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ii) We denote by R > 0 the set of positive real numbers and R ≥ 0 = R > 0 ∪{ 0 }
v) Let I be an interval of R and denote by L 1 ( I ) (resp. L 1 loc ( I ) )
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Sec. 3. We introduce the general deﬁnition of hybrid transforms, as well as the Euler-Laplace transform and the Euler-Fourier transform
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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.
Foundations of Computational Mathematics – Springer Journals
Published: Nov 22, 2022
Keywords: Topological data analysis; Integral transforms; Constructible functions; 32B20; 44A05; 55N31
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