Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation
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| Format: | Preprint |
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2024
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| _version_ | 1866917733265833984 |
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| author | Gröchenig, Karlheinz Romero, José Luis Speckbacher, Michael |
| author_facet | Gröchenig, Karlheinz Romero, José Luis Speckbacher, Michael |
| contents | We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sjöstrand class. We prove that larger symbol classes lead to Hölder continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,αΛ)$ with respect to the parameter $α>0$, where $Λ$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $α$ if $g\in M^1(\mathbb{R}^d)$, and are Hölder continuous if $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the Hölder exponent explicitly given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation Gröchenig, Karlheinz Romero, José Luis Speckbacher, Michael Functional Analysis Spectral Theory 47G30, 42C40, 47A10, 47L80, 35S05 We build on our recent results on the Lipschitz dependence of the extreme spectral values of one-parameter families of pseudodifferential operators with symbols in a weighted Sjöstrand class. We prove that larger symbol classes lead to Hölder continuity with respect to the parameter. This result is then used to investigate the behavior of frame bounds of families of Gabor systems $\mathcal{G}(g,αΛ)$ with respect to the parameter $α>0$, where $Λ$ is a set of non-uniform, relatively separated time-frequency shifts, and $g\in M^1_s(\mathbb{R}^d)$, $0\leq s\leq 2$. In particular, we show that the frame bounds depend continuously on $α$ if $g\in M^1(\mathbb{R}^d)$, and are Hölder continuous if $g\in M^1_s(\mathbb{R}^d)$, $0<s\leq 2$, with the Hölder exponent explicitly given. |
| title | Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation |
| topic | Functional Analysis Spectral Theory 47G30, 42C40, 47A10, 47L80, 35S05 |
| url | https://arxiv.org/abs/2407.18065 |