Hybrid transforms of constructible functions

TL;DR

This paper introduces hybrid integral transforms combining Lebesgue and Euler calculus for constructible functions, including new Euler-Fourier and Euler-Laplace transforms, with applications to topological data analysis and persistence modules.

Contribution

It defines a general framework for hybrid transforms, introduces two new transforms, and establishes their properties and applications to persistence and index theory.

Findings

Euler-Fourier transform has a left inverse

Euler-Laplace transform generalizes persistent magnitude

Provides expectation formulas for Gaussian filtrations

Abstract

We introduce a general definition of hybrid transforms for constructible functions. These are integral transforms combining Lebesgue integration and Euler calculus. Lebesgue integration gives access to well-studied kernels and to regularity results, while Euler calculus conveys topological information and allows for compatibility with operations on constructible functions. We conduct a systematic study of such transforms and introduce two new ones: the Euler-Fourier and Euler-Laplace transforms. We show that the first has a left inverse and that the second provides a satisfactory generalization of Govc and Hepworth's persistent magnitude to constructible sheaves, in particular to multi-parameter persistent modules. Finally, we prove index-theoretic formulae expressing a wide class of hybrid transforms as generalized Euler integral transforms. This yields expectation formulae for transforms of constructible functions associated to (sub)level-sets persistence of random Gaussian filtrations.