Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

We study the problem of finding unitary submatrices of the
$N \times N$
discrete Fourier transform matrix, in the context of interpolating a discrete bandlimited signal using an orthogonal basis. This problem is related to a diverse set of questions on idempotents on
$\mathbb {Z}_{N}$
and tiling
$\mathbb {Z}_{N}$
. In this work, we establish a graph-theoretic approach and connections to the problem of finding maximum cliques. We identify the key properties of these graphs that make the interpolation problem tractable when
$N$
is a prime power, and we identify the challenges in generalizing to arbitrary
$N$
. Finally, we investigate some connections between graph properties and the spectral-tile direction of the Fuglede conjecture.