On the computation of the SVD of Fourier submatrices

TL;DR

This paper investigates the singular value decomposition of Fourier submatrices, focusing on the properties of the associated discrete prolate spheroidal sequences and their numerical computation, with applications highlighted.

Contribution

It expands known results on stable numerical computation of SVD for Fourier submatrices and explores the properties of discrete prolate spheroidal sequences.

Findings

Singular values exhibit an initial plateau followed by rapid decay.

The paper provides methods for stable numerical computation of singular vectors.

Applications of Fourier submatrices are discussed.

Abstract

Contiguous submatrices of the Fourier matrix are known to be ill-conditioned. In a recent paper in SIAM Review A. Barnett has provided new bounds on the rate of ill-conditioning of the discrete Fourier submatrices. In this paper we focus on the corresponding singular value decomposition. The singular vectors go by the name of periodic discrete prolate spheroidal sequences (P-DPSS). The singular values exhibit an initial plateau, which depends on the dimensions of the submatrix, after which they decay rapidly. The latter regime is known as the plunge region and it is compatible with the submatrices being ill-conditioned. The discrete prolate sequences have received much less study than their continuous counterparts, prolate spheroidal wave functions, associated with continuous Fourier transforms and widely studied following the work of Slepian in the 1970's. In this paper we collect and expand known results on the stable numerical computation of the singular values and vectors of Fourier submatrices. We illustrate the computations and point out a few applications in which Fourier submatrices arise.