Uncertainty Inequalities for Fourier Series of Pairs of Reciprocal Positive Functions

In the early 1930s, Wiener proved that if f(x) is a strictlypositive periodic function whose Fourier series is absolutelyconvergent, then the Fourier series of g(x)=1/f(x) is also absolutelyconvergent [8, pp. 10–14]. This phenomenon can be easilyunderstood nowadays using Banach algebra techniques (see, forexample, [4, pp. 202–203]). In fact, these techniquesallow us to study the absolute convergence of g(x)=F(f(x)),where F is holomorphic in an open subset of C that containsthe range of f(x) (for xR). In this context, Wiener's originalproblem corresponds to the choice F(z)=1/z. In this work we want to analyse the constraints on the simultaneousrate of vanishing of the Fourier coefficients f(n) and (n) asn. We shall focus on g=1/f, but we shall also study the generalcase g=F(f). In either case, there are obviously no constraintswhen f is a constant function. Although this problem does not seem to be directly related touncertainty inequalities for the Fourier Transform, we observethat there are some analogies, both in the nature of the resultsand in the proof techniques. The general fact with which weare dealing is that f(n) and (n) cannot vanish too quickly atthe same time as n, unless f(x) is constant. The general factthat underlies uncertainty inequalities is that a non-periodicfunction (x) and its Fourier Transform circ;(u) cannot vanishtoo quickly at the same time as x and u, unless (x) is zero(almost everywhere). For a simple introduction to some aspectsof uncertainty inequalities, see [5]; for a thorough and recentintroduction to this vast subject, see [3]. 1991 MathematicsSubject Classification 42A05, 42A16, 42A99.