The Hermite type optimal quadrature formula on the Sobolev space
S.L. Sobolev considered the problem of constructing optimal lattice formulas over a space and finding the optimal coefficients reduced to the solution of a discrete Wiener-Hopf problem. SL Sobolev, investigating a discrete Wiener-Hopf type problem, proved the existence and uniqueness of the solution of this problem and created an algorithm for finding the optimal coefficients of cubature formulas. In the one-dimensional case, he reduced the finding of the optimal coefficients of quadrature formulas in space to the solution of the system from the equations. An optimal quadrature weight formula is constructed in the space of periodic functions and the norm of the error functional of the constructed quadrature formula in the dual space is calculated, and an extremal function for the given quadrature formula is found. In addition, for the error functional of a quadrature formula of Hermite type for class functions, an upper bound is obtained and the optimal coefficients of a quadrature formula of Hermite type for and are found.