Generalized and Fractional Prolate Spheroidal Wave Functions

An important problem in communication engineering is the energy concentration problem, that is the problem of finding a signal bandlimited to [−σ, σ] with maximum energy concentration in the interval [−τ, τ ], 0 < τ, in the time domain, or equivalently, finding a signal that is time limited to the interval [−τ, τ ] with maximum energy concentration in [−σ, σ] in the frequency domain. This problem was solved by a group of mathematicians at Bell Labs in the early 1960’s. The solution involves the prolate spheroidal wave functions which are eigenfunctions of a differential and an integral equations. The main goal of this talk is to present a solution to the energy concentration problem in a Hilbert space of functions. This solution will contain as a special case the solution to the energy concentration problem in both the fractional Fourier transform and the linear canonical transform domains. The solution involves a generalization of the prolate spheroidal wave functions, which when restricted to the fractional Fourier transform domain, we may call fractional prolate spheroidal wave functions.