On Generalized Gaussian Quadratures for Exponentials and Their Applications

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 , selecting an eigenvalue close to 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.  2002 Elsevier Science (USA)

Wave propagation using bases for bandlimited functions

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

“…The nodes and weights in Proposition 2 are computed as a function of the bandlimit c > 0 and the accuracy > 0 and can be viewed as the generalized Gaussian quadratures for the bandlimited functions. We note that the algorithm in [7] identifies the nodes of the generalized Gaussian quadratures as zeros of the discrete prolate spheroidal wave functions (DPSWF) corresponding to small eigenvalues. For a study of DPSWFs we refer to [5].…”

“…Although we compute the nodes and weights as in [7] by selecting first the bandlimit, c, and then computing the minimal (or nearly minimal) number of nodes, M, to achieve a given accuracy , once such quadratures are generated we use the number of nodes as the variable and c = c(M, ) to study node accumulation.…”

“…Using the method in [7], we have computed the generalized Gaussian quadratures for different accuracies and observed the rate r(M, ) at which nodes accumulate near the end points. We illustrate our results for two choices of accuracy, ≈ 10 −7 and ≈ 10 −17 .…”

“…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.…”