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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.14260 |
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| _version_ | 1866911927530160128 |
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| author | Zikkos, Elias |
| author_facet | Zikkos, Elias |
| contents | Let $\{e^{iλ_n t}\}_{n\in\mathbb{Z}}$ be an exponential Schauder Basis for $L^2 (0,1)$, for $λ_n\in\mathbb{R}$, and let $\{r_n(t)\}_{n\in\mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a non-empty subset of the integers containing exactly $M$ elements. We prove that for $α>0$ the weighted system \[ \{t^α\cdot r_n(t)\}_{n\in\mathbb{Z}\setminus A} \] is exact in the space $L^2 (0,1)$, that is, it is complete and minimal in $L^2 (0,1)$, if and only if \[ M-\frac{1}{2}\le α< M+\frac{1}{2}. \] We also show that such a system is not a Riesz Basis for $L^2 (0,1)$. In particular, the weighted trigonometric system $\{t^α\cdot e^{2πi n t}\}_{n\in\mathbb{Z}\setminus A}$ is exact in $L^2 (0,1)$, if and only if $α\in [M-\frac{1}{2}, M+\frac{1}{2})$, but it is not a Schauder Basis for $L^2 (0,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On exact systems $\{t^α\cdot e^{2πi nt}\}_{n\in\mathbb{Z}\setminus A}$ in $L^2 (0,1)$ which are not Schauder Bases and their generalizations Zikkos, Elias Functional Analysis 42C30, 42A65 Let $\{e^{iλ_n t}\}_{n\in\mathbb{Z}}$ be an exponential Schauder Basis for $L^2 (0,1)$, for $λ_n\in\mathbb{R}$, and let $\{r_n(t)\}_{n\in\mathbb{Z}}$ be its dual Schauder Basis. Let $A$ be a non-empty subset of the integers containing exactly $M$ elements. We prove that for $α>0$ the weighted system \[ \{t^α\cdot r_n(t)\}_{n\in\mathbb{Z}\setminus A} \] is exact in the space $L^2 (0,1)$, that is, it is complete and minimal in $L^2 (0,1)$, if and only if \[ M-\frac{1}{2}\le α< M+\frac{1}{2}. \] We also show that such a system is not a Riesz Basis for $L^2 (0,1)$. In particular, the weighted trigonometric system $\{t^α\cdot e^{2πi n t}\}_{n\in\mathbb{Z}\setminus A}$ is exact in $L^2 (0,1)$, if and only if $α\in [M-\frac{1}{2}, M+\frac{1}{2})$, but it is not a Schauder Basis for $L^2 (0,1)$. |
| title | On exact systems $\{t^α\cdot e^{2πi nt}\}_{n\in\mathbb{Z}\setminus A}$ in $L^2 (0,1)$ which are not Schauder Bases and their generalizations |
| topic | Functional Analysis 42C30, 42A65 |
| url | https://arxiv.org/abs/2406.14260 |