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.