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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Restricting a Schauder basis to a set of positive measure


Author: James Shirey
Journal: Trans. Amer. Math. Soc. 184 (1973), 61-71
MSC: Primary 42A64
DOI: https://doi.org/10.1090/S0002-9947-1973-0330914-0
MathSciNet review: 0330914
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Abstract: Let $ \{ {f_n}\} $ be an orthonormal system of functions on [0, 1] containing a subsystem $ \{ {f_{{n_k}}}\} $ for which (a) $ {f_{{n_k}}} \to 0$ weakly in $ {L_2}$, and (b) given $ E \subset [0,1]$, $ \vert E\vert > 0$, $ {\operatorname{Lim}}\;{\operatorname{Inf}}{\smallint _E}\vert{f_{{n_k}}}(x)\vert dx > 0$. There then exists a subsystem $ \{ {g_n}\} $ of $ \{ {f_n}\} $ such that for any set E as above, the linear span of $ \{ {g_n}\} $ in $ {L_1}(E)$ is not dense.

For every set E as above, there is an element of $ {L_p}(E)$, $ 1 < p < \infty $, whose Walsh series expansion converges conditionally and an element of $ {L_1}(E)$ whose Haar series expansion converges conditionally.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0330914-0
Article copyright: © Copyright 1973 American Mathematical Society