Translation-based completeness on compact intervals
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909331674365952 |
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| author | Liehr, Lukas |
| author_facet | Liehr, Lukas |
| contents | Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - λ) : λ\in Λ\}$ are complete in $C(I)$ if and only if the series of reciprocals of $Λ$ diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308]. An additional characterization is obtained when $Λ$ is an arithmetic progression, and the generator $f$ constitutes a linear combination of translates of a function with sufficiently fast decay. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Translation-based completeness on compact intervals Liehr, Lukas Classical Analysis and ODEs 42C30, 42A65 Given a compact interval $I \subseteq \mathbb{R}$, and a function $f$ that is a product of a nonzero polynomial with a Gaussian, it will be shown that the translates $\{ f(\cdot - λ) : λ\in Λ\}$ are complete in $C(I)$ if and only if the series of reciprocals of $Λ$ diverges. This extends a theorem in [R. A. Zalik, Trans. Amer. Math. Soc. 243, 299-308]. An additional characterization is obtained when $Λ$ is an arithmetic progression, and the generator $f$ constitutes a linear combination of translates of a function with sufficiently fast decay. |
| title | Translation-based completeness on compact intervals |
| topic | Classical Analysis and ODEs 42C30, 42A65 |
| url | https://arxiv.org/abs/2409.17563 |