Optimal Quadrature Formulas with Derivative for Calculating Integrals of Strongly Oscillating Functions

In this work, in the space of differentiable functions, we consider the construction of weighted optimal quadrature formulas with derivative for the approximate calculation of integrals of fast oscillating functions. Here, using the extremal function, the squared norm of the error functional of the quadrature formula under consideration is calculated. Minimizing this norm by the coefficients of the quadrature formula, a system of the Wiener-Hopf type is obtained. By solving this system using a discrete analogue of an differential operator, the coefficients of optimal quadrature formulas are obtained.