The Franklin system as Schauder basis for $L^ p_ \mu [0,1]$

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.