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

Abstract

In two earlier papers∗in this series, the extent to which a square-integrable function and its Fourier transform can be simultaneously concentrated in their respective domains was considered in detail. The present paper generalizes much of that work to functions of many variables. In treating the case of functions of two variables whose Fourier transforms vanish outside a circle in the two-dimensional frequency plane, we are led to consider the integral equation$\gamma \varphi(x) = \int_{0}^{1} J_{N}(cxy) \sqrt{cxy} \varohi(y) dy. \qquad \eqno{\hbox{(i)}}$It is shown that the solutions are also the bounded eigenfunctions of the differential equation$(1-x^{2}){d^{2}\varphi \over dx^{2}} - 2x {d \varphhi \over dx} +\left(x-c^{2}x^{2} + {{1 \over 4} - N^{2} \over x^{2}}\right) \varphi = 0. \qquad \eqno{\hbox{(ii)}}$a generalization of the equation for the prolate spheroidal wave functions. The functions ϕ (called “generalized prolate spheroidal functions”) and the eigenvalues of both (i) and (ii) are studied in detail here, and both analytic and numerical results are presented. Other results include a general perturbation scheme for differential equations and the reduction to two dimensions of the case of functions of D > 2 variables restricted in frequency to the D sphere.