The vector-valued Stieltjes moment problem with general exponents
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916298404921344 |
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| author | Debrouwere, Andreas Neyt, Lenny |
| author_facet | Debrouwere, Andreas Neyt, Lenny |
| contents | We characterize the sequences of complex numbers $(z_{n})_{n \in \mathbb{N}}$ and the locally complete $(DF)$-spaces $E$ such that for each $(e_{n})_{n \in \mathbb{N}} \in E^\mathbb{N}$ there exists an $E$-valued function $\mathbf{f}$ on $(0,\infty)$ (satisfying a mild regularity condition) such that $$\int_{0}^{\infty} t^{z_{n}} \mathbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb{N},$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution $\mathbf{f}$ that is smooth on $(0,\infty)$ and satisfies certain optimal growth bounds near $0$ and $\infty$. The scalar-valued case $(E = \mathbb{C})$ was treated by Durán [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00497 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The vector-valued Stieltjes moment problem with general exponents Debrouwere, Andreas Neyt, Lenny Functional Analysis Primary 30E05, 44A60, 46A63, 47A57. Secondary, 46E10, 46E40, 46G10 We characterize the sequences of complex numbers $(z_{n})_{n \in \mathbb{N}}$ and the locally complete $(DF)$-spaces $E$ such that for each $(e_{n})_{n \in \mathbb{N}} \in E^\mathbb{N}$ there exists an $E$-valued function $\mathbf{f}$ on $(0,\infty)$ (satisfying a mild regularity condition) such that $$\int_{0}^{\infty} t^{z_{n}} \mathbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb{N},$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution $\mathbf{f}$ that is smooth on $(0,\infty)$ and satisfies certain optimal growth bounds near $0$ and $\infty$. The scalar-valued case $(E = \mathbb{C})$ was treated by Durán [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result. |
| title | The vector-valued Stieltjes moment problem with general exponents |
| topic | Functional Analysis Primary 30E05, 44A60, 46A63, 47A57. Secondary, 46E10, 46E40, 46G10 |
| url | https://arxiv.org/abs/2401.00497 |