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# Higher order Lipschitz classes of functions and absolutely convergent Fourier series

Acta Mathematica Hungarica
Author: Ferenc Móricz

## Abstract

We introduce the higher order Lipschitz classes Λr(α) and λ r(α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < αr. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients in order that f belongs to one of the above classes.

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# Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

Acta Mathematica Hungarica
Author: Ferenc Móricz

## Abstract

We consider N-multiple trigonometric series whose complex coefficients c j1,...,j N, (j 1,...,j N) ∈ ℤN, form an absolutely convergent series. Then the series
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N )$$ \end{document}
converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}
N,
\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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}
:= [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., αN) and lip (α1,..., αN) for some α1,..., αN > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.
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# Fourier coefficients and Lipschitz class

Analysis Mathematica
Author: И. Пак
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# Trigonometric series of Nikol’skii classes

Acta Mathematica Hungarica
Author: S. Tikhonov

## Abstract

We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol’skii class of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients are presented.

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# On a problem of L. Leindler concerning strong approximation by Fourier series and Lipschitz classes

Analysis Mathematica
Author: Л. Д. Гоголадзе
Пустьϕ — возрастающа я непрерывная фцнкци я на [0,π],ϕ(0)=0 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} $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ \end{document}
Положим
\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} $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ \end{document}
Доказывается следую щая теорема.Пусть f∈ С[−π, π], ω(f, δ)=О(ϕ(δ))) 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} $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ \end{document}
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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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# On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

Acta Mathematica Hungarica
Author: Ferenc Móricz

series with positive coefficients and generalized Lipschitz classes Acta Sci. Math. (Szeged) 54 291 – 304 .  Tikhonov , S

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# Addendum to “Trigonometric series of Nikol’skii classes”

Acta Mathematica Hungarica
Author: S. Tikhonov

## Abstract

We describe the functions from Nikol’skii class in terms of behavior of their Fourier coefficients. Results for series with general monotone coefficients are presented. The problem of strong approximation of Fourier series is also studied.

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# Approximation in weighted generalized grand smirnov classes

Studia Scientiarum Mathematicarum Hungarica
Authors: Daniyal M. Israfilov and Ahmet Testici

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and $εp),θ(G−,ω)$, 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

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