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.
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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