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

Georges Comte; Dan J. Miller; Tamara Servi

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arXiv · 2507.06142

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

Georges Comte·Dan J. Miller·Tamara Servi

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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.

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BibTeX RIS

Georges Comte, Dan J. Miller, Tamara Servi. 2025-07-08. Decay of Fourier transforms and analytic continuation of power-constructible functions. https://arxiv.org/abs/2507.06142

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