽ \{\frac{A}{μ(E)}\}^{n−1} sup_{t∈E} |p̃(t)|.

Again we write the Hadamard factorization

p̃(z) = ce^{az} ∏_{j⩽11λ}(z − z_{j}) ∏_{j>11λ}(1 − \frac{z}{z_{j}} )e^{z/z_{j}} =^{def} ce^{az}Q(z)R(z).

The oscillation of the factors ce^{az} and R(z) on the interval [−\frac{1}{2}, \frac{1}{2}] can be majorized by e^{const(n−1)}, as before (of course, to estimate |a| now we should use the Turan lemma instead of evaluating the argument increment on vertical segments). But the information we get from Lemma 4.2 is not sufficient to investigate the set where Q(z) is small: though for every z ∈ [−\frac{1}{2}, \frac{1}{2}], except, may be, a set of measure 44h, the Kartan type estimate |z−z_{j}| ⩾ \frac{jh}{(n−1)} is still valid (as above, z_{j} are the complex zeros of the polynomial Q(z) enumerated so that the numbers |z − z_{j}| are monotone nondecreasing), Lemma 4.2 provides us with a nontrivial a priori estimate of |z − z_{j}| only for j > 4(n − 1), not for j > (n − 1) (as we need). I do not know whether in our case one can prove the inequality |z − z_{j}| ⩾ γ \frac{j−(n−1)}{(n−1)} with some universal constant γ > 0; we shall pass over this difficulty in another way.

Lemma 4.3. Let p̃(z) = ∑_{k=1}^{n} c_{k}e^{λ̃_{k}t}, where |λ̃_{k}| ⩽ n − 1 for each k = 1, …, n. Let z ∈ ℂ, and let z_{j} be complex zeros of p̃(z) enumerated so that the numbers |z − z_{j}| are monotone nondecreasing. Then

∏_{j=n}^{11(n−1)} \frac{|z − z_{j}|}{1 + |z − z_{j}|} ⩾ e^{−const(n−1)},

the constant being universal.

Proof. We choose N ∈ {0, …, 11(n−1)−1} so that |z−z_{N}| < 1 and |z−z_{N+1}| ⩾ 1 (N = 0 means that |z − z_{j}| ⩾ 1 for all j = 1, 2, …). The product ∏_{j=N+1}^{11(n−1)} \frac{|z−z_{j}|}{1+|z−z_{j}|} is obviously

greater than 2^{−(11(n−1)−N)} ⩾ 2^{−11(n−1)}, so the lemma is trivial if N < n. Otherwise we estimate the product ∏_{j=n}^{N} \frac{|z−z_{j}|}{1+|z−z_{j}|} from below comparing the maximums m and M of the

absolute value of the exponential polynomial p̃(z) over the disks D_{r} =^{def} { ζ : |ζ − z| ⩽ r } and D_{1} =^{def} { ζ : |ζ − z| ⩽ 1 }, where r =^{def} |z − z_{n}| ∈ (0, 1) (r is actually greater than 0, because a nontrivial exponential polynomial of order n cannot have a zero of multiplicity greater than n − 1).

As above (see the proof of Lemma 4.2), we derive from the Turan lemma that

M ⩽ \{\frac{4e(1 + r)}{2r}\}^{n−1}e^{(n−1)(1+r)}m ⩽ (4e^{3})^{n−1}(\frac{1}{r})^{n−1}m.

On the other hand, in the disk D_{r} the absolute value of the Blaschke factor B_{z_{j}}(z) corresponding to the root z_{j} ∈ D_{1} does not exceed

\frac{|z − z_{j}|r}{1 + |z − z_{j}|r} ⩽ \{ \begin{matrix}2r, & j = 1, …, n − 1, \\ \frac{4|z−z_{j}|}{1+|z−z_{j}|}, & j = n, …, N. \end{matrix}

Thus, m ⩽ (2r)^{n−1} ∏_{j=n}^{N} \frac{4|z−z_{j}|}{1+|z−z_{j}|}M. It remains to compare the two inequalities obtained above.

It is worth mentioning that the additional factor e^{|I| max | Re λ_{k}|} in the Turan type estimate arises because E may be concentrated near one of the endpoints of the interval I. If we want to estimate p(x) at the points x ∈ ℝ such that E is “massive enough” on the both sides of x, there is a majoration not involving the exponents λ_{k}.

Proposition 4.4. (The interpolation Turan lemma). Let p(z) be an exponential polynomial of order n with arbitrary complex exponents. Let E ⊂ ℝ and x ∈ ℝ. Then

|p(x) ⩽ \{\frac{A}{m_{E}(x)}\}^{n−1} sup_{t∈E} |p(t)|,

where m_{E}(x) = min(M_{E}^{+}(x), M_{E}^{−}(x)), and

M_{E}^{+}(x) =^{def} sup_{r>0} \frac{μ\{E ∩ (x − r, x)\}}{r}, M_{E}^{−}(x) =^{def} sup_{r>0} \frac{μ\{E ∩ (x, x + r)\}}{r}

are the “one-sided” Hardy–Littlewood maximal functions.

Proof. This is left to the reader as an easy exercise.

In the case of pure imaginary exponents one can replace m_{E}(x) with M_{E}(x) = max(M_{E}^{+}(x), M_{E}^{−}(x)).

§4.2. Functions with spectrum of finite Lebesgue measure

Let Σ ⊂ ℝ be an open set of finite measure. In Chapter 2 it was shown that for every function f ∈ L^{2}(ℝ) with spectrum spec f ⊂ Σ and for every measurable set E ⊂ ℝ of finite measure, the norm ∥f∥_{L^{2}(ℝ)} can be estimated in terms of the integral of |f|^{2} over ℝ ∖ E, namely,

(∗)  ∥f∥_{L^{2}(ℝ)}^{2} ⩽ e^{Aμ(E)μ(Σ)} ∫_{ℝ∖E} |f|^{2} dμ.

If Σ is an interval, then this statement may be generalized in two directions.

a) L^{2}(ℝ) may be replaced by any other Lebesgue function class L^{p}(ℝ) with p > 0 (if p < 1, then the inclusion spec f ⊂ Σ may be understood, for example, as the possibility of an arbitrarily good approximation of f in the sense of L^{p}-norm by functions from the Schwartz class S with spectrum in Σ).

b) The condition μ(E) < +∞ may be replaced with that of “relative density” of ℝ ∖ E (we say that ℝ ∖ E is relatively dense if there exist two numbers D > d > 0 such that μ(E ∩J) ⩽ D −d for every interval J of length D; if μ(E) > +∞, one can put D = 2μ(E), d = μ(E)).

The corresponding inequality

∥f∥_{L^{p}(ℝ)}^{p} ⩽ e^{2πp|Σ| \frac{D^{2}}{d}} ∫_{ℝ∖E} |f|^{p} dμ

is called “the theorem of Logvinenko and Sereda”. The proof (based on the methods of theory of analytic functions) is as follows. Without loss of generality Σ = [−s, s] (otherwise we consider fe^{iλx} with a properly chosen λ ∈ ℝ instead of f). Then f is an entire function of exponential type at most 2πs. Consider f(z) on the line ℝ + iD. The relative density condition imposed on ℝ ∖ E implies that the harmonic measure ω_{ℂ_{+}}(ℝ ∖ E, z) of the set ℝ ∖ E with respect to the upper half-plane ℂ_{+} and the point z ∈ ℝ + iD is at least \frac{d}{πD} =^{def} γ (to prove this, it suffices to observe that the density of the harmonic measure ω_{ℂ_{+}}(·, z) with respect to Lebesgue measure is greater than \frac{1}{2πD} on each of the two intervals (Re z − D, Re z) and (Re z, Re z + D)). Therefore, for every z ∈ ℝ + iD

log |f(z)|^{p} ⩽ psD + ∫_{ℝ} log |f(t)|^{p}P (z, t) dt = psD + ∫_{ℝ∖E} + ∫_{E}
⩽ psD + ω_{ℂ_{+}}(ℝ ∖ E, z) log ∫_{ℝ∖E} |f(t)|^{p}P (z, t) dt
+ ω_{ℂ_{+}}(E, z) log ∫_{E} |f(t)|^{p}P (z, t) dt.

Hence,

\begin{aligned}|f(z)|^{p} & ⩽ e^{psD}(∫_{ℝ∖E} |f(t)|^{p}P (z, t) dt)^{ω_{ℂ_{+}} (ℝ∖E,z)}(∫_{ℝ} |f(t)|^{p}P (z, t) dt)^{ω_{ℂ_{+}} (E,z)} \\ & ⩽ e^{psD}(∫_{ℝ∖E} |f(t)|^{p}P (z, t) dt)^{γ}(∫_{ℝ} |f(t)|^{p}P (z, t) dt)^{1−γ}. \end{aligned}

Thus,

\begin{matrix}∫_{ℝ} |f(x + iD)|^{p} dx \\ ⩽ e^{psD} ∫_{ℝ}(∫_{ℝ∖E} |f(t)|^{p}P (x + iD, t) dt)^{γ}(∫_{ℝ} |f(t)|^{p}P (x + iD, t) dt)^{1−γ} dx. \end{matrix}

By the Hölder inequality, the last expression does not exceed

\begin{matrix}e^{psD}(∫_{ℝ}(∫_{ℝ∖E} |f(t)|^{p}P (x + iD, t) dt) dx)^{γ} \\ × (∫_{ℝ}(∫_{ℝ} |f(t)|^{p}P (x + iD, t) dt) dx)^{1−γ} = e^{psD}(∫_{ℝ∖E} |f|^{p})^{γ}(∫_{ℝ} |f|^{p})^{1−γ}, \end{matrix}

and, therefore,

∥f(· + iD)∥_{L^{p}(ℝ)}^{p} ⩽ e^{psD}∥f(·)∥_{L^{p}(ℝ)}^{p(1−γ)}(∫_{ℝ∖E} |f|^{p} dμ)^{γ}.

Since f is an entire function, not only can we raise the real line, but lower it as well. We get

∥f(·)∥_{L^{p}(ℝ)}^{p} ⩽ e^{psD}∥f(· + iD)∥_{L^{p}(ℝ)}^{p}.

Thus,

∥f∥_{L^{p}(ℝ)}^{p} ⩽ e^{2psD}∥f∥_{L^{p}(ℝ)}^{p(1−γ)}(∫_{ℝ∖E} |f|^{p} dμ)^{γ},

whence ∥f∥_{L^{p}(ℝ)}^{p} ⩽ e^{2psD/γ} ∫_{ℝ∖E} |f|^{p} dμ, proving the desired statement.

To what extent is such a generalization possible in the case of arbitrary measurable set Σ? We shall restrict ourselves to the question as to whether it is possible to replace the L^{2}-norm with the L^{p}-norm in (∗). If 0 < p < 2, then the answer is affirmative and the corresponding estimate is

(∗_{p})  ∥f∥_{L^{p}(ℝ)}^{p} ⩽ e^{2Apμ(E)μ(Σ)} ∫_{ℝ∖E} |f|^{p} dμ,

where A is the same constant as in (∗). It should be noted that this inequality is rather sharp, namely, there exists an absolute constant a > 0 such that for every p ∈ (0, 2) and every two intervals E and Σ, one can find a function f ∈ L^{p}(ℝ) with spectrum in Σ satisfying

∥f∥_{L^{p}(ℝ)}^{p} ⩾ e^{apμ(E)μ(Σ)} ∫_{ℝ∖E} |f|^{p} dμ.

Proof. Let f ∈ S. We denote by Φ the decreasing rearrangement of |f|. (Φ is the function defined on (0, +∞) by Φ^{−1}(y) = μ(|f| > y).) Inequality (∗) is equivalent to each of the following estimates:

∫_{0}^{t} Φ(x)^{2} dx ⩽ (1 − e^{−Aμ(Σ)t})∥f∥_{L^{2}(ℝ)}^{2} ⟺ ∫_{t}^{∞} Φ(x)^{2} dx ⩾ e^{−Aμ(Σ)t}∥f∥_{L^{2}(ℝ)}^{2},

for all t > 0. The first of them implies, by the Hölder inequality, that

∫_{0}^{t} Φ(x)^{2} dx ⩽ t(\frac{1 − e^{−Aμ(Σ)t}}{t} )^{p/2}∥f∥_{L^{2}(ℝ)}^{p}.

To get some information from the second one, we shall apply the following classical lemma on convex functions.

Lemma. Let φ, ψ: (0, +∞) ⟶ ℝ be nonnegative convex functions such that lim_{+∞} φ = lim_{+∞} ψ = 0. If φ(t) ⩾ ψ(t) for every t > 0, then for every q ∈ (0, 1)

∫_{t}^{∞} |φ′(x)|^{q} dx ⩾ ∫_{t}^{∞} |ψ′(x)|^{q} dx (t > 0).

Putting here φ(t) =^{def} ∫_{t}^{∞} Φ(x)^{2} dx, ψ(t) =^{def} e^{−Aμ(Σ)t}∥f∥_{L^{2}(ℝ)}^{2}, q =^{def} \frac{p}{2}, we get

\begin{aligned}∫_{t}^{∞} Φ(x)^{p} dx & ⩾ \{Aμ(Σ)\}^{p/2}∥f∥_{L^{2}(ℝ)}^{p} ∫_{t}^{∞} e^{−Aμ(Σ)px} dx \\ & = \frac{\{Aμ(Σ)\}^{p/2−1}}{p} ∥f∥_{L^{2}(ℝ)}^{p}e^{−Aμ(Σ)pt}. \end{aligned}

Obviously, (∗_{p}) ⟺ ∫_{0}^{t} Φ(x)^{p} dx ⩽ (e^{2Aμ(Σ)pt} − 1) ∫_{t}^{∞} Φ(x)^{p} dx. Taking the above estimates into account, we see that it suffices to prove the inequality

t(\frac{1 − e^{−Aμ(Σ)t}}{t} )^{p/2} ⩽ (e^{2Aμ(Σ)pt} − 1)\frac{\{Aμ(Σ)\}^{p/2−1}}{p} ∥f∥_{L^{2}(ℝ)}^{p}e^{−Aμ(Σ)pt},

which is equivalent to

(\frac{1 − e^{−Aμ(Σ)t}}{Aμ(Σ)t} )^{p/2} ⩽ \frac{e^{Aμ(Σ)pt} − e^{−Aμ(Σ)pt}}{Aμ(Σ)pt} .

But the last estimate is obvious because the quantity on the left is always less than 1, while that on the right is always greater than 1.

Thus, for 0 < p ⩽ 2 the estimate is essentially the same as in the Logvinenko–Sereda theorem. For p > 2, the situation becomes quite different: we are going to construct a set Σ of finite measure such that the estimate ∥f∥_{L^{p}(ℝ)}^{p} ⩽ C ∫_{ℝ∖E} |f|^{p} dμ is impossible if E is a set of positive measure. The following simple and well-known lemma will play a crucial role in our construction.

Lemma. Let φ ∈ S, A ≠ 0. Then

|∫_{ℝ} φ(x)e^{2πiAx^{2}} dx| ⩽ \frac{1}{\sqrt{|A|}}∥φ′∥_{L^{1}(ℝ)}.

Proof. Putting v(x) =^{def} ∫_{x} e^{2πit^{2}} dt (x ∈ ℝ), we easily check that |v(x)| ⩽ 1 for every x ∈ ℝ. We assume, for definiteness, that A > 0. Then

∫_{ℝ} φ(x)e^{2πiAx^{2}} dx = ∫_{ℝ} φ(x)v′(\sqrt{A}x) dx = −\frac{1}{\sqrt{A}} ∫_{ℝ} φ′(x)v(\sqrt{A}x) dx.

It remains to observe that |∫_{ℝ} φ′(x)v(\sqrt{A}x) dx| ⩽ ∥φ′∥_{L^{1}(ℝ)}.

Corollary. For every λ, b ∈ ℝ

|∫_{ℝ} φ(x)e^{2πi(Ax^{2}+λx+b)} dx| ⩽ \frac{1}{\sqrt{|A|}}∥φ′∥_{L^{1}(ℝ)}.

Proof. A linear change of variable.

Now we fix ε > 0 and choose an arbitrary C^{∞}-smooth 1-periodic function g_{ε} satisfying ∫_{0}^{1} g_{ε} dμ = 1, g_{ε} ≡ 0 on (−\frac{1}{2}, \frac{1}{2}) ∖ (−\frac{ε}{2}, \frac{ε}{2}), and |g_{ε}| ⩽ \frac{2}{ε}. Let J ⊂ ℝ be an interval, f ∈ S, and spec f ⊂ J. We fix some numbers A > 0 and b ∈ [0, 1] and put F̂(λ) = f̂(λ)g_{ε}(Aλ^{2}+b). Clearly, F ∈ S and spec F ⊂ { λ ∈ J : Aλ^{2} + b ∈ supp g_{ε} } =^{def} Σ(b). We have ∫_{0}^{1} μ(Σ(b)) ⩽ ε|J|, and, therefore, we can choose b so that μ(Σ(b)) ⩽ ε|J|. Besides, we always have ∥F ∥_{L^{2}(ℝ)} = ∥F̂∥_{L^{2}(ℝ)} ⩽ \frac{2}{ε}∥f∥_{L^{2}(ℝ)}. To estimate the difference |F (x) − f(x)| (x ∈ ℝ), we expand g_{ε} in Fourier series: g_{ε}(t) = ∑_{k∈ℤ} ĝ_{ε}(k)e^{2πikt}. Our choice of g_{ε} immediately yields ĝ_{ε}(0) = 1, ∑_{k∈ℤ} |ĝ_{ε}(k)| < +∞. Therefore,

|F (x) − f(x)| = |∫_{ℝ}(F̂(λ) − f̂(λ))e^{2πiλx} dλ|

= |∫_{ℝ}( ∑_{k∈ℤ∖{0}} f̂(λ)ĝ_{ε}(k)e^{2πik(Aλ^{2}+b)})e^{2πiλx} dλ|

⩽ ∑_{k∈ℤ∖{0}}|ĝ_{ε}(k)||∫_{ℝ} f̂(λ)e^{2πik(Ax^{2}+λx+b)} dλ|

⩽ ∑_{k∈ℤ∖\{0\}}|ĝ_{ε}(k)|\frac{1}{\sqrt{|k|A}}∥(f̂)′∥_{L^{1}(ℝ)} ⩽ \frac{1}{\sqrt{a}}∥(f̂)′∥_{L^{1}(ℝ)} ∑_{k∈ℤ}|ĝ_{ε}(k)|.

For fixed ε and f, the last expression can be made arbitrarily small if we take A large enough. So, if p > 2, then the norm ∥f − F ∥_{L^{p}(ℝ)}^{p} is also arbitrarily small provided that A is large enough, because

\begin{aligned}∥f − F ∥_{L^{p}(ℝ)}^{p} & ⩽ ∥f − F ∥_{L^{2}(ℝ)}^{2}∥f − F ∥_{L^{∞}(ℝ)}^{p−2} \\ & ⩽ (\frac{2}{ε} + 1)^{2}∥f∥_{L^{2}(ℝ)}^{2}\{\frac{1}{\sqrt{A}}∥(f̂)′∥_{L^{1}(ℝ)} ∑_{k∈ℤ} |ĝ_{ε}(k)|\}^{p−2}. \end{aligned}

We get the following proposition.

Proposition. Let p > 2, f ∈ S, spec f ⊂ J. Then for every μ, δ > 0 there exists a set Σ = Σ(f, μ, δ) ⊂ J and a function F ∈ S with spectrum contained in Σ such that μ(Σ) ⩽ μ and ∥f − F ∥_{L^{p}(ℝ)} ⩽ δ.

Now the construction of a “bad” set Σ is easy. Let f_{0} ∈ S be a function with spectrum contained in J_{0} =^{def} (1, 2). We fix an arbitrary sequence of positive numbers μ_{0}, μ_{1}, μ_{2}, … satisfying ∑_{k=0}^{∞} μ_{k} < +∞. If f_{k}(x) =^{def} 2^{k/p}f_{0}(2^{k}x) (k = 0, 1, 2, …), then spec f_{k} ⊂ J_{k} =^{def} (2^{k}, 2^{k+1}) and ∥f_{k}∥_{L^{p}(ℝ)} = ∥f_{0}∥_{L^{p}(ℝ)}. Also, we fix a sequence δ_{k} ↓ 0. By the above proposition, we can choose F_{k} ∈ S and Σ_{k} ⊂ J_{k} satisfying the inequalities μ(Σ_{k}) ⩽ μ_{k},

∥F_{k} − f_{k}∥_{L^{p}(ℝ)} ⩽ δ_{k}. We put Σ =^{def} ⋃_{k=0}^{∞} Σ_{k}.

Now let E ⊂ ℝ, μ(E) > 0. Assume that 0 is a density point of E. (This leads to no loss of generality because a shift does not change the spectrum.) The functions F_{k} satisfy

∥F_{k}∥_{L^{p}(ℝ)} ⩾ ∥f_{k}∥_{L^{p}(ℝ)} − δ_{k} = ∥f_{0}∥_{L^{p}(ℝ)} − δ_{k} ⟶ ∥f_{0}∥_{L^{p}(ℝ)},
(∫_{ℝ∖E} |F_{k}|^{p} dμ)^{1/p} ⩽ δ_{k} + (∫_{ℝ∖E} |f_{k}|^{p} dμ)^{1/p} = δ_{k} + (∫_{ℝ∖2^{k}E} |f_{0}|^{p} dμ)^{1/p} ⟶ 0.

So, certain functions with spectrum contained in Σ may be almost supported on E in the L^{p}-sense.

Finally, we note that the set Σ constructed above is very simple from the topological point of view: the intersection of E with every bounded interval is merely a union of a finite number of disjoint closed intervals (therefore, it does not matter how one understands the inclusion spec f ⊂ Σ). Besides, it follows from our construction that Σ may thin out arbitrarily fast at infinity, i.e., it can be ensured that the function t → μ(Σ ∖ [−t, t]) be majorized by a prescribed positive monotone decreasing function.

§4.3. The Mandelbrojt theorem

In this section we discuss the sharpness of Theorem 3.6 from §4 of Chapter 3. We recall a well-known theorem of Mandelbrojt on the behavior of functions with “thin” spectrum on small arcs.

Theorem. Let 0 < σ < 1, Λ ⊂ ℤ. The following statements are equivalent:

(a) Among every N successive integers there is at most constN^{σ} elements of Λ;

(b) For every function f ∈ L^{2}(𝕋) with spectrum contained in Λ and every 𝔏 ⊂ 𝕋

∫𝔏 |f|^{2} dμ ⩽ const|𝔏|^{1−σ}∥f∥_{L^{2}(𝕋)}^{2};

(c) For every function f ∈ L^{2}(𝕋) with spectrum contained in Λ and every 𝔏 ⊂ 𝕋

∥f∥_{L^{2}(𝕋)}^{2} ⩽ exp\{const|𝔏|^{− \frac{σ}{1−σ}} \} ∫𝔏 |f|^{2} dμ.

Clearly, this theorem implies that to prove the sharpness of Theorem 3.6 (stated in §4 of Chapter 3) it suffices to construct a set Λ ⊂ ℤ satisfying the Zygmund condition and such that card Λ ∩ [0, ℕ] ⩾ cN^{\frac{1}{2} −ε} for each N = 1, 2, . . .. Simple computations show that as Λ we can take the random set { n ∈ ℕ : ξ_{n} = 1 }, where ξ_{n} are independent random variables defined by P (ξ_{n} = 1) = n^{− \frac{1}{2} −ε}, P (ξ_{n} = 0) = 1 − n^{− \frac{1}{2} −ε}.

References

1. A. S. Belov, Quasianalyticity of the sum of a lacunary series, Mat. Sb. 99 (1976), no. 3, 433–467.

2. V. H. Logvinenko, Yu. F. Sereda, Equivalent norms in the space of entire functions of exponential type, Funktsional Anal. i Prilozhen. (1973), no. 19, 234–246.

3. S. Mandelbrojt, Quasianalytic classes of functions, GIITL, Moscow–Leningrad, 1937.

4. I. M. Mikheev, On series with gaps, Mat. Sb. 98 (1975), no. 4, 538–563.

5. W. O. Amrein and A. M. Berthier, On support properties of L^{p}-functions and their Fourier transform, J. Functional. Anal. 24 (1977), no. 3, 258–267.

6. M. Benedics, Fourier transform of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl. 106 (1985), no. 1, 180–183.

7. D. Gaier, Bemerkungen zum Turanshen Lemma, A. Math. Sem. Univ. Hamburg 35 (1970), no. 1–2, 1–7.

8. V. Havin and B. Jöricke, The uncertainty principle in harmonic analysis, Springer-Verlag (to appear).

9. G. W. Morgan, A note on Fourier transforms, J. London Math. Soc. 9 (1934), no. 3, 187–192.

10. W. Rudin, Trigonometrical series with gaps, J. Math. and Mech. 9 (1960), no. 2, 203–227.

11. P. Turan, Eine neue Methode in der Analyses und deren Anwendungen, Acad. Kiado, Budapest, 1953.

12. A. Zygmund, Trigonometrical Series, vol. 1–2, Cambridge Univ. Press, 1959.

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