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Indiana Math. J. 27 127 – 142 10.1512/iumj.1978.27.27011 . [2] Bustamante , J. 2008 Estimates of positive linear operators in

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486 Anastassiou, G. A. and Duman, O. , Statistical fuzzy approximation by fuzzy positive linear operators, Comput. Math. Appl. , 55 (2008), no. 3, 573–580. MR

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Abstract  

A sequence of positive linear operators which approximate each continuous function on [0,1] while preserving the functione 2 (x) =x 2 is presented. Quantitative estimates are given and are compared with estimates of approximation by Bernstein polynomials. Connections with summability are discussed.

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343 Gonska, H. , On the composition and decomposition of positive linear operators, in: Approximation Theory and its Applications (Ukrainian) (Kiev, 1999), 161–180, Pr. Inst. Mat. Nats

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Abstract  

We consider certain modified Szsz-Mirakyan operators A n (f;r) in polynomial weight spaces of functions of one variable and we study approximation properties of these operators.

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Abstract  

A Riesz-type representation theorem is given for all positive linear endomorphisms of the space of continuous functions on a compact interval.

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ПустьМ[а, b] — множество вещественных функци йf, измеримых и ограниче нных на отрезке [а, b], иL: M[a, b] → M[a, b] — лин ейный положительный оператор. В статье пол учены оценки погрешностей аппроксимации вL p -нор ме, выраженные в термина х так называемого усредненного локаль ного модуля гладкост и, илиτ-модуля. ПустьL облад ает свойствами: (e i,i(t):=t i дляi=0,1,2)Le 0=e 0,Le 1=e 1,Le 2 (х)=х 2 +β(х), x∈[a,b] и i)A ≦ min (1, (b-a)2/4) для 1 <p<∞, ii)A ≦ min (e−1, (b — a)2) дляр=1 приA:=∥βt8. Тогда выполнены оцен ки:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{gathered} \left\| {f - Lf} \right\|_p \leqq C\frac{p}{{p - 1}}\tau _2 (f;\sqrt A )_p 1< p< \infty \hfill \\ \left\| {f - Lf} \right\|_1 \leqq C\tau _2 \left( {f;\sqrt {A\ln \frac{1}{A}} } \right)_1 , \hfill \\ \end{gathered}$$ \end{document}
где положительные по стоянныеС не завися т от оператораL, функцииfи параметрар.
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The main object of this paper is to define the q -Laguerre type positive linear operators and investigate the approximation properties of these operators. The rate of convegence of these operators are studied by using the modulus of continuity, Peetre’s K -functional and Lipschitz class functional. The estimation to the difference | M n +1, q ( ƒ ; χ )− M n , q ( ƒ ; χ )| is also obtained for the Meyer-König and Zeller operators based on the q -integers [2]. Finally, the r -th order generalization of the q -Laguerre type operators are defined and their approximation properties and the rate of convergence of this r -th order generalization are also examined.

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] Volkov , Yu.I. , Certain positive linear operators , Mat. Zametki , 23 ( 1978 ), 363 – 368 .

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