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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.11339 |
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| _version_ | 1866909786953482240 |
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| author | di Dio, Philipp J. |
| author_facet | di Dio, Philipp J. |
| contents | In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence $(s_α)_{α\in\mathbb{N}_0^n}$ there exists a Schwartz function $f\in\mathcal{S}(\mathbb{R}^n,\mathbb{C})$ with $\mathrm{supp}\, f\subseteq [0,\infty)^n$ such that $s_α= \int x^α\cdot f(x)~\mathrm{d}x$ for all $α\in\mathbb{N}_0^n$. It has been claimed to be a generalization of the result by T. Sherman [Rend. Circ. Mat. Palermo 13 (1964), 273-278], that every complex sequences is represented by a complex measure on $[0,\infty)^n$. In the present work we use the convolution of sequences and measures to show, that Duran's result is a trivial consequence of Sherman's result. We use our easy proof to extend the Schwartz function result and to show the flexibility in choosing very specific functions $f$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Absolutely continuous representing measures of complex sequences di Dio, Philipp J. Functional Analysis In 1989, A. J. Duran [Proc. Amer. Math. Soc. 107 (1989), 731-741] showed, that for every complex sequence $(s_α)_{α\in\mathbb{N}_0^n}$ there exists a Schwartz function $f\in\mathcal{S}(\mathbb{R}^n,\mathbb{C})$ with $\mathrm{supp}\, f\subseteq [0,\infty)^n$ such that $s_α= \int x^α\cdot f(x)~\mathrm{d}x$ for all $α\in\mathbb{N}_0^n$. It has been claimed to be a generalization of the result by T. Sherman [Rend. Circ. Mat. Palermo 13 (1964), 273-278], that every complex sequences is represented by a complex measure on $[0,\infty)^n$. In the present work we use the convolution of sequences and measures to show, that Duran's result is a trivial consequence of Sherman's result. We use our easy proof to extend the Schwartz function result and to show the flexibility in choosing very specific functions $f$. |
| title | Absolutely continuous representing measures of complex sequences |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2509.11339 |