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Prolate spheroidal wave functions (PSWFs) play an important role in variousareas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g.signal processing, filter design). One of the principal reasons for theimportance of PSWFs is that they are a natural and efficient tool for computingwith bandlimited functions, that frequently occur in the abovementioned areas.This is due to the fact that PSWFs are the eigenfunctions of the integraloperator, that represents timelimiting followed by lowpassing.  Needless to say, the behavior of this operator is governed by the decay rateof its eigenvalues. Therefore, investigation of this decay rate plays a crucialrole in the related theory and applications - for example, in construction ofquadratures, interpolation, filter design, etc.  The significance of PSWFs and, in particular, of the decay rate of theeigenvalues of the associated integral operator, was realized at least half acentury ago. Nevertheless, perhaps surprisingly, despite vast numericalexperience and existence of several asymptotic expansions, a non-trivialexplicit upper bound on the magnitude of the eigenvalues has been missing fordecades.  The principal goal of this paper is to close this gap in the theory of PSWFs.We analyze the integral operator associated with PSWFs, to derive fairly tightnon-asymptotic upper bounds on the magnitude of its eigenvalues. Our resultsare illustrated via several numerical experiments.

Certain upper bounds on the eigenvalues associated with prolate spheroidal wave functions