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 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 wavefunctions 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.

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Generalized Gaussian quadratures appear to have been introduced by Markov [11,12] late in the last century, and have been studied in great detail as a part of modern analysis (see [2,8,9]). They have not been widely used as a computational tool, in part due to absence of effective numerical schemes for their construction. Recently, a numerical scheme was introduced for the design of such quadratures (see [10]); numerical results presented in [10] indicate that such quadratures dramatically reduce the computational cost of the evaluation of integrals under certain conditions. In this paper, we modify the approach of [10], improving the stability of the scheme and extending its range of applicability. The performance of the method is illustrated with several numerical examples.

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