  As demonstrated by Slepian et. al. in a sequence of classical papers, prolate
spheroidal wave functions (PSWFs) provide a natural and efficient tool for
computing with bandlimited functions defined on an interval. Recently, PSWFs
have been becoming increasingly popular in various areas in which such
functions occur - this includes physics (e.g. wave phenomena, fluid dynamics),
engineering (signal processing, filter design), etc.
  To use PSWFs as a computational tool, one needs fast and accurate numerical
algorithms for the evaluation of PSWFs and related quantities, as well as for
the construction of corresponding quadrature rules, interpolation formulas,
etc. During the last 15 years, substantial progress has been made in the design
of such algorithms. However, many of the existing algorithms tend to be
relatively slow when $c$ is large (e.g. c>10^4).
  In this paper, we describe several numerical algorithms for the evaluation of
PSWFs and related quantities, and design a class of PSWF-based quadratures for
the integration of bandlimited functions. While the analysis is somewhat
involved and will be published separately, the resulting numerical algorithms
are quite simple and efficient in practice. For example, the evaluation of the
$n$th eigenvalue of the prolate integral operator requires $O(n+c \cdot \log
c)$ operations; the construction of accurate quadrature rules for the
integration (and associated interpolation) of bandlimited functions with band
limit $c$ requires $O(c)$ operations. All algorithms described in this paper
produce results essentially to machine precision. Our results are illustrated
via several numerical experiments.
