Compactness of Fourier concentration operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929762201501696 |
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| author | Samuelsen, Helge Jørgen |
| author_facet | Samuelsen, Helge Jørgen |
| contents | We present a sufficient condition on sets $E$ and $F$ in $\mathbb{R}^d$ to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets $E$ and $F$ are both very thin at infinity, then the associated Fourier concentration operator is compact on $L^2(\mathbb{R}^d)$. The proof relies on a combination of the Logvinenko-Sereda uncertainty principle together with an uncertainty principle due to Shubin, Vakilian and Wolff. This provides a partial answer to a question posed by Katsnelson and Machluf on truncated Fourier operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13122 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Compactness of Fourier concentration operators Samuelsen, Helge Jørgen Classical Analysis and ODEs Functional Analysis 42B10, 47B07, 47G10 We present a sufficient condition on sets $E$ and $F$ in $\mathbb{R}^d$ to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets $E$ and $F$ are both very thin at infinity, then the associated Fourier concentration operator is compact on $L^2(\mathbb{R}^d)$. The proof relies on a combination of the Logvinenko-Sereda uncertainty principle together with an uncertainty principle due to Shubin, Vakilian and Wolff. This provides a partial answer to a question posed by Katsnelson and Machluf on truncated Fourier operators. |
| title | Compactness of Fourier concentration operators |
| topic | Classical Analysis and ODEs Functional Analysis 42B10, 47B07, 47G10 |
| url | https://arxiv.org/abs/2503.13122 |