  Generalized prolate spheroidal functions (GPSFs) arise naturally in the study
of bandlimited functions as the eigenfunctions of a certain truncated Fourier
transform. In one dimension, the theory of GPSFs (typically referred to as
prolate spheroidal wave functions) has a long history and is fairly complete.
Furthermore, more recent work has led to the development of numerical
algorithms for their computation and use in applications. In this paper we
consider the more general problem, extending the one dimensional analysis and
algorithms to the case of arbitrary dimension. Specifically, we introduce
algorithms for efficient evaluation of GPSFs and their corresponding
eigenvalues, quadrature rules for bandlimited functions, formulae for
interpolation via GPSF expansion, and various analytical properties of GPSFs.
We illustrate the numerical and analytical results with several numerical
examples.
