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.

翻译：广义长椭球函数（GPSFs）作为某截断傅里叶变换的特征函数，自然而然地出现在带限函数的研究中。在一维情形下，广义长椭球函数（通常称为长椭球波函数）的理论历史悠久且较为完备。此外，近期研究已推动了其数值算法的发展及其在应用中的使用。本文考虑更一般的问题，将一维分析与算法推广至任意维情形。具体而言，我们引入了用于高效计算GPSFs及其对应特征值的算法、带限函数的求积法则、基于GPSF展开的插值公式以及GPSFs的多种分析性质。我们通过若干数值示例阐述了数值与分析结果。