2025/07/08 by Georges Comte, D.J. Miller, Comte, Georges +3
Mathematics · #14P15 #26B15 #32B20 #42A38 (Primary) #42B20 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2507.06142
openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a subfield K of C, we denote by CK 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 CK, including the fact that f in CK(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.