Bounded sequence-to-function Hausdorff transformations

Georgakis, Constantine

doi:10.1090/S0002-9939-1988-0943080-1

Bounded sequence-to-function Hausdorff transformations
HTML articles powered by AMS MathViewer

by Constantine Georgakis PDF

Proc. Amer. Math. Soc. 103 (1988), 531-542 Request permission

Abstract:

Let \[ \left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0\] 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 $\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )$ if and only if $C\left ( {p,s} \right ) < \infty$, and that for $1 < p < \infty ,{\left \| {Ta} \right \|_{p,s}} < C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}$ 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 $\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}$. 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.

References

D. Borwein, On a scale of Abel-type summability methods, Proc. Cambridge Philos. Soc. 53 (1957), 318–322. MR 86158
G. H. Hardy, An inequality for Hausdorff means, J. London Math. Soc. 18 (1943), 46–50. MR 8854, DOI 10.1112/jlms/s1-18.1.46
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620

Elementary theorems concerning power series and moment constants

157

Inequalities

Otto Henriksson, Über die Hausdorffschen Limitierungsverfahren. die schwächer sind als das Abelsche, Math. Z. 39 (1935), no. 1, 501–510 (German). MR 1545514, DOI 10.1007/BF01201370
Amnon Jakimovski, The sequence-to-function analogues to Hausdorff transformations, Bull. Res. Council Israel Sect. F 8F (1960), 135–154 (1960). MR 126095

Über Reihen mit positiven Griedern

30

Dany Leviatan and Lee Lorch, A characterization of totally regular $[J,\,f(x)]$ transforms, Proc. Amer. Math. Soc. 23 (1969), 315–319. MR 246014, DOI 10.1090/S0002-9939-1969-0246014-9
A. Rényi, Summation methods and probability theory, Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 (1959), 389–399 (English, with Russian and Hungarian summaries). MR 130701
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923