Estimating the Norm of the Error Functional in Sobolev Space of Periodic Functions

This article is devoted to obtaining an upper estimate of the error of an optimal quadrature formula for approximating the integral of periodic functions in the Sobolev space \(\widetilde{W_{2}^{(2,1,0)}}(0,1]\). In the quadrature formulas, a complex exponential weight function of the form \({{e}^{2\pi i\omega x}}\) is used. To minimize the norm of the error functional of the quadrature formula, a corresponding extremal function is found, and using it, an expression for the norm of the error functional is derived. The optimal coefficients that give the smallest value to this norm are obtained. Using Fourier analysis and the extremal function method, explicit formulas for the optimal coefficients are derived. These results extend the classical theory of quadrature formulas to the exponential-weight and oscillatory cases, providing efficient schemes for the numerical integration of periodic functions.