On an optimal quadrature formula for approximation of Fourier integrals in the space $W_2^{(1,0)}$

TL;DR

This paper develops an optimal quadrature formula for Fourier integrals in a specific Hilbert space, minimizing the error norm and providing explicit formulas for coefficients, with analysis of convergence order.

Contribution

It introduces a new optimal quadrature formula for Fourier integrals in $W_2^{(1,0)}$, with explicit coefficient formulas and convergence analysis.

Findings

Derived explicit formulas for optimal coefficients.

Established the order of convergence of the formula.

Validated the effectiveness of the quadrature in approximating Fourier integrals.

Abstract

The present paper is devoted to construction of an optimal quadrature formula for approximation of Fourier integrals in the Hilbert space $W_2^{(1,0)}[a,b]$ of non-periodic, complex valued functions. Here the quadrature sum consists of linear combination of the given function values on uniform grid. The difference between integral and quadrature sum is estimated by the norm of the error functional. The optimal quadrature formula is obtained by minimizing the norm of the error functional with respect to coefficients. In addition, analytic formulas for optimal coefficients are obtained using the discrete analogue of the differential operator $d^2/d x^2-1$. Further, the order of convergence of the optimal quadrature formula is studied.