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• Author or Editor: Ferenc Móricz
Clear All Modify Search  # 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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# пОРьДОк РОстА АРИФМЕ тИЧЕскИх сРЕДНИх ДВО ИНОгО ОРтОгОНАльНОгО РьДА

Analysis Mathematica
Author: Ferenc Móricz
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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

## Abstract.

We consider complex-valued functions fL 1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform and the cosine Fourier transform belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.

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# Интегрируемость дво йных тригонометриче ских рядов со специальными коэф фициентами

Analysis Mathematica
Author: Ferenc Móricz
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# Fejér type theorems for Fourier-Stieltjes series

Analysis Mathematica
Author: Ferenc Móricz
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# ltivariate Hausdorff operators on the spaces H 1(R n ) and BMO(R n )

Analysis Mathematica
Author: Ferenc Móricz

Summary A multivariate Hausdorff operator H = H(µ, c, A) is defined in terms of a s-finite Borel measure µ on Rn, a Borel measurable function c on Rn, and an × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular µ-a.e. The operator H*:= H (µ, c | det A -1|, A -1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H 1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.

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# Приэнак типа приэнака Дини поточечной сходимости двойных интегралов Фурье

Analysis Mathematica
Author: Ferenc Móricz

<a name="abs1"/>Abstract??We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued functionf?L 1(?2) with bounded support at a given point (x 0,y 0) ? ?2. It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functionsf(x,y 0),x? ?, andf(x 0,y),y? ?, at the pointsx:=x 0andy:=y 0, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.

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# Приэнаки типа Прингсхейма для поточечной сходимости двойных тригонометрически сопряженных интегралов Фурье

Analysis Mathematica
Author: Ferenc Móricz

## Abstract

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function fL 1 (ℝ2) with bounded support at a given point (x 0, g 0) ∈ ℝ2. It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y 0), x ∈ ℝ, and f(x 0, y), y ∈ ℝ, at x:= x 0 and y:= y 0, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.

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# Bundle Convergence of Cesàro Means of Orthogonal Sequences in Noncommutative L 2-Spaces

Periodica Mathematica Hungarica
Author: Ferenc Móricz
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