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  • Author or Editor: Ferenc Móricz x
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We prove that the maximal conjugate and Hilbert operators are not bounded from the real Hardy space H 1 to L 1, where the underlying spaces may be over T or R. We also draw corollaries for the corresponding spaces over T 2 and R 2.

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Abstract  

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H 1, which may be considered over

\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} $$\mathbb{T}$$ \end{document}
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} $$\mathbb{R}$$ \end{document}
. We also draw corollaries for the corresponding Hardy spaces over
\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} $$\mathbb{T}$$ \end{document}
2 and
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2.

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The first named author has recently proved necessary and sufficient Tauberian conditions under which statistical convergence (introduced by H. Fast in 1951) follows from statistical summability (C, 1). The aim of the present paper is to generalize these results to a large class of summability methods (N¯,p) by weighted means.

Let p = (p k : k = 0,1, 2,...) be a sequence of nonnegative numbers such that po > 0 and Pn:k=0npkasn Let (x k) be a sequence of real or complex numbers and set tn:Pn1k=0npkxk for n = 0,1, 2,.... We present necessary and sufficient conditions under which the existence of the limit st-lim x k = L follows from that of st-lim t n = L, where L is a finite number. If (x k) is a sequence of real numbers, then these are one-sided Tauberian conditions. If (x k) is a sequence of complex numbers, then these are two-sided Tauberian conditions.

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