Parametric Fourier and Mellin transforms of power-constructible functions
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| Format: | Preprint |
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2023
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| _version_ | 1866917854875484160 |
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| author | Cluckers, Raf Comte, Georges Servi, Tamara |
| author_facet | Cluckers, Raf Comte, Georges Servi, Tamara |
| contents | We enrich the class of power-constructible functions, introduced in [CCRS23], to a class of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of the class. We furthermore identify a subclass which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This subclass is also stable under parametric integration, under taking pointwise and $L^p$limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in this subclass. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_15958 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parametric Fourier and Mellin transforms of power-constructible functions Cluckers, Raf Comte, Georges Servi, Tamara Classical Analysis and ODEs Algebraic Geometry Logic 26B15, 14P15, 32B20, 42B20, 42A38 (Primary) 03C64, 14P10, 33B10 (Secondary) We enrich the class of power-constructible functions, introduced in [CCRS23], to a class of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of the class. We furthermore identify a subclass which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This subclass is also stable under parametric integration, under taking pointwise and $L^p$limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in this subclass. |
| title | Parametric Fourier and Mellin transforms of power-constructible functions |
| topic | Classical Analysis and ODEs Algebraic Geometry Logic 26B15, 14P15, 32B20, 42B20, 42A38 (Primary) 03C64, 14P10, 33B10 (Secondary) |
| url | https://arxiv.org/abs/2308.15958 |