In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.

第一类零阶角球面波函数和非负整数特征指数除了是受限傅里叶算子的本征函数外，也是奇异自伴Sturm-Liouville问题的解。用于 Sturm-Liouville 特征值数值评估的标准算法的运行时间随着带限和特征指数的增加而增长。在这里，我们描述了一种新方法，其运行时间不受这些参数的限制。尽管 Sturm-Liouville 特征值本身没有什么意义，但我们的算法是其中一位作者开发的用于对长椭球波函数进行数值评估的快速方案的一个组成部分。我们通过数值实验说明了我们方法的性能。