Prolate spheroidal wavefunctions, quadrature and interpolation

Abstract

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.

Wave propagation using bases for bandlimited functions

“…[8]). We note that an erroneous comment was made in [6] about the rate of accumulation of nodes, suggesting that (in our terms) r(M,…”

“…Recently, the generalized Gaussian quadratures for bandlimited exponentials were developed in [6,7]. w k e icθ k x <…”

“…The phases θ k are chosen as nodes of the generalized Gaussian quadratures ([7, Theorem 6.1], see also [6]). Following [7], we use the quadrature nodes and weights to construct bases for E c .…”

“…[8]. As it turns out, the nodes of the generalized Gaussian quadratures for exponentials do not concentrate excessively (the rate reported in [6] is in error, see Section 2.2) and the sampling rate asymptotically approaches the rate for periodic functions.…”

“…A basis for bandlimited functions, the prolate spheroidal wave functions (PSWFs), was introduced in the 1960s by Slepian et al in a series of papers [1][2][3][4][5]. Recently the generalized Gaussian quadratures became available in [6,7], making it possible to construct efficient numerical algorithms for such functions.…”