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This book presents a systematic overview of approximation by linear combinations of positive linear operators, a useful tool used to increase the order of approximation. Fundamental and recent results from the past decade are described with their corresponding proofs. The volume consists of eight chapters that provide detailed insight into the representation of monomials of the operators Ln , direct and inverse estimates for a broad class of positive linear operators, and case studies involving finite and unbounded intervals of real and complex functions. Strong converse inequalities of Type A in terminology of Ditzian–Ivanov for linear combinations of Bernstein and Bernstein–Kantorovich operators and various Voronovskaja-type estimates for some linear combinations are analyzed and explained. Graduate students and researchers in approximation theory will find the list of open problems in approximation of linear combinations useful. The book serves as a reference for graduate and postgraduate courses as well as a basis for future study and development.



Chapter 1. Moments and Combinations of Positive Linear Operators

The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the important areas of researchers related to approximation theory. Apart from the earlier known examples several new sequences of p.l.o. were introduced and their approximation properties have been discussed in the last few decades. There are several books in approximation theory, which deal with the linear and nonlinear operators of different kind. We mention here some of the books available in the related area, which are due to DeVore [42], DeVore–Lorentz [43], Ditzian–Totik [50] and Pǎltǎnea [153].

Vijay Gupta, Gancho Tachev

Chapter 2. Direct Estimates for Approximation by Linear Combinations

The aim of this chapter is to collect the known results for the error of approximation by linear combinations L n, r , measured in different norms L p (B) and usually in terms of Ditzian–Totik moduli of smoothness ω φ r(f, t) p , or the ordinary moduli of continuity ωr(f, t) p . We will see the importance of the information about central moments of the p.l.o. (respectively of L n, r ).

Vijay Gupta, Gancho Tachev

Chapter 3. Inverse Estimates and Saturation Results for Linear Combinations

Let the Ditzian–Totik moduli of smoothness ω φ r(f, t) p be given as (2.1.2) and the equivalent K-functional K r, φ (f, tr) p be given as (2.1.4). As in Section 2.1, we suppose the same definitions for the weight function φ(x) and for the domain D of the operators Bn,Sn,Vn,B̂n,Ŝn,V̂n. $$B_{n},S_{n},V _{n},\widehat{B}_{n},\widehat{S}_{n},\widehat{V }_{n}.$$ To establish the inverse results for approximation by L n, r we need two Bernstein type inequalities and the Berens–Lorentz lemmaBerens–Lorentz lemma, which results we formulate as follows:

Vijay Gupta, Gancho Tachev

Chapter 4. Voronovskaja-Type Estimates

The subject of this paragraph is recent development in quantitative estimates of Voronovskaja type. Voronovskaja type In 1932 Elena Voronovskaja in [193] proved the following famous result:

Vijay Gupta, Gancho Tachev

Chapter 5. Pointwise Estimates for Linear Combinations

It was pointed out by Feilong and Zongben [56] that the Baskokov operators with the weight function φ2(x) = x(1 + x) are non-concave on [0, ∞). By using the Ditzian–Totik modulus Ditzian–Totik modulusωφλr(f,t) $$\omega _{\varphi ^{\lambda }}^{r}(f,t)$$ of r-th order with r∈ℕ,0≤λ<1 $$r \in \mathbb{N},0 \leq \lambda <1$$ they established the following four main results for the classical Baskakov operators.

Vijay Gupta, Gancho Tachev

Chapter 6. Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Let E be a subspace of C[0, ∞) which contains the polynomials and L n : E → C[0, ∞) be a sequence of linear positive operators. The weighted modulus of continuity, considered by Acar–Aral–Rasa in [7] is denoted by Ω(f;δ) $$\Omega (f;\delta )$$ and given by Ω(f;δ)=sup0≤h<δ,x∈[0,∞)|f(x+h)−f(x)|(1+h2)(1+x2) $$\displaystyle{\Omega (f;\delta ) =\sup _{0\leq h<\delta,x\in [0,\infty )}\frac{\vert f(x + h) - f(x)\vert } {(1 + h^{2})(1 + x^{2})} }$$

Vijay Gupta, Gancho Tachev

Chapter 7. Direct Estimates for Some New Operators

In this chapter we deal with direct estimates for some integral type operators, established in the recent years.

Vijay Gupta, Gancho Tachev

Chapter 8. Convergence for Operators Based on Pǎltǎnea Basis

In the year 1987 Chen [36] and Goodman–Sharma [81] introduced the genuine Bernstein polynomials, which preserve linear functions. Some other generalizations of Bernstein polynomials have been introduced and studied in [6, 18, 90, 100, 102, 108, 152] and [171] etc., but they only reproduce constant functions.

Vijay Gupta, Gancho Tachev


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