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Bounded sequence-to-function generalized Hausdorff transformations

Author(s): B. E. Rhoades
Journal: Proc. Amer. Math. Soc. 126 (1998), 1769-1782.
MSC (1991): Primary 40C10, 40G05
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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.

References:

1.: K. Endl, Untersuchungen über Momentprobleme bei Verfahren von Hausdorffschen Typus, Math. Annal. 139 (1960), 403-432. MR 22:12329
2.: C. Georgakis, Bounded sequence-to-function Hausdorff transformations, Proc. Amer. Math. Soc. 103 (1988), 531-542. MR 89i:40003
3.: G. H. Hardy and J. E. Littlewood, Elementary theorems concerning power series and moment constants, J. für Reine und Ang. Math. 157 (1927), 141-158.
4.: A. Jakimovski, The product of summability methods; new classes of transformations and their properties, Contract No. Air Force 61(1052)-187(1959), 1-76.


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