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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A note on unconditionally converging series in $ L\sb{p}$


Author: Peter Ørno
Journal: Proc. Amer. Math. Soc. 59 (1976), 252-254
MSC: Primary 46E30; Secondary 40H05
MathSciNet review: 0458156
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Abstract: Theorem. A series $ \sum {{f_i}} $ in $ {L_p}[0,\,1](1 \leqslant p \leqslant 2)$ is unconditionally convergent if and only if for each $ i$ and for all $ t \in [0,\,1],\;{f_i}(t) = {\alpha _i}g(t){w_i}(t)$ where $ ({\alpha _i}) \in {l_2},\;g \in {L_2}[0,\,1]$ and $ ({w_i})$ is an orthonormal sequence in $ {L_2}[0,\,2]$. This characterization allows the generalization (to u.c. series in $ {L_p}[0,\,1]$) of several classical theorems concerning almost everywhere convergence of orthogonal series in $ {L_2}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0458156-7
PII: S 0002-9939(1976)0458156-7
Article copyright: © Copyright 1976 American Mathematical Society