Optimal quadrature formulas with derivatives in Sobolev space

TL;DR

This paper constructs optimal quadrature formulas with derivatives in Sobolev spaces, deriving explicit coefficients and error norms, and demonstrates the optimality of classical formulas for specific cases while introducing new formulas for higher orders.

Contribution

It develops a method to find optimal quadrature formulas involving derivatives in Sobolev spaces, extending classical results and providing explicit formulas for arbitrary N and m ≥ 4.

Findings

Optimal quadrature formulas are derived for arbitrary N and m ≥ 4.

Classical Euler-Maclaurin formula is shown to be optimal for m=4, 5.

New optimal quadrature formulas are introduced for m ≥ 6.

Abstract

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal quadrature formulas are found and the norm of the optimal error functional is calculated for arbitrary natural number $N$ and for any $m\geq 4$ using S.L. Sobolev method which is based on discrete analogue of the differential operator $d^{2m}/dx^{2m}$. In particular, for $m=4,\ 5$ optimality of the classical Euler-Maclaurin quadrature formula is obtained. Starting from $m=6$ new optimal quadrature formulas are obtained.