Statistical convergence of a non-positive approximation process

Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is inherited by the new sequence. Also, our result includes information about the uniform convergence. Two applications in q-Calculus are presented. We study q-analogues both of Meyer-König and Zeller operators and Stancu operators.     

On two classes of approximation processes of integral type

The paper aims to study two classes of linear positive operators representing modifications of Picard and Gauss operators. The new operators reproduce both constants and a given exponential function. Approximation properties in polynomial weighted spaces are investigated and the speed of convergence is measured using a certain weighted modulus of smoothness. Also, the asymptotic behavior of the integral operators are established. Finally, aspects on generalized convexity are analyzed.

     

Uniform approximation of some classes of linear positive operators expressed by series

The paper deals with a general class of linear positive approximation processes designed using series. For continuous and bounded functions defined on unbounded interval we give rate of convergence in terms of the usual modulus of smoothness. The main goal is to identify functions for which these operators provide uniform approximation over unbounded intervals. Particular cases are delivered.     

Approximation with arbitrary order by certain linear positive operators

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence \((\lambda _n)_{n\ge 1}\) such that \(\lim \limits _{n\rightarrow \infty }\lambda _n=0\), how fast we want. Particular cases are presented.     

Statistical convergence of integral operators generated by a single kernel

We introduce a class of double-complex integral linear operators. Some geometric properties are investigated and a statistical approximation theorem is obtained. In a particular case, our operators turn into the complex Picard operators.     

Asymptotic behaviour of Jain operators

The topic of the present paper are certain approximation operators acting on the space of continous functions on [0,+∞) having polynomial growth. The operators which were defined by Jain in 1972 are based on a probability distribution which is called generalized Poisson distribution. As a main result we derive a complete asymptotic expansion for the sequence of these operators.     

On a Double Complex Sequence of Linear Operators

The main goal of the article is to introduce a class of double complex linear operators of integral type. The technique is based by extension into the complex domain of a real positive approximation process. Involving the first modulus of continuity, we investigate their geometric and approximation properties. The statistical convergence of our sequence is proved. In a particular case, our operators turn into the double complex Gauss-Weierstrass integral operators.     

On statistical approximation in spaces of continuous functions

Our goal is to present approximation theorems for sequences of positive linear operators defined on C(X), where X is a compact metric space. Instead of the uniform convergence we use the statistical convergence. Examples and special cases are also provided.      

Approximation properties of a class of linear operators

This work focuses on a class of linear positive operators of discrete type. We present the relationship between the local smoothness of functions and the local approximation. Also, the degree of approximation in terms of the moduli of smoothness is established, and the statistical convergence of the sequence is studied. Copyright © 2013 John Wiley & Sons, Ltd.