Vol. 37, No. 2, 1971

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ISSN: 0030-8730
The asymptotic behavior of norms of powers of absolutely convergent Fourier series

Dennis Michael Girard

Vol. 37 (1971), No. 2, 357–381
Abstract

Let f(t) have an absolutely convergent Fourier series f(t) = akeikt and set f= |ak|. In this paper we will study the asymptotic behavior of fnas n →∞. THEOREM. Let f be absolutely continuous and let fbe of bounded variation on the real line modulo 2π. Let f(0) = 1 but |f(t)| < 1 for t0 and suppose that for t near 0, f(t) = ϕ(eit) 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(A0)
|ϕ(eit)| = 1 βtq + o(tq),t 0(β0).
Then, (a) for pq
∥fn∥ ∼ (2∕π)1∕2−δ(p)q−1[p(p − 1)]1∕2Γ (p∕2q)|A |1∕2β−p∕2qn(1−p∕q)∕2

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

 lim  ∥fn∥ = (1∕2π)∥ˆF∥1
n→ ∞

where F is the Fourier transform of F(t) = exp(Atp). 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 ReA0;p = q implies β = ReA;

(1) ϕ (eit) = exp[iαt+ Atp + 𝒪 (tp+1)],t → 0 if p = q,

and

(2) ϕ(eit) = exp[
p  p    ∑q    r    q
iαt+ (− 1) At + i    crt − βt
r=p+1
+       ]
𝒪(tq+1), as t 0,c r real, if pq.
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 < fnn(1p∕q)2 < C

Mathematical Subject Classification
Primary: 42.12
Milestones
Received: 4 September 1969
Published: 1 May 1971
Authors
Dennis Michael Girard