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  • Author or Editor: Ferenc Móricz x
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Abstract  

The well-known characterization indicated in the title involves the moving maximal dyadic averages of the sequence (X k: k = 1, 2, …) of random variables in Probability Theory. In the present paper, we offer another characterization of the SLLN which does not require to form any maximum. Instead, it involves only a specially selected sequence of moving averages. The results are also extended for random fields (X k: k, ℓ = 1, 2, …).

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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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Abstract  

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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Abstract  

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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Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \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} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \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} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)} $ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \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} $\sigma _t (f;pd\nu ) \to Last \to \infty $ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

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Abstract  

We introduce the concept of the statistical limit (at ∞) of a measurable function in several variables and recall the concept of the statistical convergence of a multiple sequence. Then we extend a classical theorem of Schoenberg (which characterizes statistical convergence) from single to multiple sequences, and prove an analogous theorem on statistical limit. These theorems even may be extended to vector-valued sequences or functions, respectively.

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Abstract  

We prove [1, Theorems 1 and 2] under weaker conditions and in a simpler way than we did in the cited paper.

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Abstract

We consider the double Walsh orthonormal system

ea
on the unit square , where {w m(x)} is the ordinary Walsh system on the unit interval in the Paley enumeration. Our aim is to give sufficient conditions for the absolute convergence of the double Walsh–Fourier series of a function for some 1<p≦2. More generally, we give best possible sufficient conditions for the finiteness of the double series
eb
where {a mn} is a given double sequence of nonnegative real numbers satisfying a mild assumption and 0<r<2. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity of f.

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Abstract  

First, we consider integrals of the form

\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} $$\int {a(x)m^{ - x} dx for} m = 2,3,...$$ \end{document}
over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 2}^\infty {a_m m^{ - x} for x > 1} ,$$ \end{document}
, where the numbersa m≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature. Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)2 or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)2. The case wherea(x, y) is integrable on (0, ∞)2 is also included.

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