Additive energy, uncertainty principle and signal recovery mechanisms
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| Format: | Preprint |
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2025
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| _version_ | 1866917992482209792 |
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| author | Aldahleh, K. Iosevich, A. Iosevich, J. Jaimangal, J. Mayeli, A. Pack, S. |
| author_facet | Aldahleh, K. Iosevich, A. Iosevich, J. Jaimangal, J. Mayeli, A. Pack, S. |
| contents | Given a signal $f:G\to\mathbb{C}$, where $G$ is a finite abelian group, under what reasonable assumptions can we guarantee the exact recovery of $f$ from a proper subset of its Fourier coefficients? In 1989, Donoho and Stark established a result \cite{DS89} using the classical uncertainty principle, which states that $|\text{supp}(f)|\cdot|\text{supp}(\hat{f})|\geq |G|$ for any nonzero signal $f$. Another result, first proven by Santose and Symes \cite{SS86}, was based on the Logan phenomenon \cite{L65}. In particular, the result showcases how the $L^1$ and $L^2$ minimizing signals with matching Fourier frequencies often recovers the original signal.
The purpose of this paper is to relate these recovery mechanisms to additive energy, a combinatorial measure denoted and defined by $$Λ(A)=\left| \left\{ (x_1, x_2, x_3, x_4) \in A^4 \mid x_1 + x_2 = x_3 + x_4 \right\} \right|,$$ where $A\subset\mathbb{Z}_N^d$. In the first part of this paper, we use combinatorial techniques to establish an improved variety of the uncertainty principle in terms of additive energy. In a similar fashion as the Donoho-Stark argument, we use this principle to establish an often stronger recovery condition. In the latter half of the paper, we invoke these combinatorial methods to demonstrate two $L^p$ minimizing recovery results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive energy, uncertainty principle and signal recovery mechanisms Aldahleh, K. Iosevich, A. Iosevich, J. Jaimangal, J. Mayeli, A. Pack, S. Classical Analysis and ODEs Information Theory 94A12, 42B10 Given a signal $f:G\to\mathbb{C}$, where $G$ is a finite abelian group, under what reasonable assumptions can we guarantee the exact recovery of $f$ from a proper subset of its Fourier coefficients? In 1989, Donoho and Stark established a result \cite{DS89} using the classical uncertainty principle, which states that $|\text{supp}(f)|\cdot|\text{supp}(\hat{f})|\geq |G|$ for any nonzero signal $f$. Another result, first proven by Santose and Symes \cite{SS86}, was based on the Logan phenomenon \cite{L65}. In particular, the result showcases how the $L^1$ and $L^2$ minimizing signals with matching Fourier frequencies often recovers the original signal. The purpose of this paper is to relate these recovery mechanisms to additive energy, a combinatorial measure denoted and defined by $$Λ(A)=\left| \left\{ (x_1, x_2, x_3, x_4) \in A^4 \mid x_1 + x_2 = x_3 + x_4 \right\} \right|,$$ where $A\subset\mathbb{Z}_N^d$. In the first part of this paper, we use combinatorial techniques to establish an improved variety of the uncertainty principle in terms of additive energy. In a similar fashion as the Donoho-Stark argument, we use this principle to establish an often stronger recovery condition. In the latter half of the paper, we invoke these combinatorial methods to demonstrate two $L^p$ minimizing recovery results. |
| title | Additive energy, uncertainty principle and signal recovery mechanisms |
| topic | Classical Analysis and ODEs Information Theory 94A12, 42B10 |
| url | https://arxiv.org/abs/2504.14702 |