Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.
The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.
We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia .
Authors:Nazim Mahmudov, Mehmet Özarslan, and Pembe Sabancigil
In this paper we study I-approximation properties of certain class of linear positive operators. The two main tools used in this paper are I-convergence and Ditzian-Totik modulus of smoothness. Furthermore, we define q-Lupaş-Durrmeyer operators and give local and global approximation results for such operators.
Authors:M. Potapov, F. Berisha, М. Потапов, and Ф. Берища
An asymmetric operator of generalized translation is introduced in this paper. Using this operator, we define a generalized
modulus of smoothness and prove direct and inverse theorems of approximation theory for it.