Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ Space

TL;DR

This paper derives explicit optimal quadrature formulas with positive coefficients in Sobolev space, providing error norms and comparing them with existing formulas to improve numerical integration accuracy.

Contribution

The paper introduces explicit forms of optimal quadrature coefficients with positive weights in Sobolev space and analyzes their error norms, enhancing numerical integration methods.

Findings

Explicit optimal coefficients are derived for quadrature formulas.

Error norms of the formulas are calculated.

Optimal formulas with positive coefficients are compared with existing methods.

Abstract

In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing parameter $\eta_0$ in (1.5) the optimal quadrature formulas with positive coefficients are obtained and compared with well known optimal formulas.