On the Computation of the SVD of Fourier Submatrices

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 (Barnett in SIAM Rev 64:105–131, 2022). In this paper we focus on the corresponding singular value decomposition. The singular vectors can be computed from the so-called periodic discrete prolate spheroidal sequences , named in analogy to spheroidal wave functions which are associated with the continuous Fourier transform. Their numerical computation is hampered by the clustering of singular values. We collect and expand known results on the stable numerical computation of the singular value decomposition of Fourier submatrices. The prolate sequences are eigenvectors of a tridiagonal matrix whose spectrum is free of clusters and this enables their computation. We collect these observations in a simple and convenient algorithm. The corresponding singular values can be accurately computed as well, except when they are small. Even then, small singular values can be computed in high-precision arithmetic with modest computational effort, even for large and extremely ill-conditioned submatrices. We illustrate the computations and point out a few applications in which Fourier submatrices arise.