ALGEBRAIC AND TRIGONOMETRIC GENERALIZED INTERPOLATION OF HERMITE-BIRKHOFF TYPE FOR OPERATORS DEFINED ON FUNCTIONAL SPACES AND FUNCTIONS OF MATRIX VARIABLE, AND THEIR APPLICATIONS

For operators defined in function spaces, the algebraic interpolation formula of Hermite type is constructed. The interpolation formula of similar type, containing the value of the Gateaux differential of an arbitrary order, is constructed for operators on the set of matrices. Matrix analogues of the Leibniz formula are obtained. The formula for approximate calculation of the Gateaux differential of an arbitrary order of the matrix argument function is constructed. Based on the matrix interpolation formula of the Hermite type, the approximate method for solving the Cauchy problem for the matrix-differential equation is obtained. The illustrative example of approximate solving the Cauchy problem for a first-order matrix-differential equation is constructed. A parametric family of trigonometric matrix interpolation polynomials of Hermite-Birkhoff type is constructed and investigated.