On an optimal quadrature formula in Sobolev space $L_2^{(m)} (0,1)$

TL;DR

This paper develops optimal quadrature formulas in Sobolev space $L_2^{(m)}(0,1)$, incorporating function values and derivatives at endpoints, and identifies when Euler-Maclaurin formulas are optimal.

Contribution

It constructs and analyzes optimal quadrature formulas with derivative information in Sobolev spaces, providing explicit coefficients and error norms for any fixed number of nodes.

Findings

Optimal coefficients derived for arbitrary N and m≥2.

Euler-Maclaurin formula is optimal for m=2 and m=3.

Error norms explicitly calculated for the quadrature formulas.

Abstract

In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end points of integration interval. The optimal coefficients are found and norm of the error functional is calculated for arbitrary fixed $N$ and for any $m\geq 2$. It is shown that when $m=2$ and $m=3$ the Euler-Maclaurin quadrature formula is optimal.