The Square of The Norm of The Error Functional Concerning One Quadrature Formula in Sobolev Space

Abstract

Quadrature formulas are fundamental tools in numerical integration, used to approximate integrals with high accuracy. Their efficiency depends on minimizing the error functional, which is critical in functional analysis and numerical approximation theory. The study focuses on Sobolev spaces, which provide a rigorous framework for analyzing quadrature formulas. In these spaces, error functionals play a crucial role in assessing the accuracy of numerical integration methods. Prior research has explored extremal functions and optimal quadrature formulas, but precise error norm calculations remain an ongoing challenge. Although optimal quadrature formulas have been studied extensively, the explicit calculation of the square of the norm of the error functional in Sobolev spaces requires further exploration. A deeper understanding of extremal functions and their impact on error minimization is necessary to advance numerical integration techniques. This research aims to determine the square of the norm of the error functional for a given quadrature formula in Sobolev spaces, utilizing extremal function analysis and functional optimization methods. The study derives an explicit formula for the norm of the error functional, proving its dependence on the coefficients of the quadrature formula. Using Riesz representation and Green’s function techniques, an extremal function corresponding to the error functional is obtained, leading to a rigorous calculation of the error norm. The research presents a precise computation of the error functional norm, contributing to the optimization of quadrature formulas in Sobolev spaces. The findings enhance the theoretical understanding of numerical integration and provide a foundation for developing more accurate computational methods. The results are significant for improving numerical integration techniques used in applied mathematics, physics, and engineering. The optimized quadrature formulas can enhance computational efficiency in solving integral equations, reducing numerical errors in scientific computing applications.