Regular Article
On Generalized Gaussian Quadratures for Exponentials and Their Applications

https://doi.org/10.1006/acha.2002.0380Get rights and content
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Abstract

We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions.

We use a generalization of a representation theorem due to Carathéodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ϵ, selecting an eigenvalue close to ϵ yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm.

These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval.

Cited by (0)

Communicated by Vladimir, Rokhlin

1

Research supported in part by DARPA Grants F49620-98-1-0491 and F30602-98-1-0154 and University of Virginia subcontract MDA972-00-1-0016.

2

Research supported in part by DARPA Grant F30602-98-1-0154.