Optimal quadrature formulas for approximate calculation of rapidly oscillating integrals

Abstract

In this paper, we study the problem of constructing optimal formulas for approximate integration in the Sobolev space $\widetilde{L_2^{\left(m\right)}}\left(0,1\right)$ of periodic functions. Using the functional approach, we obtain optimal quadrature formulas for the approximate calculation of rapidly oscillating integrals. Then, we obtain explicit formulas for the coefficients of the optimal quadrature formulas and we get the sharp estimation of the error of the constructed formulas.