A formal definition of ǫ-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov’s idea of the complexity of an object. A definition of ǫ-complexity for a class of continuous functions with a given modulus of continuity is also proposed. Additionally, an explicit formula for the ǫ-complexity of a functional class is o...

Introduction. The object of this note is to present a short and simple proof of the following two theorems. Theorem 1. Suppose that C is a closed rectifiable Jordan curve in the complex l-plane and that C has a continuously turning tangent in a neighborhood of one of its points, XoSuppose, furthermore, that the tangent angle t(s) as function of the arc length s has a modulus of continuity (¡)(t...

We estimate the constants sup x∈(0,1) sup f∈C[0,1]\Π1 |Bn(f,x)−f(x)| ω2 f, x(1−x) n and inf x∈(0,1) sup f∈C[0,1]\Π1 |Bn(f,x)−f(x)| ω2 f, x(1−x) n , where Bn is the Bernstein operator of degree n and ω2 is the second order modulus of continuity. 2000 Mathematical Subject Classification: 41A36, 41A10, 41A25,

In this paper we introduce Stancu type generalization of Gamma operators based on the concept of q -integers. We first establish local approximation theorems for these operators. Next, we investigate the weighted approximation properties and give an estimate for the rate of convergence using classical modulus of continuity. Lastly, we obtain a Voronovskaya type theorem. Mathematics subject clas...

We give estimates with explicit constants of the degree of approximation by general positive linear operators on the interval [0,∞), using a weighted modulus of continuity. In particular we obtain a quantitative version of a result of Totik concerning Szász-Mirakjan operators. Mathematics Subject Classification (2010): 41A36.

In this present paper, we study an approximation properties of modified Baskakov-Gamma operator. Using Korovkin type theorem first give Secondly, compute the rate convergence operator by means modulus continuity and weighted spaces. Finally, Voronovskaya

The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with...