Discrete Sampling: A Graph Theoretic Approach to Orthogonal Interpolation

Abstract

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.