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Journal Articles Foundations of Computational Mathematics Year : 2022

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 to (sub)level-sets persistence of random Gaussian filtrations.
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hal-03474277 , version 1 (30-01-2023)





Vadim Lebovici. Hybrid transforms of constructible functions. Foundations of Computational Mathematics, 2022, ⟨10.1007/s10208-022-09596-2⟩. ⟨hal-03474277⟩
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