The restricted discrete Fourier transform
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909312532611072 |
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| author | Casper, W. Riley Yakimov, Milen |
| author_facet | Casper, W. Riley Yakimov, Milen |
| contents | We investigate the restriction of the discrete Fourier transform $F_N : L^2(\mathbb{Z}/N \mathbb{Z}) \to L^2(\mathbb{Z}/N \mathbb{Z})$ to the space $\mathcal C_a$ of functions with support on the discrete interval $[-a,a]$, whose transforms are supported inside the same interval. A periodically tridiagonal matrix $J$ on $L^2(\mathbb{Z}/N \mathbb{Z})$ is constructed having the three properties that it commutes with $F_N$, has eigenspaces of dimensions 1 and 2 only, and the span of its eigenspaces of dimension 1 is precisely $\mathcal C_a$. The simple eigenspaces of $J$ provide an orthonormal eigenbasis of the restriction of $F_N$ to $\mathcal C_a$. The dimension 2 eigenspaces of $J$ have canonical basis elements supported on $[-a,a]$ and its complement. These bases give an interpolation formula for reconstructing $f(x)\in L^2(\mathbb{Z}/N\mathbb{Z})$ from the values of $f(x)$ and $\widehat f(x)$ on $[-a,a]$, i.e., an explicit Fourier uniqueness pair interpolation formula. The coefficients of the interpolation formula are expressed in terms of theta functions. Lastly, we construct an explicit basis of $\mathcal C_a$ having extremal support and leverage it to obtain explicit formulas for eigenfunctions of $F_N$ in $C_a$ when $\dim \mathcal C_a \leq 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_20379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The restricted discrete Fourier transform Casper, W. Riley Yakimov, Milen Spectral Theory Classical Analysis and ODEs 42A16, 47B36 We investigate the restriction of the discrete Fourier transform $F_N : L^2(\mathbb{Z}/N \mathbb{Z}) \to L^2(\mathbb{Z}/N \mathbb{Z})$ to the space $\mathcal C_a$ of functions with support on the discrete interval $[-a,a]$, whose transforms are supported inside the same interval. A periodically tridiagonal matrix $J$ on $L^2(\mathbb{Z}/N \mathbb{Z})$ is constructed having the three properties that it commutes with $F_N$, has eigenspaces of dimensions 1 and 2 only, and the span of its eigenspaces of dimension 1 is precisely $\mathcal C_a$. The simple eigenspaces of $J$ provide an orthonormal eigenbasis of the restriction of $F_N$ to $\mathcal C_a$. The dimension 2 eigenspaces of $J$ have canonical basis elements supported on $[-a,a]$ and its complement. These bases give an interpolation formula for reconstructing $f(x)\in L^2(\mathbb{Z}/N\mathbb{Z})$ from the values of $f(x)$ and $\widehat f(x)$ on $[-a,a]$, i.e., an explicit Fourier uniqueness pair interpolation formula. The coefficients of the interpolation formula are expressed in terms of theta functions. Lastly, we construct an explicit basis of $\mathcal C_a$ having extremal support and leverage it to obtain explicit formulas for eigenfunctions of $F_N$ in $C_a$ when $\dim \mathcal C_a \leq 4$. |
| title | The restricted discrete Fourier transform |
| topic | Spectral Theory Classical Analysis and ODEs 42A16, 47B36 |
| url | https://arxiv.org/abs/2407.20379 |