No abstract

A complete set of bandlimited functions is described which possesses the curious property of being orthogonal over a given finite interval as well as over (– ∞, ∞). Properties of the functions are derived and several applications to the representation of signals are made.

A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum concentrated on a subinterval of the fundamental period of the spectrum. Key to the analysis are certain sequences, called discrete prolate spheroidal sequences, and certain functions of frequency called discrete prolate spheroidal functions. Their mathematical properties are investigated in great detail, and many applications to signal analysis are pointed out.

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 It is shown that the solutions are also the bounded eigenfunctions of the differential equation 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.

Investigation of the problem of simultaneously concentrating a function and its Fourier transform has led to some interesting special functions that have widespread applications in engineering. They provide a means of proving a rigorous version of an engineering folk theorem called the 2WT-Theorem. Many generalizations of these ideas seem to possess a similar elegant mathematical structure. A brief descriptive review is given of these developments.