Minimization of the norm of the error functional of one quadrature formula

The problem of constructing optimal quadrature and interpolation formulas for approximate calculation of definite integrals and approximation of functions is one of the important problems of computational mathematics. These problems have been studied by many mathematicians and there are several methods for constructing optimal quadrature and interpolation formulas. One of these methods is the method proposed by S. L. Sobolev with the concept of an extremal function of the error functional. In this paper, in a specific Hilbert space, finding an extremal function of the error functional of quadrature formulas is considered. The square of the norm of the error functional is calculated. To find the optimal coefficients, a system of linear algebraic equations is obtained. The existence and uniqueness of the resulting system are investigated.