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Hybrid Transforms of Constructible Functions

Hybrid Transforms of Constructible Functions 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 with (sub)level-sets persistence of random Gaussian filtrations. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Computational Mathematics Springer Journals

Hybrid Transforms of Constructible Functions

Foundations of Computational Mathematics , Volume OnlineFirst – Nov 22, 2022

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References (40)

Publisher
Springer Journals
Copyright
Copyright © SFoCM 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
ISSN
1615-3375
eISSN
1615-3383
DOI
10.1007/s10208-022-09596-2
Publisher site
See Article on Publisher Site

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 with (sub)level-sets persistence of random Gaussian filtrations.

Journal

Foundations of Computational MathematicsSpringer Journals

Published: Nov 22, 2022

Keywords: Topological data analysis; Integral transforms; Constructible functions; 32B20; 44A05; 55N31

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