The Fourier transform with Henstock--Kurzweil and continuous primitive integrals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910802678644736 |
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| author | Talvila, Erik |
| author_facet | Talvila, Erik |
| contents | For each $f\!:\!\mathbb{R}\to\mathbb{C}$ that is Henstock--Kurzweil integrable on the real line, or is a distribution in the completion of the space of Henstock--Kurzweil integrable functions in the Alexiewicz norm, it is shown that the Fourier transform is the second distributional derivative of a Hölder continuous function. The space of such Fourier transforms is isometrically isomorphic to the completion of the Henstock--Kurzweil integrable functions. There is an exchange theorem, inversion in norm and convolution results. Sufficient conditions are given for an $L^1$ function to have a Fourier transform that is of bounded variation. Pointwise inversion of the Fourier transform is proved for functions in $L^p$ spaces for $1<p<\infty$. The exchange theorem is used to evaluate an integral that does not appear in published tables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fourier transform with Henstock--Kurzweil and continuous primitive integrals Talvila, Erik Classical Analysis and ODEs 42A38, 26A39 (Primary) 46B04, 46F10 (Secondary) For each $f\!:\!\mathbb{R}\to\mathbb{C}$ that is Henstock--Kurzweil integrable on the real line, or is a distribution in the completion of the space of Henstock--Kurzweil integrable functions in the Alexiewicz norm, it is shown that the Fourier transform is the second distributional derivative of a Hölder continuous function. The space of such Fourier transforms is isometrically isomorphic to the completion of the Henstock--Kurzweil integrable functions. There is an exchange theorem, inversion in norm and convolution results. Sufficient conditions are given for an $L^1$ function to have a Fourier transform that is of bounded variation. Pointwise inversion of the Fourier transform is proved for functions in $L^p$ spaces for $1<p<\infty$. The exchange theorem is used to evaluate an integral that does not appear in published tables. |
| title | The Fourier transform with Henstock--Kurzweil and continuous primitive integrals |
| topic | Classical Analysis and ODEs 42A38, 26A39 (Primary) 46B04, 46F10 (Secondary) |
| url | https://arxiv.org/abs/2501.17118 |