Let f(t) have an absolutely
convergent Fourier series f(t) = ∑
a_{k}e^{ikt} and set ∥f∥ = ∑
a_{k}. In this paper we will
study the asymptotic behavior of ∥f^{n}∥ 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)  = z^{α} + Ai^{p}(z − 1)^{p} + o(1)(z − 1)^{p},z → 1(A≠0)  
 ϕ(e^{it})  = 1 − βt^{q} + o(t^{q}),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(e^{it})  = 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 < ∥f^{n}∥n^{−(1−p∕q)∕2} < C
