Schauder frames of discrete translates in $L^p(\mathbb{R})$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912757012496384 |
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| author | Lev, Nir Tselishchev, Anton |
| author_facet | Lev, Nir Tselishchev, Anton |
| contents | For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{λ_n\}_{n=1}^\infty$ satisfying $|λ_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals $\{g^*_n\}_{n=1}^\infty$ on $L^p(\mathbb{R})$, such that every $f \in L^p(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-λ_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schauder frames of discrete translates in $L^p(\mathbb{R})$ Lev, Nir Tselishchev, Anton Classical Analysis and ODEs Functional Analysis 42A10, 42C15, 46B15 For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{λ_n\}_{n=1}^\infty$ satisfying $|λ_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals $\{g^*_n\}_{n=1}^\infty$ on $L^p(\mathbb{R})$, such that every $f \in L^p(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-λ_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative. |
| title | Schauder frames of discrete translates in $L^p(\mathbb{R})$ |
| topic | Classical Analysis and ODEs Functional Analysis 42A10, 42C15, 46B15 |
| url | https://arxiv.org/abs/2402.09915 |