Search Results

You are looking at 1 - 10 of 11 items for :

  • "Walsh system" x
Clear All

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ɛ n↓0u .Тогд а существует множест во Е⊂[0, 1]с μЕ=0 такое что:1. Существует ряд с к оеффициентами ¦а n¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥an∥>0.2. Если b n ¦=о(ε n)и ряд сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b n=0для всех п. Теорема 3.Пусть ɛ n↓0u Тогд а существует множест во E⊂[0, 1] с υ E=0 такое, что:1.Существует ряд кот орый сходится к нулю в сюду вне E и ¦an≦¦n¦ для n=±1, ±2, ...2.Если ряд сходится к нулю всюду вне E и ¦bv¦=о(ε ¦n¦), то bn=0 для всех я. Теорема 5. Пусть послед овательности S(1)={ε0 (1), ε1 (1), ε2 (1), ...} u S20 (2), ε1 (2). ε2 (2) монотонно стремятся к нулю, , причем . Тогда для каждого ε>O н айдется множество Е⊂ [-π,π], μE >2π — ε, которое является U(S1), но не U(S1) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

Restricted access

Abstract  

We study the uniform convergence of Walsh-Fourier series of functions on the generalized Wiener class BV (p(n)↑∞)

Restricted access

Abstract

The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let such that a j(n+1)≧δsupkn a j(n) (j=1,2, n∊ℕ) for some δ>0 and a 1(+∞)=a 2(+∞)=+∞. Then for each integrable function fL 1(I 2) we have the a.e. relation . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].

Restricted access

Abstract  

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function fL 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

Restricted access

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.

Restricted access

Abstract  

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L p[0, 2π]2 was studied by M.I.Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim's sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L P[0, 1]2, p=(p 1, p 2), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.

Restricted access

Abstract  

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space H p([0, 1)d 1×…×[0, 1)d l to L p ([0, 1)d) if 1/2<p<∞ and m j≥0, ‖k j‖≤m j+1. By an interpolation theorem, we get that the maximal operator is also of weak type (

\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} $$(H_1^{\# _i } ,L_1 )$$ \end{document}
) (i=1,…,l), where the Hardy space
\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} $$H_1^{\# _i }$$ \end{document}
is defined by a hybrid maximal function and
\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} $$H_1^{\# _i } \supset L(\log L)^{l - 1}$$ \end{document}
. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f
\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} $$H_1^{\# _i }$$ \end{document}
and converge in a cone if fεL 1.

Restricted access

. 1977 29 155 164 Schipp, F. , Certain rearrangements of series in the Walsh system, Mat. Zametki , 18

Restricted access

. Hungar. 29 ( 1977 ), no. 1–2 , 155 – 164 . [12] Simon , P. , Cesáro summability with respect to two-parameter Walsh systems , Monatsh. Math. , 131 ( 2000

Restricted access

, P. , Cesàro summability with respect to two-parameter Walsh systems . Monatsh. Math. , 131 : 321 – 334 , 2000 . [20] Simon

Restricted access