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Bounded sequence-to-function Hausdorff transformations

Author: Constantine Georgakis
Journal: Proc. Amer. Math. Soc. 103 (1988), 531-542
MSC: Primary 40G05; Secondary 26D10, 47B38
MathSciNet review: 943080
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Abstract: Let

$\displaystyle \left( {Ta} \right)\left( y \right) = \sum\limits_{n = 0}^\infty ... ...rac{{{g^{\left( n \right)}}\left( y \right)}}{{n!}}{a_{n,\quad }}\quad y \geq 0$

be the sequence-to-function Hausdorff transformation generated by the completely monotone function $ g$ or, what is equivalent, the Laplace transform of a finite positive measure $ \sigma $ on $ [0,\infty )$. It is shown that for $ 1 \leq p \leq \infty $, $ T$ is a bounded transformation of $ {l^p}$ with weight $ \Gamma \left( {n + s + 1} \right) / n!$ into $ {L^p}[0,\infty )$ with weight $ {y^s},s > - 1$, whose norm $ \left\Vert T \right\Vert = \int_0^\infty {{t^{ - \left( {1 + s} \right) / p}}} d\sigma \left( t \right) = C\left( {p,s} \right)$ if and only if $ C\left( {p,s} \right) < \infty $, and that for $ 1 < p < \infty ,{\left\Vert {Ta} \right\Vert _{p,s}} < C\left( {p,s} \right){\left\Vert a \right\Vert _{p,s}}$ unless $ {a_n}$ is a null sequence. Furthermore, if $ 1 < p < r < \infty ,\,\;0 < \lambda < 1$ and $ \sigma $ is absolutely continuous with derivatives $ \psi $ such that the function $ {\psi _r}\left( t \right) = {t^{ - 1 / r}}\psi \left( t \right)$ belongs to $ {L^{1 / \lambda }}[0,\infty )$, then the transformation $ \left( {{T_\lambda }a} \right)\left( y \right) = {y^{1 - \lambda }}\left( {Ta} \right)\left( y \right)$ is bounded from $ {l^p}$ to $ {L^r}[0,\infty )$ and has norm $ \left\Vert {{T_\lambda }} \right\Vert \leq {\left\Vert {{\psi _r}} \right\Vert _{1 / \lambda }}$. The transformation $ T$ includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on $ {l^p}$ which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.

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Keywords: Hausdorff, transformation, bounded, Abel means, Borel transform, completely monotone, power series, moments
Article copyright: © Copyright 1988 American Mathematical Society

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