An Asymptotic Formula for the Sequence ||exp(i n h(t))||_A

Abstract

Given a function f with an absolutely convergent Fourier series, we define the norm of f as ||f||_A = the sum of absolute values of the Fourier coefficients of f. We study the behavior of ||f^n||_A as n goes to infinity, for f of the form exp(ih(t)) where h is a real, odd and twice continuously differentiable function such that h(t + 2\pi) = h(t) + 2k\pi for some integer k. We obtain a remarkably simple asymptotic formula for the case when h'' has no zeros in (0,\pi) and satisfies an additional condition near 0 and near \pi. Corollaries of our formula are an asymptotic formula due to D.Girard, and a formula on Bessel functions, due to G.Stey.