Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1031668
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] R. Daniel Mauldin (ed.), The Scottish Book, Birkhäuser, Boston, Mass., 1981. Mathematics from the Scottish Café; Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979. MR 666400
  • [2] S. Kaczmarz, O zupelności ukladów ortogonalnych, in Archiwum Towarzystwa Naukowego we Lwowie, Dzial III, Tom VIII, Zeszyt 5, 431-436.
  • [3] Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 0094840
  • [4] A. M. Olevskiĭ, An orthonormal system and its applications, Mat. Sb. (N.S.) 71 (113) (1966), 297–336 (Russian). MR 0203351

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42C99

Retrieve articles in all journals with MSC: 42C99

Additional Information

Keywords: Uniformly bounded orthonormal system, orthogonal completion
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society