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Weighted approximation by the Goodman–Sharma Operators East J. Approx. 15 473 – 486 . [12] Lupas , A. 1967 Some properties of the linear positive operators (I

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] Vechhia Della , B. 2003 Weighted approximation by rational operators Result. Math. 43 79 – 87 . [6

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Weighted approximation of functions by Szász--Mirakyan-type operators

Sz\'asz--Mirakyan operator, weighted modulus of smoothness, direct and converse results

Acta Mathematica Hungarica
Authors: Biancamaria Della Vecchia, Giuseppe Mastroianni, and Szabados J

Summary  

We give error estimates for the weighted approximation of functions with singularities at the endpoints on the semiaxis by some modifications of Sz\'asz--Mirakyan operators. To do so, we define a new weighted modulus of smoothness and prove its equivalence to the weighted K-functional. Also, the class of functions for which the modified Sz\'asz--Mirakyan operator can be defined will be extended to a much wider set than for the original operator.

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Vecchia , B., Mastroianni, G. and Szabados, J. , Weighted approximation of functions with endpoint or inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004), no. 1–2, 19–41. MR 2005e :41016

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Abstract  

We give direct and converse results for the weighted approximation of functions with endpoint or inner singularities by Bernstein operators.

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order Cesàro means in Jacobi spaces and weighted approximation on [−1, 1], 2004, Habilitationsschrift, Seminarberichte aus dem Fachbereich Mathematik der FernUniversität in Hagen (ISSN 0944-5838), Band 75, pp. 1

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Abstract  

Let X,X 1,X 2,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≦ t ≦ 1} where S n = Σj=1 n X j, under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n, 0 ≦ t ≦ 1}, where V n 2 = Σj=1 n X j 2. L p approximations of self-normalized partial sum processes are also discussed.

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1988 Kroó , A. and Szabados , J., On weighted approximation by lacunary polynomials and rational functions on the half-axis, East

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пУсть

\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} $$S_n (f,x) = \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{n}} \right)p_k (nx)\left( {p_k (t) = e^{ - t} \frac{{t^k }}{{k!}}} \right)}$$ \end{document}
— ОпЕРАтОРы сАсА-МИРА кььНА И
\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} $$C_\varrho = \{ f:f \in C[0,\infty ),\left\| f \right\|_\varrho = \mathop {\sup }\limits_{x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0} |\varrho (x)f(x)|< \infty \} ,$$ \end{document}
,
\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} $$\omega _2 (f,\delta ) = \mathop {\sup }\limits_{0< h< \delta } \left\| {\Delta _{h\sqrt x }^2 (f;x)} \right\|_\varrho ,$$ \end{document}
, гДЕ ФУНкцИьϱ(x) НЕпРЕР ыВНА И НЕ ВОжРАстАЕт Н А [0, ∞), 0<ϱ(x)≦1, И
\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} $$\mathop {\sup }\limits_{x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0} \{ {{\varrho (x)} \mathord{\left/ {\vphantom {{\varrho (x)} \varrho }} \right. \kern-\nulldelimiterspace} \varrho }(x + h\sqrt x )\}< \infty .$$ \end{document}
. ДОкАжАНО, ЧтО Дль ФУНк цИИf C p,pΦ 2 УслОВИь
\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} $$\left\| {S_n (f) - f} \right\|_\varrho = O\left( {p\left( {\frac{1}{{\sqrt n }}} \right)} \right),n \to \infty ,$$ \end{document}
, И
\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} $$\begin{array}{*{20}c} {\omega _2 (f.\delta ) = O(p(\delta )),} & {\Delta _\delta ^2 (f,0) = O(p(\sqrt \delta )),} & {\delta \to 0,} \\ \end{array}$$ \end{document}
, ЁкВИВАлЕНтНы.

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