Generalized prolate spheroidal functions: algorithms and analysis

Abstract: 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.