The asymptotic behavior of norms of powers of absolutely convergent Fourier series

Dennis Michael Girard

Vol. 37 (1971), No. 2, 357–381

DOI: 10.2140/pjm.1971.37.357

Abstract

Let f(t) have an absolutely convergent Fourier series f(t) = ∑ a_ke^ikt and set ∥f∥ = ∑ |a_k|. In this paper we will study the asymptotic behavior of ∥fⁿ∥ as n →∞. THEOREM. Let f be absolutely continuous and let f′ be of bounded variation on the real line modulo 2π. Let f(0) = 1 but |f(t)| < 1 for t≠0 and suppose that for t near 0, f(t) = ϕ(e^it) where ϕ is defined and analytic near z = 1. Define the parameters α,p,q, A and β as follows

α	= ϕ(1)
ϕ(z)	= zα + Aip(z − 1)p + o(1)(z − 1)p,z → 1(A≠0)
|ϕ(eit)|	= 1 − βtq + o(tq),t → 0(β≠0).

Then, (a) for p≠q

where δ(p) = 0 if p is even and, = 1 if p is odd; (b) for p = q

where F is the Fourier transform of F(t) = exp(At^p). The following results about these parameters are known and easily verifiable: p and q are positive integers, 2 ≦ p ≦ q, q is even, β > 0,ReA ≦ 0;p = q if and only if ReA≠0;p = q implies β = −ReA;

and

(2) ϕ(eit)	= exp
	+ , as t → 0,c r real, if p≠q.

We outline in §5 how it is possible to relax the condition of analyticity at t = 0 and replace it with conditions (1) and (2) where the 𝒪 terms satisfy certain smoothness conditions. G. W. Hedstrom proved in 1966 that under these same hypotheses, there exist two constants c,C such that c < ∥fⁿ∥n^{−(1−p∕q)∕2} < C

Mathematical Subject Classification

Primary: 42.12

Milestones

Received: 4 September 1969

Published: 1 May 1971