Optimal quadrature formulas in the Sobolev space of complex-valued functions

Abstract

In this work, the extremal function of the error functional in the Sobolev space of complex-valued functions is derived. Using the obtained extremal function, the squared norm of the error functional of quadrature formulas is computed. By minimizing the squared norm of the error functional with respect to the coefficients of the quadrature formulas, a system is derived for determining the optimal coefficients of the considered quadrature formula in the Sobolev space. In addition, an analogue of I. Babu\v{s}ka's theorem is proved.