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Subsystems of the Walsh orthogonal system whose multiplicative completions are quasibases for $L^{p}[0,1]$, $1\leq p < +\infty $

Authors: M. G. Grigorian and Robert E. Zink
Journal: Proc. Amer. Math. Soc. 131 (2003), 1137-1149
MSC (2000): Primary 42C10
Published electronically: July 25, 2002
MathSciNet review: 1948105
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Abstract: If one discards some of the elements from the Walsh family, an ancient example of a system that serves as a Schauder basis for each of the $L^{p}$-spaces, with $1< p<+\infty $, the residual system will not be a Schauder basis for any of those spaces. Nevertheless, Price has shown that each member of a large class of such subsystems is complete on subsets of $[0,1]$ that have measure arbitrarily close to $1$. In the present work, it is shown that subsystems of this kind can be multiplicatively completed in such a way that the resulting systems are quasibases for each space $L^{p}[0,1]$, $1\le p<+\infty $, from which the earlier completeness result follows as a corollary.

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Additional Information

M. G. Grigorian
Affiliation: Department of Mathematics, Erevan State University, Alex Manoogian Str., 375049 Yerevan, Armenia

Robert E. Zink
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1968

Received by editor(s): August 22, 2001
Received by editor(s) in revised form: November 2, 2001
Published electronically: July 25, 2002
Communicated by: Andreas Seeger
Article copyright: © Copyright 2002 American Mathematical Society