Colloquium Schedule for 1996-7
Conjectures on partial orderings of distribution functionsof exponential sums on the circle
Ivo Klemes
McGill University
The Dirichlet sum (kernel) with N terms was conjectured by Hardy and Littlewood to have the smallestL1 norm among all exponential sums with N terms. Similarsharp inequalities were conjectured for Lp norms if0 < p < 2, and the reverse inequalities if 2 < p . Weconsider the question of what stronger properties of thedistribution or density functions could possibly implyall of these inequalities. One candidate is the well -known majorisation ordering (see Hardy, Littlewood andPolya), but here this fails to hold when N = 4 . Wepropose a related ordering, which was motivated by atheorem of Szego and the notion of predictability of thesequence of Fourier coefficients.