The restricted discrete Fourier transform

We investigate the restriction of the discrete Fourier transform FN : L²(Z/N Z) → L²(Z/N Z) to the space Cₐ of functions with support on the discrete interval $[-a,a]$, whose transforms are supported inside the same interval. A periodically tridiagonal matrix J on L²(Z/N Z) is constructed having the three properties that it commutes with FN, has eigenspaces of dimensions 1 and 2 only, and the span of its eigenspaces of dimension 1 is precisely Cₐ. The simple eigenspaces of J provide an orthonormal eigenbasis of the restriction of FN to Cₐ. 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)∈ L²(Z/NZ) from the values of $f(x)$ and 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 Cₐ having extremal support and leverage it to obtain explicit formulas for eigenfunctions of FN in Cₐ when Cₐ ≤ 4.