Prolate Spheroidal Wave Functions, Quadrature, and Interpolation.

摘要

Polynomials are one of 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 treatment appropriate to such a polynomial is applied. We introduce analogous techniques based on the assumption that the function to be dealt with is band-limited, and use the well-developed apparatus of Prolate Spheroidal Wave Functions to construct quadratures, interpolation and differentiation formulae, etc. for band-limited functions. Since band-limited functions are often encountered in physics, engineering, statistics, etc. the apparatus we introduce appears to be natural in many environments. Our results are illustrated with several numerical examples.