Fourier transforms of bounded functions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866915750677053440 |
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| author | Talvila, Erik |
| author_facet | Talvila, Erik |
| contents | The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a Hölder continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16912 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fourier transforms of bounded functions Talvila, Erik Classical Analysis and ODEs Primary 42A38, 26A42, Secondary 46B04, 46F12 The Fourier transform of a bounded measurable function, $f$, on the real line is shown to be the second distributional derivative of a Hölder continuous function. The Fourier transform is written as the difference of $\int_{-1}^1 e^{-ist}f(t)\,dt$ and the second distributional derivative of the integral $\int_{\lvert{t}\rvert>1}e^{-ist}f(t)\,dt/t^2$. The space of such Fourier transforms is isometrically isomorphic to $L^\infty(\mathbb{R})$. There is an exchange theorem, inversion and convolution results. The Fourier transform of the functions $x\mapsto\cos^m(a/x)$ for each natural number $m$ are computed. Also for $x\mapsto x\sin(a/x)$ and $x\mapsto\arctan(x/a)$. |
| title | Fourier transforms of bounded functions |
| topic | Classical Analysis and ODEs Primary 42A38, 26A42, Secondary 46B04, 46F12 |
| url | https://arxiv.org/abs/2601.16912 |