The Franklin system as Schauder basis for $L^ p_ \mu [0,1]$
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- by Robert E. Zink
- Proc. Amer. Math. Soc. 103 (1988), 225-233
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938673-1
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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.References
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Bibliographic Information
- © Copyright 1988 American Mathematical Society
- 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