Discrete Sampling: A graph theoretic approach to Orthogonal Interpolation

TL;DR

This paper introduces a graph-theoretic framework for identifying orthogonal submatrices of the discrete Fourier transform matrix to facilitate bandlimited signal interpolation, especially for prime power sizes.

Contribution

It establishes a novel connection between unitary submatrices, graph theory, and tiling problems, providing new insights for prime power dimensions and highlighting challenges for general N.

Findings

Graph approach links submatrix selection to maximum cliques

Key properties enable tractable interpolation for prime power N

Connections to Fuglede conjecture and spectral tiling

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.