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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2508.12194 |
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| _version_ | 1866912540811853824 |
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| author | Bhowmik, P. Deodhar, S. Iosevich, A. |
| author_facet | Bhowmik, P. Deodhar, S. Iosevich, A. |
| contents | A classical result due to Agranovsky and Narayanan (\cite{AN04}) says that if the support of the Fourier transform of $f: {\mathbb R}^n \to {\mathbb C}$ is carried by a smooth measure on a $d$-dimensional manifold $M$, and $f \in L^p({\mathbb R}^d)$ for $p \leq \frac{2n}{d}$, then $f$ is identically equal to $0$. In this paper, we investigate an analogous problem for functions $f: {\mathbb Z}_N^d \to {\mathbb C}$. Bourgain's celebrated result on $Λ_p$ sets (\cite{Bou89}), random constructions (\cite{Bab89}), and connections with the theory of exact signal recovery (\cite{DS89}, \cite{MS73}, \cite{IKLM24}, \cite{IM24}) play an important role. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12194 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Results in Spectral Synthesis Over ${\mathbb Z}_N^d$ Bhowmik, P. Deodhar, S. Iosevich, A. Classical Analysis and ODEs Combinatorics 42B A classical result due to Agranovsky and Narayanan (\cite{AN04}) says that if the support of the Fourier transform of $f: {\mathbb R}^n \to {\mathbb C}$ is carried by a smooth measure on a $d$-dimensional manifold $M$, and $f \in L^p({\mathbb R}^d)$ for $p \leq \frac{2n}{d}$, then $f$ is identically equal to $0$. In this paper, we investigate an analogous problem for functions $f: {\mathbb Z}_N^d \to {\mathbb C}$. Bourgain's celebrated result on $Λ_p$ sets (\cite{Bou89}), random constructions (\cite{Bab89}), and connections with the theory of exact signal recovery (\cite{DS89}, \cite{MS73}, \cite{IKLM24}, \cite{IM24}) play an important role. |
| title | Some Results in Spectral Synthesis Over ${\mathbb Z}_N^d$ |
| topic | Classical Analysis and ODEs Combinatorics 42B |
| url | https://arxiv.org/abs/2508.12194 |