Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): M. G. Grigorian; Robert E. Zink
Journal: Proc. Amer. Math. Soc. 131 (2003), 1137-1149.
MSC (2000): Primary 42C10
Posted: 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 that have measure arbitrarily close to . 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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