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Approximation with Positive Linear Operators and Linear Combinations

Part of the Developments in Mathematics Book Series (volume 50)

DOI: 10.1007/978-3-319-58795-0

ISBN: 978-3-319-58794-3

Publisher: Springer International Publishing - Springer Journals

Jun 28, 2017

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Chapter 1Moments and Combinations of Positive Linear Operators

Gupta, Vijay; Tachev, Gancho

Pages: 1 –14

[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...
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Chapter 2Direct Estimates for Approximation by Linear Combinations

Gupta, Vijay; Tachev, Gancho

Pages: 15 –23

[The aim of this chapter is to collect the known results for the error of approximation by linear combinations Ln, r, measured in different norms Lp(B) and usually in terms of Ditzian–Totik moduli of smoothness ωφr(f, t)p, or the ordinary moduli of continuity ωr(f, t)p. We will see the importance...
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Chapter 3Inverse Estimates and Saturation Results for Linear Combinations

Gupta, Vijay; Tachev, Gancho

Pages: 25 –36

[Let the Ditzian–Totik moduli of smoothness ωφr(f, t)p be given as (2.1.2) and the equivalent K-functional Kr, 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...
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Chapter 4Voronovskaja-Type Estimates

Gupta, Vijay; Tachev, Gancho

Pages: 37 –65

[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:]
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Chapter 5Pointwise Estimates for Linear Combinations

Gupta, Vijay; Tachev, Gancho

Pages: 67 –87

[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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...
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Chapter 6Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Gupta, Vijay; Tachev, Gancho

Pages: 89 –116

[Let E be a subspace of C[0, ∞) which contains the polynomials and Ln: 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)\documentclass[12pt]{minimal} \usepackage{amsmath}...
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Chapter 7Direct Estimates for Some New Operators

Gupta, Vijay; Tachev, Gancho

Pages: 117 –155

[In this chapter we deal with direct estimates for some integral type operators, established in the recent years.]
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Chapter 8Convergence for Operators Based on Pǎltǎnea Basis

Gupta, Vijay; Tachev, Gancho

Pages: 157 –173

[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...

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