Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Dimensions; Generalized Prolate Spheroidal Functions

In two earlier papers 1,2 in this series, the extent to which a square-integrable function and its Fourier transform can be simultaneously concentrated was considered in detail. In that analysis, the eigenfunctions and eigenvalues of the finite Fourier transform played a key role. These functions, defined for | x | ^ 1 by the integral equation «,*,(*) = £ eicx"My)dy, (1)