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On a problem of S. Banach from The Scottish book


Author: K. S. Kazarian
Journal: Proc. Amer. Math. Soc. 110 (1990), 881-887
MSC: Primary 42C99
DOI: https://doi.org/10.1090/S0002-9939-1990-1031668-0
MathSciNet review: 1031668
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Abstract: Denote by $ B\left( \Phi \right)$ the closure in $ {L^2}$ of linear combinations of functions of the orthonormal system $ \Phi = \left\{ {{\Phi _n}} \right\}_{n = 1}^\infty $. Denote by $ B{(\Phi )^ \bot }$ the orthogonal complement of $ B(\Phi )$ in $ {L^2}$. We prove the following theorem: Let $ \varphi $ be a nonnegative measurable function defined on $ \left[ {0, + \infty } \right)$ such that $ {\text{li}}{{\text{m}}_{x \to + \infty }}{x^2}/\varphi \left( x \right) = 0$ and $ N$ is a positive integer or $ N = + \infty $. Then there is a uniformly bounded orthonormal system $ {\Phi ^{\left( N \right)}} = \left\{ {{\Phi _n}} \right\}_{n = 1}^\infty $ such that $ \operatorname{dim} B{({\Phi ^{(N)}})^ \bot } = N$ and, for every nontrivial function $ f$ from $ B{({\Phi ^{(N)}})^ \bot },\int {\varphi \left( {\left\vert {f\left( t \right)} \right\vert} \right)} dt = + \infty $.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1031668-0
Keywords: Uniformly bounded orthonormal system, orthogonal completion
Article copyright: © Copyright 1990 American Mathematical Society