Weighted Optimal Formulas for Approximate Integration

Abstract

Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of the differential operator 1−12π2d2dx2m in the Hilbert space H2μR, called Dmβ. We modify the Sobolev algorithm to construct optimal quadrature formulas using a discrete operator. We provide a weighted optimal quadrature formula, using this algorithm for the case where m=1. Finally, we construct an optimal quadrature formula in the Hilbert space H2μR for the weight functions px=1 and px=e2πiωx when m=1.