The restricted discrete Fourier transform

Abstract

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. The collections of simple eigenvalues of $J$ are proved to be strictly greater than the double eigenvalues. 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 $\mathcal C_a$ when $\dim \mathcal C_a \leq 4$.