TO THE THEORY OF HERMITE – BIRKHOFF INTERPOLATION OF NONLINEAR ORDINARY DIFFERENTIAL OPERATORS

This article is devoted to the problem of construction and research of generalized interpolation formulas of Hermite – Birkhoff type for operators given on the function spaces. The construction of operator interpolation formulas is based on interpolation polynomials for scalar functions with respect to the arbitrary Chebyshev system of functions. The given formulas contain the Stieltjes integrals and the Gateaux differentials of an interpolated operator and are invariant for the special-class operator polynomials. An explicit representation of the error of the generalized Hermite – Birkhoff operator interpolation is obtained. On the basis of the generalized interpolation Hermite – Birkhoff formulas the interpolation polynomials for ordinary differential operators of arbitrary order given in the space of continuously differentiable functions are constructed. Some special cases of the Hermite – Birkhoff formulas of this type for various Chebyshev systems of scalar functions are also considered. The obtained results can be used in theoretical research as a basis for constructing approximate methods of solving some nonlinear operator equations that occur in nonlinear dynamics, mathematical physics.

operator polynomial, operator interpolation, generalized Hermite – Birkhoff-type interpolation, differential operator, differential of Gateaux, integral of Stieltjes, interpolation error

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