Error Estimates for Gauss--Christoffel Quadrature under Reduced Regularity Conditions

TL;DR

This paper derives new error bounds for Gauss--Christoffel quadrature under weaker regularity assumptions, expanding applicability to functions with limited endpoint regularity by using a novel derivative identity.

Contribution

It introduces a new identity for Chebyshev polynomial derivatives and establishes less restrictive error estimates for Gauss--Christoffel quadrature under reduced regularity conditions.

Findings

New error bounds under weakened regularity assumptions

Improved decay estimates for Chebyshev coefficients

Extension to Gauss--Gegenbauer quadrature

Abstract

Gauss--Christoffel quadrature is a fundamental method for numerical integration, and its convergence analysis is closely related to the decay of Chebyshev expansion coefficients. Classical estimates, including those due to Trefethen, are based on weighted bounded variation assumptions involving the singular weight $(1-x^{2})^{-1/2}$, which may be too restrictive for functions with limited regularity at the endpoints. In this paper, we establish a new error bound for Gauss--Christoffel quadrature under weakened regularity assumptions. The analysis relies on a new identity for higher-order derivatives of Chebyshev polynomials. As a consequence, we obtain an improved decay estimate for Chebyshev coefficients, where the classical weighted condition \[ V_{r}=\int_{-1}^{1}\frac{|f^{(r+1)}(x)|}{\sqrt{1-x^{2}}}\,dx \] is replaced by the weaker condition \[ U_{r}=\int_{-1}^{1}|f^{(r+1)}(x)|\,dx. \] This result leads to a corresponding error estimate for the Gauss--Christoffel quadrature rule, which is less restrictive than previous bounds. The approach is also extended to the Gauss--Gegenbauer case. Numerical experiments are provided to illustrate the theoretical results.