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# On the integral modulus of continuity of Fourier series

Acta Mathematica Hungarica
Author: Bogdan Szal

## 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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# On Proinov’s bound for the quadrature error of functions having a given modulus of continuity

Analysis Mathematica
Author: Knut Petras

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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# A functional CLT for the L 2 modulus of continuity of local time

Periodica Mathematica Hungarica
Author: Jay Rosen

## 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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# The modulus of continuity and the best approximation over the dyadic group

Acta Mathematica Hungarica
Author: J. Tateoka

## Without Abstract

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# On the modulus of continuity with respect to functions defined on Vilenkin groups

Acta Mathematica Hungarica
Author: S. Fridli
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# On rational approximation of convex functions with a given modulus of continuity

Analysis Mathematica
Authors: A. P. Bulanov and A. Hatamov
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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# On the uniform modulus of continuity of the operator of best approximation in the space of periodic functions

Acta Mathematica Hungarica
Author: A. Kroó
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# On positive operators involving a certain class of generating functions

Studia Scientiarum Mathematicarum Hungarica
Authors: O. Doğru, M. A. Özarslan, and F. Taşdelen

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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# q -Laguerre type linear positive operators

Studia Scientiarum Mathematicarum Hungarica
Author: Mehmet Özarslan

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 . 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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# Estimate of the approximation of periodic functions by negative Cesàro means

Acta Mathematica Hungarica
Author: D. Tsirekidze

## 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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