Optimal quadrature formulas of closed type in the space $L_2^{(m)}(0,1)$

TL;DR

This paper develops optimal quadrature formulas with equally spaced nodes in the Sobolev space $L_2^{(m)}(0,1)$, providing explicit coefficient representations for any order and number of nodes.

Contribution

It introduces explicit formulas for optimal quadrature coefficients with equally spaced nodes in $L_2^{(m)}(0,1)$, extending previous results to arbitrary natural numbers $m$ and $N$.

Findings

Explicit formulas for optimal coefficients are derived.

The formulas apply to any natural numbers $m$ and $N$.

The approach optimizes quadrature in the sense of Sard.

Abstract

It is discussed the problem on construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$, when the nodes of quadrature formulas are equally spaced. Here the representations of optimal coefficients for any natural numbers $m$ and $N$ are found.