  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 $\dim \mathcal C_a \leq 4$.
