On quadrature error expansions. Part I

Numerical Quadrature and Asymptotic Expansions

Quadrature Error Functional Expansions for the Simplex When the Integrand Function

In a previous paper by Lyness, an asymptotic expansion for the quadrature error functional for the «-dimensional simplex, valid when the integrand is an analytic function, was given. In this paper we develop the analogue of this result to cover the cases in which the integrand function has singularities of a specified nature at one or more vertices of the simplex.

Sharp Error Bounds for Jacobi Expansions and Gegenbauer-Gauss Quadrature of Analytic Functions

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and val...

Truncation Error on Wavelet Sampling Expansions

(see [9] and [10]). Throughout this work, we assume that the function satis®es the following conditions: (i) j …x†j …cons:† jB…x†j=jxj1‡"; where " 0 and jB…x†j is bounded and 1-periodic function on R. (ii) P n2Z …n† eÿin converges absolutely to a function that has no zeros on ‰ÿ ; Š. It is known that conditions (i) and (ii) imply that fK…x; n†; n 2 Zg is a Riesz basis on V , with a unique biort...

An error analysis of two related quadrature methods for computing zeros of analytic functions, Part II

We consider the quadrature method developed by Kravanja, Sakurai and Van Barel (BIT 39 (1999), no. 4, 646–682) for computing all the zeros of an analytic function that lie inside the unit circle. A new perturbation result for generalized eigenvalue problems allows us to obtain a detailed upper bound for the error between the zeros and their approximations. To the best of our knowledge, it is th...