  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.
