Totally Positive Functions and Gabor Frames over Rational Lattices
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910453101232128 |
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| author | Gröchenig, Karlheinz |
| author_facet | Gröchenig, Karlheinz |
| contents | We show that for an arbitrary totally positive function $g\in L^1(\mathbb{R} )$ and
$αβ$ rational, the Gabor family $\{e^{2πi βl
t} g(t-αk): k,l \in \mathbb{Z} \}$ is a frame for $L^2(\mathbb{R})$, if and only if $αβ<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00857 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Totally Positive Functions and Gabor Frames over Rational Lattices Gröchenig, Karlheinz Functional Analysis 42C15, 42C40, 94A20, 42A82 We show that for an arbitrary totally positive function $g\in L^1(\mathbb{R} )$ and $αβ$ rational, the Gabor family $\{e^{2πi βl t} g(t-αk): k,l \in \mathbb{Z} \}$ is a frame for $L^2(\mathbb{R})$, if and only if $αβ<1$. |
| title | Totally Positive Functions and Gabor Frames over Rational Lattices |
| topic | Functional Analysis 42C15, 42C40, 94A20, 42A82 |
| url | https://arxiv.org/abs/2301.00857 |