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arXiv · 2308.15958

Parametric Fourier and Mellin transforms of power-constructible functions

Raf Cluckers·Georges Comte·Tamara Servi
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Abstract

We enrich the class of power-constructible functions, introduced in [CCRS23], to a class of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of the class. We furthermore identify a subclass which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This subclass is also stable under parametric integration, under taking pointwise and $L^p$limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in this subclass.

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BibTeXRIS

Raf Cluckers, Georges Comte, Tamara Servi. 2023-08-30. Parametric Fourier and Mellin transforms of power-constructible functions. https://doi.org/10.1017/fms.2024.128

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