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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Prolate Spheroidal Wave Functions (PSWFs) are a well-studied subject with applications in signal processing, wave propagation, antenna theory, etc. Originally introduced in the context of separation of variables for certain partial differential equations, PSWFs became an important tool for the analysis of band-limited functions after the famous series of articles by Slepian et al. The popularity of PSWFs seems likely to increase in the near future, as band-limited functions become a numerical (as well as an analytical) tool.The classical theory of PSWFs is based primarily on their connection with Legendre polynomials: the coefficients of the Legendre series for a PSWF are the coordinates of an eigenvector of a certain tridiagonal matrix, and the latter becomes diagonally dominant when the order of the function is large compared to the band-limit. This apparatus (historically formulated in terms of three-term recursions, and referred to as the Bouwkamp algorithm) leads to an effective numerical scheme for the evaluation of the PSWFs, and yields a number of analytical properties of PSWFs. When the order of the PSWF is not large compared to the band-limit, the scheme still can be used as a numerical tool (though it becomes less efficient), but does not supply very much analytical information.In this article, we observe that the coefficients of the Hermite expansion of a PSWF also satisfy a three-term recursion; the latter becomes diagonally dominant when the band-limit is large compared to the order of the function (i.e., in the regime where the classical recursion loses its simplicity), and leads to asymptotic (for large band-limits) expressions for the PSWFs, their corresponding eigenvalues, and a number of related quantities.We present several such asymptotic expressions, and illustrate their behavior with numerical examples.Math Subject Classifications. 33E10, 35C20, 42C10, 35S30.