Decay of Fourier transforms and analytic continuation of power-constructible functions

TL;DR

This paper explores the relationship between the holomorphic extension properties of certain globally subanalytic functions and the decay of their Fourier transforms, establishing criteria for meromorphic extension and integrability.

Contribution

It characterizes the exponential decay of Fourier transforms in terms of holomorphic extension widths and links analytic continuation to rationality of functions in the class C^K.

Findings

f in C^K(R) extends meromorphically iff f is rational

Decay rate of Fourier transform linked to holomorphic extension width

Fourier transform F[f] is integrable if f is integrable and continuous

Abstract

For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f in C^\K(R), we study links between holomorphic extensions of f and the decay of its Fourier transform F[f] by using tameness properties of the globally subanalytic functions from which f is constructed. We first prove a number of theorems about analytic continuation of functions in C^K, including the fact that f in C^K(R) extends meromorphically to C if and only if f is rational. We then characterize the exponential rate of decay of F[f] by the maximal width of a horizontal strip in the plane about the real axis to which f extends holomorphically. Finally, we show that F[f] is integrable if f is integrable and continuous.