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：广义长球面函数（GPSFs）在研究带限函数时自然出现，作为某种截断傅里叶变换的特征函数。在一维情形下，GPSF理论（通常称为长球面波函数）历史悠久且较为完善。此外，近期研究已发展出用于其计算和应用的数值算法。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，我们介绍了GPSF及其对应特征值的有效评估算法、带限函数的高斯求积规则、基于GPSF展开的插值公式以及GPSF的若干解析性质。通过多个数值算例展示了数值与解析结果。