Prolate spheroidal wavefunctions, quadrature and interpolation

Polynomials are one of the principal tools of classical numerical analysis. When a function needs to be interpolated, integrated, differentiated, etc, it is assumed to be approximated by a polynomial of a certain fixed order (though the polynomial is almost never constructed explicitly), and a treat ment appropriate to such a polynomial is applied. We introduce analogous te chniques based on the assumption that the function to be dealt with is band -limited, and use the well developed apparatus of prolate spheroidal wavefu nctions to construct quadratures, interpolation and differentiation formula e, etc, for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc, the apparatus we intr oduce appears to be natural in many environments. Our results are illustrat ed with several numerical examples.