There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

On (p, q)-analogue of Bernstein operators

In this paper, we introduce a new analogue of Bernstein operators and we call it as (p, q)-Bernstein operators which is a generalization of q-Bernstein operators. We also study approximation properties based on Korovkin's type approximation theorem of (p, q)-Bernstein operators and establish some direct theorems. Furthermore, we show comparisons and some illustrative graphics for the convergence of operators to a function.

On (p,q)-analogue of Bernstein Operators

Some approximation results by (p, q)-analogue of Bernstein-Stancu operators

In this paper, we have given a corrigendum to our paper "Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators" published in Applied Mathematics and Computation $264 (2015) 392-402.$ We introduce a new analogue of Bernstein-Stancu operators and we call it as $(p,q)$-Bernstein-Stancu operators. We study approximation properties based on Korovkin's type approximation theorem of $(p,q)$-Bernstein-Stancu operators. We also establish some direct theorems.

Some Approximation Results by $(p,q)$-analogue of Bernstein-Stancu Operators (Revised)

Introduction of q-calculus.- q-Discrete operators and their results.- q-Integral operators.- q-Bernstein type integral operators.- q-Summation-integral operators.- Statistical convergence of q-operators.- q-Complex operators.

Applications of q-Calculus in Operator Theory

In the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.

https://www.degruyter.com/downloadpdf/j/ms.2011.61.issue-4/s12175-011-0032-3/s12175-011-0032-3.xml

Generalized q -Baskakov operators

A generalization of Szász-Mirakyan operators based on q-integers

In the present paper, we introduce Kantorovich modifications of (p, q)-Bernstein operators for bivariate functions using a new (p, q)-integral. We first estimate the moments and central moments. We give the uniform convergence of new operators, rate of convergence in terms of modulus of continuity. The approximations behaviours of the operators for functions having continuous partial derivatives and for functions belong to Lipschitz class are investigated as well.

Approximation by Bivariate ( p , q )-Bernstein–Kantorovich Operators

The concern of this paper is to introduce a Kantorovich modification of $( p,q ) $
 -Baskakov operators and investigate their approximation behaviors. We first define a new $( p,q ) $
 -integral and construct the operators. The rate of convergence in terms of modulus of continuities, quantitative and qualitative results in weighted spaces, and finally pointwise convergence of the operators for the functions belonging to the Lipschitz class are discussed.

/pdf/on-kantorovich-modification-of-p-q-baskakov-operators-4wy1aj60up.pdf

Ali Aral

Papers

Applications of q-Calculus in Operator Theory

Generalized q -Baskakov operators

A generalization of Szász-Mirakyan operators based on q-integers

Approximation by Bivariate ( p , q )-Bernstein–Kantorovich Operators

On Kantorovich modification of (p,q)-Baskakov operators