Hybrid transforms of constructible functions

Vadim Lebovici

Submitted on 15 November 2021, last revised on 16 November 2022


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.


Comment: 44 pages. Some sections are clarified and proofs simplified following the reviews. We thank anonymous referees for their suggestions

Subjects: Mathematics - Algebraic Topology; Computer Science - Computational Geometry; Mathematics - Algebraic Geometry; 32B20, 44A05, 55N31