Parametric Fourier and Mellin transforms of power-constructible functions

TL;DR

This paper extends the class of power-constructible functions to include complex powers of subanalytic functions and their transforms, providing detailed asymptotic analysis and stability properties under various operations.

Contribution

It introduces a new algebra of functions containing all power-constructible functions and stable under parametric Fourier transforms, integration, and limits, with explicit asymptotic expansions.

Findings

The class contains all complex powers of subanalytic functions.

The class is stable under parametric Fourier transforms and integration.

A full asymptotic expansion in the power-logarithmic scale is provided.

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.