An Optimal Quadrature Formula with Derivative in a Hilbert Space

Abstract

This article presents the derivation and analysis of a weighted optimal quadrature formula in the
Hilbert space
(2,1)
2 W (0,1)
. The quadrature formula is expressed as a linear combination of function values
and their first-order derivatives at equidistant nodes in the interval
[0,1]
.The coefficients are determined
by minimizing the norm of the error functional in the dual space
(2,1)*
2 W (0,1)
, which measures the
difference between the integral of a function over the interval and the quadrature approximation. Key
results include explicit expressions for the coefficients and the norm of the error functional