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Proceedings of the American Mathematical Society

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

 
 

 

A note on unconditionally converging series in $L_{p}$


Author: Peter Ørno
Journal: Proc. Amer. Math. Soc. 59 (1976), 252-254
MSC: Primary 46E30; Secondary 40H05
DOI: https://doi.org/10.1090/S0002-9939-1976-0458156-7
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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Article copyright: © Copyright 1976 American Mathematical Society