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.
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Notes
Since \(f_{|\text {supp}(\varphi )}\) is proper and Y is Hausdorff and locally compact, it is a closed map.
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Acknowledgements
The author is grateful to François Petit and Steve Oudot for their continuous support and their scientific advice through the development of this paper. We also express our gratitude to Nicolas Berkouk for taking the time to discuss and offer useful explanations on sheaf theory.
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Communicated by Peter Bubenik.
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Lebovici, V. Hybrid Transforms of Constructible Functions. Found Comput Math 24, 539–585 (2024). https://doi.org/10.1007/s10208-022-09596-2
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DOI: https://doi.org/10.1007/s10208-022-09596-2