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.
We give a theorem of Vijayaraghavan type for summability methods for double sequences, which allows a conclusion from boundedness
in a mean and a one-sided Tauberian condition to the boundedness of the sequence itself. We apply the result to certain power
series methods for double sequences improving a recent Tauberian result by S. Baron and the author .
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.
Nathanson, M. B. and Tenenbaum, G., Inversetheorems and the number of sums and products, in: Structure theory of set addition, Astérisque 258 (Deshouillers et al., ed.), SMF (Paris, 1999), pp. 195-204. MR 2000h :11110