On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866916167114817536 |
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| author | Hughes, Kevin Israel, Arie Mayeli, Azita |
| author_facet | Hughes, Kevin Israel, Arie Mayeli, Azita |
| contents | Let $F$, $S$ be bounded measurable sets in $\mathbb{R}^d$. Let $P_F : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d) $ be the orthogonal projection on the subspace of functions with compact support on $F$, and let $B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on $S$. In this paper, we derive improved distributional estimates on the eigenvalue sequence $1 \geq λ_1(F,S) \geq λ_2(F,S) \geq \cdots > 0$ of the \emph{spatio-spectral limiting operator} $B_S P_F B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$. The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy.
Our proof is based on the decomposition techniques developed in \cite{MaRoSp23}. The novelty of our approach is in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of the results in \cite{ArieAzita23} on the eigenvalues of spatio-spectral limiting operators associated to cubical domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13092 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II Hughes, Kevin Israel, Arie Mayeli, Azita Classical Analysis and ODEs Spectral Theory 42B10, 47A75 Let $F$, $S$ be bounded measurable sets in $\mathbb{R}^d$. Let $P_F : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d) $ be the orthogonal projection on the subspace of functions with compact support on $F$, and let $B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on $S$. In this paper, we derive improved distributional estimates on the eigenvalue sequence $1 \geq λ_1(F,S) \geq λ_2(F,S) \geq \cdots > 0$ of the \emph{spatio-spectral limiting operator} $B_S P_F B_S : L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$. The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. Our proof is based on the decomposition techniques developed in \cite{MaRoSp23}. The novelty of our approach is in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of the results in \cite{ArieAzita23} on the eigenvalues of spatio-spectral limiting operators associated to cubical domains. |
| title | On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II |
| topic | Classical Analysis and ODEs Spectral Theory 42B10, 47A75 |
| url | https://arxiv.org/abs/2403.13092 |