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The study of linear positive operators is an area of mathematical studies with significant relevance to studies of computer-aided geometric design, numerical analysis, and differential equations. This book focuses on the convergence of linear positive operators in real and complex domains. The theoretical aspects of these operators have been an active area of research over the past few decades. In this volume, authors Gupta and Agarwal explore new and more efficient methods of applying this research to studies in Optimization and Analysis. The text will be of interest to upper-level students seeking an introduction to the field and to researchers developing innovative approaches.



Chapter 1. Preliminaries

Approximation theory has been an established field of mathematics in the past century. This chapter deals with the basic definitions and standard theorems of approximation theory, which are important for the convergence point of view.
Vijay Gupta, Ravi P. Agarwal

Chapter 2. Approximation by Certain Operators

In the theory of approximation following the well-known Weierstrass theorem, the study on direct results was initiated by Jackson’s classical work [160] on algebraic and trigonometric polynomials of best approximation.
Vijay Gupta, Ravi P. Agarwal

Chapter 3. Complete Asymptotic Expansion

Another topic of interest on linear positive operators is the asymptotic expansion. The commendable work on complete asymptotic expansion for different operators was done in the last two decades by Abel and collaborators (see, e.g., [1–3,11–13]). In this chapter, we discuss the asymptotic expansion of some of the operators.
Vijay Gupta, Ravi P. Agarwal

Chapter 4. Linear and Iterative Combinations

The linear positive operators are conceptually simpler, and easier to construct and study, but they lack rapidity of convergence for sufficiently smooth functions.
Vijay Gupta, Ravi P. Agarwal

Chapter 5. Better Approximation

Many well-known approximating operators L n , preserve the constant as well as the linear functions, that is, L n (e 0, x) = e 0(x) and L n (e 1, x) = e 1(x) for e i (x) = x i (i = 0, 1). These conditions hold specifically for the Bernstein polynomials, Szász–Mirakjan operators, Baskakov operators, Phillips operators, genuine Durrmeyer-type operators, and so on.
Vijay Gupta, Ravi P. Agarwal

Chapter 6. Complex Operators in Compact Disks

If \(f: G \rightarrow \mathbb{C}\) is an analytic function in the open set \(G \subset \mathbb{C}\), with \(\overline{D_{1}} \subset G\) (where \(D_{1} =\{ z \in \mathbb{C}: \vert z\vert < 1\}\)), then S. N. Bernstein proved that the complex Bernstein polynomials converge uniformly to f in \(\overline{D_{1}}\) (see, e.g., Lorentz [182], p. 88). The main contributions for the complex operators are due to S. G. Gal; in fact, several important results have been compiled in his 2009 monograph [77]. Since that publication, several important results on Durrmeyer-type operators and other extensions of the known operators have been discussed. In this chapter, we present some of the important results on certain complex operators that were not discussed in [77].
Vijay Gupta, Ravi P. Agarwal

Chapter 7. Rate of Convergence for Functions of Bounded Variation

By Jordan’s theorem, a function is with bounded variation (BV) if and only if it can be represented as the difference of two increasing (decreasing) functions.
Vijay Gupta, Ravi P. Agarwal

Chapter 8. Convergence for Bounded Functions on Bézier Variants

The various Bézier variants (BV) of the approximation operators are important research topics in approximation theory. They have close relationships with geometry modeling and design. Let \(p_{n,k}(x) = \left (\begin{array}{c} n\\ k \end{array} \right ){x}^{k}{(1-x)}^{n-k},(0 \leq k \leq n)\) be Bernstein basis functions. The Bézier basis functions, which were introduced in 1972 by Bézier [39], are defined as \(J_{n,k}(x) =\sum _{ j=k}^{n}p_{n,j}(x)\).
Vijay Gupta, Ravi P. Agarwal

Chapter 9. Some More Results on the Rate of Convergence

Shaw et al. [210] investigated the problem for the general family of positive linear operators, which includes Bernstein, Kantorovich, and Durrmeyer operators as special cases. They investigated their results for the classes of functions BV [a, b] and DBV [a, b]. Also, Hua and Shaw [156] extended this problem for linear integral operators with a not necessarily positive kernel.
Vijay Gupta, Ravi P. Agarwal

Chapter 10. Rate of Convergence in Simultaneous Approximation

In the theory of approximation, the study of the rate of convergence in simultaneous approximation is also an interesting area of research. Several researchers have worked in this direction; some of them have obtained the rate of convergence for bounded/bounded variation functions in simultaneous approximation.
Vijay Gupta, Ravi P. Agarwal

Chapter 11. Future Scope and Open Problems

In 1983, based on two parameters α, β satisfying the conditions 0 ≤ αβ, Stancu [222] proposed a generalization of the classical Bernstein polynomials. In more recent papers, some approximation properties of the Stancu-type generalization on different operators were discussed (see, e.g., [50, 133, 187, 238]). Future studies could address defining the Stancu-type generalization of other operators and the convergence behavior, asymptotic formulas, and rate of convergence for functions of BV and for functions having derivatives of BV.
Vijay Gupta, Ravi P. Agarwal


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