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## Über dieses Buch

The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric series, complex analysis, and particle physics. ​​This monograph is an introduction to combining approximation theory and q-Calculus with applications, by using well- known operators. The presentation is systematic and the authors include a brief summary of the notations and basic definitions of q-calculus before delving into more advanced material. The many applications of q-calculus in the theory of approximation, especially on various operators, which includes convergence of operators to functions in real and complex domain​ forms the gist of the book.

This book is suitable for researchers and students in mathematics, physics and engineering, and for professionals who would enjoy exploring the host of mathematical techniques and ideas that are collected and discussed in the book.

## Inhaltsverzeichnis

### Chapter 1. Introduction of q-Calculus

Abstract
In the field of approximation theory, the applications of q-calculus are new area in last 25 years. The first q-analogue of the well-known Bernstein polynomials was introduced by Lupas in the year 1987. In 1997 Phillips considered another q-analogue of the classical Bernstein polynomials.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 2. q-Discrete Operators and Their Results

Abstract
This chapter deals with the q-analogue of some discrete operators of exponential type. We study some approximation properties of the q-Bernstein polynomials, q-Szász–Mirakyan operators, q-Baskakov operators, and q-Bleimann, Butzer, and Hahn operators. Here, we present moment estimation, convergence behavior, and shape-preserving properties of these discrete operators.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 3. q-Integral Operators

Abstract
For many years scientists have been investigating to develop various aspects of approximation results of above operators. The recent book written by Anastassiou and Gal [18] includes great number of results related to different properties of these type of operators and also includes other references on the subject. For example, in Chapter 16 of [18], Jackson-type generalization of these operators is one among other generalizations, which satisfy the global smoothness preservation property (GSPP). It has been shown in [19] that this type of generalization has a better rate of convergence and provides better estimates with some modulus of smoothness. Beside, in [22, 23], Picard and Gauss–Weierstrass singular integral operators modified by means of nonisotropic distance and their pointwise approximation properties in different normed spaces are analyzed. Furthermore, in [40, 110], Picard and Gauss Weierstrass singular integrals were considered in exponential weighted spaces for functions of one or two variables.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 4. q-Bernstein-Type Integral Operators

Abstract
In order to approximate integrable functions on the interval [0,1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein–Durrmeyer operators.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 5. q-Summation–Integral Operators

Abstract
Aral and Gupta [32], proposed a q-analogue of the Baskakov operators and investigated its approximation properties. In continuation of their work they introduced Durrmeyer-type modification of q-Baskakov operators. These operators, opposed to Bernstein–Durrmeyer operators, are defined to approximate a function f on $$\left [0,\ \infty \right )$$. The Durrmeyer-type modification of the q-Bernstein operators was first introduced in [48].
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 6. Statistical Convergence of q-Operators

Abstract
One of the most recently studied subject in approximation theory is the approximation of function by linear positive operators using A-statistical convergence or a matrix summability method.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Chapter 7. q-Complex Operators

Abstract
In the recent years applications of q-calculus in the area of approximation theory and number theory are an active area of research. Several researchers have proposed the q-analogue of exponential, Kantorovich- and Durrmeyer-type operators. Also Kim [106] and [105] used q-calculus in the area of number theory.
Ali Aral, Vijay Gupta, Ravi P. Agarwal

### Backmatter

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