  Prolate spheroidal wave functions (PSWFs) play an important role in various
areas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g.
signal processing, filter design). One of the principal reasons for the
importance of PSWFs is that they are a natural and efficient tool for computing
with bandlimited functions, that frequently occur in the abovementioned areas.
This is due to the fact that PSWFs are the eigenfunctions of the integral
operator, that represents timelimiting followed by lowpassing.
  Needless to say, the behavior of this operator is governed by the decay rate
of its eigenvalues. Therefore, investigation of this decay rate plays a crucial
role in the related theory and applications - for example, in construction of
quadratures, interpolation, filter design, etc.
  The significance of PSWFs and, in particular, of the decay rate of the
eigenvalues of the associated integral operator, was realized at least half a
century ago. Nevertheless, perhaps surprisingly, despite vast numerical
experience and existence of several asymptotic expansions, a non-trivial
explicit upper bound on the magnitude of the eigenvalues has been missing for
decades.
  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 tight
non-asymptotic upper bounds on the magnitude of its eigenvalues. Our results
are illustrated via several numerical experiments.

Functional Analysis;Classical Analysis and ODEs;Numerical Analysis