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Proceedings of the American Mathematical Society

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The Franklin system as Schauder basis for $ L\sp p\sb \mu[0,1]$


Author: Robert E. Zink
Journal: Proc. Amer. Math. Soc. 103 (1988), 225-233
MSC: Primary 46E30; Secondary 42C10
DOI: https://doi.org/10.1090/S0002-9939-1988-0938673-1
MathSciNet review: 938673
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Abstract: The present work is devoted to a characterization of those spaces $ L_\mu ^p[0,1],p \geq 1,\mu $ totally finite, for which $ \mathcal{F}$, the Franklin system, is a Schauder basis. Because, in such cases, the measure $ \mu $ must be absolutely continuous with respect to the Lebesgue measure, $ m$, the necessary and sufficient condition is expressible in terms of a weight function, $ W$, the Radon-Nikodym derivative of $ \mu $ with respect to $ m$. One finds that the spaces $ L_\mu ^p[0,1]$ for which $ \mathcal{F}$ serves as Schauder basis are precisely those for which $ W$ satisfies the $ {A_p}$ condition introduced by Muckenhoupt. On the other hand, Krancberg has shown that a much less restrictive condition on $ \mu $ is both necessary and sufficient for the Haar system, $ \mathcal{H}$, to be a Schauder basis for $ L_\mu ^p[0,1]$. Thus, as a mildly surprising corollary of the theorem contained herein, one finds that the class of spaces $ L_\mu ^p[0,1]$, for which $ \mathcal{H}$ is a Schauder basis, properly contains the corresponding class of spaces for which the Franklin system so serves.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0938673-1
Article copyright: © Copyright 1988 American Mathematical Society