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  • Author or Editor: Ferenc Móricz x
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<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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Abstract  

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function f with absolutely convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients of f which ensure the belonging of f either 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. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J. Németh and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.

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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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We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function fL p(ℝ) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.

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We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients
\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} $$\hat f$$ \end{document}
(n), n ∈ ℕ, of functions fL p for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σn=1 a 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\hat f$$ \end{document}
(n)|r, where {a n} is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones.
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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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