INTERPOLATION HERMITE – BIRKHOFF-TYPE FORMULAS WITH RESPECT TO THE ALGEBRAIC AND TRIGONOMETRIC SYSTEMS OF FUNCTIONS WITH ONE SPECIAL NODE

This article is devoted to the problem of construction and research of the generalized interpolation Hermite –Birkhofftype formulas. For the scalar argument functions, the algebraic and trigonometric interpolation Hermite – Birkhoff-type polynomials, containing the value of the differential operator of special form at one of the nodes, are constructed. In the both cases, the differential operator of special form annuls the first basic functions of the corresponding Chebyshev system. Furthermore, the order of the differential operator does not depend on the number of nodes. For interpolation polynomials, the satisfaction theorems of interpolation conditions are proved. The classes of the polynomials, for which the interpolation formulas are exact, are determined. The trigonometric analogue of the Leibniz formula is constructed. This formula is used to prove the satisfaction theorem of interpolation conditions in the trigonometric case. The represenations and estimates of the interpolation error are obtained. In algebraic case, to obtain the representations and estimates of interpolation error, the consequence of Rolle’s theorem is used. In the trigonometric case, the integral representation of the interpolation error is utilized. The illustrative example of application of the trigonometric interpolation formula is constructed. The results can be used in the theoretical research as a basis for constructing both approximation methods of linear operators and approximate methods of solving some nonlinear operator equations that are available in nonlinear dynamics, mathematical physics.

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