Clifford Prolate Spheroidal Wavefunctions and Associated Shift Frames

Prolate spheroidal wave functions have long been used in mathematical physics as a basis in which to expand solutions of the Helmholtz equation in prolate spheroidal coordinates. They are simultaneous eigenfunctions of a Sturm-Liouville differential operator and a truncation of the Fourier transform to an interval. In 1961, Slepian and Pollak exploited the connection to the truncated Fourier transform to show that the prolates provide the solution to the spectral concentration problem. Shortly after, Slepian constructed eigenfunctions of a higher-dimensional truncation of the Fourier transform (in which the truncations are performed with respect to balls in n-dimensional Euclidean space) by a separation of variables approach. The radial parts of these solutions were shown to be eigenfunctions of a Sturm-Liouville differential operator with a singularity at the origin. In this chapter we first review a Clifford analysis-based approach to the construction of higher-dimensional prolates associated with the ball-truncated Fourier transform. A non-singular Clifford differential operator acting on multidimensional Clifford-valued functions is shown to commute with the ball-truncated Fourier transform, and the associated eigenfunctions (Clifford prolate spheroidal wave functions, or CPSWFs) are constructed numerically. Properties of these functions and their associated eigenvalues are explored. The advantages of this approach are that the singular Sturm-Liouville operator plays no role, replaced by a better-behaved operator, and the CPSWFs take values in a multi-channel algebra, allowing for analysis of vector-valued signals such as those which arise in electromagnetic theory. Finally, we apply this theory to the construction of frames for the space of Clifford-valued bandlimited functions generated from the translates of a finite collection of CPSWFs. This generalizes the one-dimensional scalar-valued construction of Hogan and Lakey.