In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha >-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both an integral operator, and a Sturm–Liouville differential operator. Different from existing works on multi-dimensional PSWFs, the ball PSWFs are defined as a generalization of orthogonal ball polynomials in primitive variables with a tuning parameter $c>0$, through a “perturbation” of the Sturm–Liouville equation of the ball polynomials. From this perspective, we can explore some interesting intrinsic connections between the ball PSWFs and the finite Fourier and Hankel transforms. We provide an efficient and accurate algorithm for computing the ball PSWFs and the associated eigenvalues, and present various numerical results to illustrate the efficiency of the method. Under this uniform framework, we can recover the existing PSWFs by suitable variable substitutions.

在本文中，我们介绍了实阶的扁长球面波函数（PSWF） $\alpha >-\mathrm{1个}$在任意尺寸的单位球上，称为球PSWF。它们既是积分算子又是Sturm-Liouville微分算子的本征函数。与关于多维PSWF的现有工作不同，球形PSWF被定义为具有调整参数的原始变量中正交球形多项式的推广$C>0$，通过球多项式的Sturm-Liouville方程的“摄动”。从这个角度来看，我们可以探索球形PSWF与有限傅里叶和汉克尔变换之间的一些有趣的内在联系。我们提供了一种有效，准确的算法来计算球PSWF和相关的特征值，并提供了各种数值结果来说明该方法的有效性。在此统一框架下，我们可以通过适当的变量替换来恢复现有的PSWF。