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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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Abstract  

Let f: R NC be a periodic function with period 2π in each variable. We prove suffcient conditions for the absolute convergence of the multiple Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to multiple Fourier series.

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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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In this paper the author studies classesH q Ω of periodic functions of several variables whose mixed moduli of continuity do not exceed a given modulus of continuity ω(t 1 ...,t d). Necessary and sufficient conditions of belonging of a functionf(x 1, ...,x d) to the classH q Ω are considered (Theorem 1). These necessary and sufficient conditions are proved under some additional assumptions on ω(t 1, ...,t d). It is shown that additional assumptions cannot be omitted (Theorem 3). Besides, the estimates of best approximations of classesH q Ω with some special ω(t 1, ...,t d) are given (Theorems 4 and 5).

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Реэюме  

Получены точные неравенства типа Джексона-Стечкина для ос-редненных с весом модулей непрерывности m-го (m ∈ ℕ) порядка. Для классов функций, определенных при помоши мажорант и укаэанных осредненных величин, вычислены точные эначения раэличных n-поперечников при выполнении определенных ограничений на мажоранты.

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Abstract  

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szász, Zygmund and others related to absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.

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Abstract  

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C β q H ω of Poisson integrals of functions from the class H ω generated by convex upwards moduli of continuity ω(t) which satisfy the condition ω(t)/t → ∞ as t → 0. As an implication, a solution of the Kolmogorov-Nikol’skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H α, 0 < α < 1, is obtained.

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Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:1.IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;2.Iff(x 1,x 2, ...,x m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD⊂R m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝D¦gradf¦dx≦2π m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;3.LetE=Π={z∶¦z¦=1}, and letc m(R) be the mth Laurent coefficient ofR onΠ,C m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C m(n) ≦ min {n/¦m¦, 1}.

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83 207 217 CSÁKI, E. and CSÖRGÖ, M., Inequalities for increments of stochastic processes and moduli of continuity, Ann. Probab. 20 (1992), 1031

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