We investigate the Slepian spatiospectral localization problem within subdomains of the $d$-dimensional ball. Opposed to the more classical setups of the Euclidean space or the sphere, the ball lacks a standard or universally accepted definition of bandwidth. Here, we consider a Fourier-Jacobi function system, decoupling the spherical and radial contributions via spherical harmonics and Jacobi polynomials. Special cases of this setup are of interest for various inverse problems in geophysics and medical imaging, since they relate to the underlying non-uniqueness, as well as in optics, where they represent the widely used Zernike polynomials. Bandwidth can be prescribed separately for the spherical and the radial contributions, where the particular choice of coupling between the two contributions determines the spectral shape, i.e., the overall notion of bandlimit. Understanding the effects of the spectral shape on the eigenvalue distribution of the Slepian spatiospectral localization problem can provide hints on particularly suitable notions of bandwidth for different applications. We provide rigorous asymptotic results for the spectral shape being defined via the overall polynomial degree as well as for being defined via sequential limits for the spherical and radial contributions. For various other spectral shapes, we provide numerical illustrations of the asymptotic eigenvalue distribution. Furthermore, we demonstrate a direct connection of the spectral shape to common indexing schemes for Zernike polynomials.

翻译：我们研究了$d$维球体子域内的Slepian空谱局域化问题。与欧几里得空间或球面等更经典的设定不同，球体缺乏标准或普遍接受的带宽定义。本文采用傅里叶-雅可比函数系统，通过球谐函数和雅可比多项式解耦球面分量与径向分量。该框架的特殊情形对地球物理学和医学成像中的多种反问题具有重要意义，因其与内在的非唯一性相关；在光学领域则对应广泛使用的泽尼克多项式。带宽可分别针对球面分量和径向分量进行规定，而两个分量间耦合关系的特定选择决定了频谱形状，即带限的整体定义。理解频谱形状对Slepian空谱局域化问题特征值分布的影响，可为不同应用场景中特别适用的带宽定义提供启示。我们针对通过总多项式次数定义的频谱形状，以及通过球面与径向分量序列极限定义的频谱形状，给出了严格的渐近结果。对于其他多种频谱形状，我们提供了特征值渐近分布的数值图示。此外，我们论证了频谱形状与泽尼克多项式常见索引方案之间的直接关联。