Remark to function approximation by modules of smoothness

Abstract

Recently in the theory of approximation modules of smoothness, which were introduced by Z. Ditsian and V.Totik for functions continuous on the interval [-1; 1], are widely used. In works of Dzyadyk V.K, Alibekov G.A., Volkov Yu.I. the constructive characteristic of the uniform approximation of functions on sets of a complex plane with a piecewise smooth boundary considered in terms of the modules of smoothness for the function related with the conformal mapping of the circle appearance on the appearance of the considered set. We have the actual question of the function approximation on sets of a complex plane with modules analogous to the DT-modules of smoothness.

Great attention was paid to the function approximation on sets of complex plan in the works of Dzyadik V.K., Shevchuk I.O., Tamrazov P.M. and others. In the terms of the modules of smoothness, direct and inverse theorems were considered, which resulted in constructive characterization of the uniform approximation of functions on continuums of the complex plane.

The analysis of the theory of smoothness  by Z. Ditsian and V.Totik resulted in its extension to sets of complex planes with a piecewise smooth boundary. As a result, an analog of the DT-modules of smoothness was introduced in regions with angles and its properties were investigated for the further construction of a constructive characteristic of uniform function approximation in terms of the introduced smoothness module.

In the conducted research, methods of uniform function approximation were used, in particular interpolation of functions by Lagrange polynomials, separated differences, polynomial kernels of the Jackson and Dzyadyk, and the identity of Popovichiu.

A closed set is considered, the boundary of which is a Jordan curve and consists of a finite number of smooth curves which form external angles α, 0 <α <2π at the joint points. The function f, which is continuous on the boundary of Γ, is considered, and a Lagrange polynomial is written which interpolates this function in k different points of the complex plane.

The definition of the D-module of the order of k smoothness is introduced as a supremum of the difference between the values of function and the Lagrange polynomial, where the inner supremum is taken over all sets of points that satisfy certain conditions.

For some functions which are continuous on the piecewise smooth curves of the complex plane, some important properties of the D-module of smoothness are investigated. The main result of the study is the property of the normality of the introduced smoothness module, which is formulated as a theorem and proved. Note that this property is similar to the classical modulus of smoothness. In the proof of properties, secondary geometric lemmas are used.

Сonsequently, the definition of the modules of smoothness by Z. Ditsian and V.Totik expands in the article on functions which are continuous on the piecewise smooth curves of the complex plane, and the main properties of the introduced D-module of smoothness are studied. The considered module 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.

Keywords: function approximation, complex plane, piecewise smooth curves, modules of smoothness