In the early 1930s, Wiener proved that if f(x)
is a strictly positive periodic function
whose Fourier series is absolutely convergent, then the Fourier series of
g(x)=1/f(x)
is also absolutely convergent [8, pp. 10–14].
This phenomenon can be easily
understood nowadays using Banach algebra techniques (see, for example, [4, pp.
202–203]). In fact, these techniques allow us to study the absolute convergence of
g(x)=F(f(x)), where F
is holomorphic in an open subset of [Copf ] that contains the
range of f(x) (for x∈ℝ). In this
context, Wiener's original problem corresponds to the
choice F(z)=1/z.
In this work we want to analyse the constraints on the simultaneous rate of
vanishing of the Fourier coefficients fˆ(n) and
ĝ(n) as n→∞. We shall focus on
g=1/f, but we shall also study the general case
g=F(f). In either case, there are
obviously no constraints when f is a constant function.
Although this problem does not seem to be directly related to uncertainty
inequalities for the Fourier Transform, we observe that there are some analogies,
both in the nature of the results and in the proof techniques. The general fact with
which we are dealing is that fˆ(n) and ĝ(n)
cannot vanish too quickly at the same time as n→∞, unless
f(x) is constant. The general fact that underlies uncertainty
inequalities is that a non-periodic function ϕ(x) and its Fourier Transform
ϕcirc;(u) cannot vanish too quickly at the same time as
x→∞ and u→∞, unless ϕ(x) is zero
(almost everywhere). For a simple introduction to some aspects of uncertainty
inequalities, see [5]; for a thorough and recent introduction
to this vast subject, see [3].
