Error bounds of a quadrature formula with multiple nodes for the Fourier-Chebyshev coefficients for analytic functions

TL;DR

This paper derives three effective error bounds for a generalized quadrature formula with multiple nodes, applicable to analytic functions within certain regions, and illustrates their calculation with a numerical example.

Contribution

It introduces new error bounds for a generalized quadrature formula extending Micchelli-Rivlin, specifically for analytic functions in confocal ellipse regions.

Findings

Three types of error bounds are established.

Error bounds are applicable to functions analytic in confocal ellipses.

Numerical example demonstrates the calculation of these bounds.

Abstract

Three kinds of effective error bounds of the quadrature formulas with multiple nodes that are generalizations of the well known Micchelli-Rivlin quadrature formula, when the integrand is a function analytic in the regions bounded by confocal ellipses, are given. A numerical example which illustrates the calculation of these error bounds is included.