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Abstract  

The main aim of this paper is to prove that the maximal operator

\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} $$\sigma _0^* : = \mathop {\sup }\limits_n \left| {\sigma _{n,n} } \right|$$ \end{document}
of the Fejr mean of the double Walsh-Fourier series is not bounded from the Hardy space H 1/2 to the space weak-L 1/2.

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Abstract  

The aim of this paper is to prove that for an arbitrary set of measure zero there exists a bounded function for which the Fejér means of the Walsh-Fourier series of the function diverge.

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Abstract  

The aim of this paper is to prove that the Cesàro means of order α (0 < α < 1) of the Fourier series with respect to representative product systems converge to the function in L 1-norm, only for certain values of α which depend on some parameter of the representative product system.

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Abstract  

Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H p to the space L p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ * α,k is bounded from the martingale Hardy space H 1/(1+α) to the space weak-L 1/(1+α).

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Abstract

Properties of Fourier–Haar coefficients of continuous functions are studied. It is established that Fourier–Haar coefficients of continuous functions are monotonic in a certain sense for convex functions. Questions of quasivariation of Fourier–Haar coefficients of continuous functions are also considered.

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Abstract  

The main aim of this paper is to prove that the maximal operator σ 0 k*:= supnσ n,n k∣ of the Fej�r means of double Fourier series with respect to the Kaczmarz system is not bounded from the Hardy space H 1/2 to the space weak-L 1/2.

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Abstract  

The main aim of this paper is to prove that there exists a martingale fH 1 2/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier series does not belong to the space weak-L 1/2.

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This paper is devoted to the study of Θ-summability of Fourier-Jacobi series. We shall construct such processes (using summations) that are uniformly convergent in a Banach 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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } } ,\parallel \cdot \parallel _{w_{\gamma ,\delta } }$$ \end{document}
) of continuous functions. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. Our aim is to give such conditions with respect to Jacobi weights w γ,δ , w α,β and to summation matrix Θ for which the uniform convergence holds for all 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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } }$$ \end{document}
. Order of convergence will also be investigated. The results and the methods are analogues to the discrete case (see [16] and [17]).

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Abstract  

The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l 2-norm for some general Mercer kernel matrices are provided. As an example, we give the l 2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l 2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.

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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].

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