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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Bounded sequence-to-function generalized
Hausdorff transformations

Author: B. E. Rhoades
Journal: Proc. Amer. Math. Soc. 126 (1998), 1769-1782
MSC (1991): Primary 40C10, 40G05
MathSciNet review: 1443852
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Abstract: Georgakis (1988) obtained the norm of the transformation

\begin{equation*}(Ta)(y) = \sum ^{\infty }_{n=0} (-y)^{n} \frac{g^{(n)}(y)}{n!} a_{n},\quad y\geq 0, \end{equation*}

considered as an operator from the sequence space $\ell ^{p}$, with weights
$\Gamma (n+s+1)/n!$ to $L^{p}[0{,}\infty )$, with weight $y^{s},\ s>-1$. As corollaries he obtained inequality statements for Borel and generalized Abel transformations. He also obtained the best constants possible for several weighted norm inequalities of Hardy and Littlewood. In this paper Georgakis' results are extended to the Endl generalized Hausdorff matrices.

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Additional Information

B. E. Rhoades
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-5701

Keywords: Generalized Hausdorff, transformation, bounded operator, weighted $\ell ^{p}$-spaces
Received by editor(s): March 9, 1995
Received by editor(s) in revised form: December 6, 1996
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1998 American Mathematical Society