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Abstract

We study L p-integrability (1<p<∞) of a sum ϕ of trigonometric series under the assumptions that the sequence of coefficients of ϕ belongs to the class . Then we discuss the relations between the properties of ϕ and the properties of the sequence (λ n)∊GM(β,r), and deduce an estimate for modulus of continuity of ϕ in L p norm.

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Summary We discuss the question, under which conditions Proinov’s upper bound for the quadrature error of functions having a given modulus of continuity is not improvable. Improvability usually holds if the quadrature formula is not a Riemannian sum. We derive a condition on the second Peano kernel yielding always the sharpness of Proinov’s result. This condition is applicable to Gaussian quadrature.

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Abstract  

We show that as processes in (c, d, t) ∈ C(R 2 × R + 1)

\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} $$\frac{{\int_c^d {(L_t^{x + h} - L_t^x )^2 dx - 4h} \int_c^d {L_t^x dx} }} {{h^{3/2} }}\mathop \Rightarrow \limits^\mathcal{L} \left( {\frac{{64}} {3}} \right)^{1/2} \int_c^d {L_t^x d\eta (x)}$$ \end{document}
as h → 0 for Brownian local time L t x. Here η(x) is an independent two-sided Brownian motion.

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Доказывается, что для наименьших равномер ных рациональных уклоне нийR n(f) выпуклой на [0,1] функции с модулем непрерывно сти, не превосходящемω(δ), сп раведлива оценка
\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} $$R_n (f) \leqq c\frac{{\ln ^2 n}}{{n^2 }}\mathop {\max }\limits_{e^{ - n} \leqq \theta< 1} \left\{ {\omega (\theta )\ln \frac{1}{\theta }} \right\},$$ \end{document}
гдес — абсолютная по стоянная.
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In this paper we introduced the general sequence of linear positive operators via generating functions. Approximation properties of these operators are obtained with the help of the Korovkin Theorem. The order of convergence of these operators computed by means of modulus of continuity Peetre’s K-furictiorial and the elements of the usual Lipschitz class. Also we introduce the r-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally we give some applications to differential equations.

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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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Abstract  

The problem of approximation of continuous functions by Cesàro (C,α)-means, −1 < α < 0, in terms of L p and C-modulus of continuity is studied.

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