REMARK TO FUNCTION APPROXIMATION BY MODULES OF SMOOTHNESS
Journal Title: Енергетика і автоматика - Year 2018, Vol 0, Issue 6
Abstract
Recently, in the approximation theory, the DT-modules of smoothness, which was introduced by Z. Ditsian and V.Totik are widely used for functions continuous on the segment [-1; 1]. The question of the approximation of functions on complexes of complexes was actual by means of modules analogous to the DT-modules of smoothness. As a result, an analog of the DT-module of smoothness was introduced in the regions of a complex plane with a piecewise smooth boundary and its properties were investigated for the further construction of the constructive characteristic of the uniform approximation of functions in terms of the introduced smoothness module. The definition of the D-module of the order of k smoothness is introduced as a supremum of the difference between the values of a function and the Lagrange polynomial, where the inner supremum is taken over all sets of points that satisfy certain conditions. In the conducted researches methods of uniform approximation of functions were used, in particular interpolation of functions by Lagrange polynomials, separated differences, polynomials of the Jackson and Dzyadyk nuclei, and the identity of Popovichiu. The main result of the research is the property of the normality of the introduced smoothness module, which is formulated as a theorem and proved. The considered modulus of smoothness is planned to be used to construct a constructive characteristic of the uniform approximation of classes of continuous functions on regions of a complex plane with a piecewise smooth boundary.
Authors and Affiliations
О. Ю. ДЮЖЕНКОВА
Keywords
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