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

arXiv0805.1699MaRDI QIDQ6209482

Publication date: 12 May 2008

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 + 2pi) = h(t) + 2kpi 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.

Mathematics Subject Classification ID

Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (41A60) Fourier coefficients, Fourier series of functions with special properties, special Fourier series (42A16)

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