On a problem of S. Banach from The Scottish book
Abstract: Denote by the closure in of linear combinations of functions of the orthonormal system . Denote by the orthogonal complement of in . We prove the following theorem: Let be a nonnegative measurable function defined on such that and is a positive integer or . Then there is a uniformly bounded orthonormal system such that and, for every nontrivial function from .
-  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
-  S. Kaczmarz, O zupelności ukladów ortogonalnych, in Archiwum Towarzystwa Naukowego we Lwowie, Dzial III, Tom VIII, Zeszyt 5, 431-436.
-  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
-  A. M. Olevskiĭ, An orthonormal system and its applications, Mat. Sb. (N.S.) 71 (113) (1966), 297–336 (Russian). MR 0203351
- R. D. Mauldin, ed., The Scottish book, Birkhäuser, 1981. MR 666400 (84m:00015)
- S. Kaczmarz, O zupelności ukladów ortogonalnych, in Archiwum Towarzystwa Naukowego we Lwowie, Dzial III, Tom VIII, Zeszyt 5, 431-436.
- Ulf Grenander and Gabor Szegö, Teoplitz forms and their applications, Univ. of California Press, Los Angeles, CA, 1958. MR 0094840 (20:1349)
- A. M. Olevskii, On an orthonormal system and its applications, Mat. Sb. 71(1966), 297-336; English transl., Transl. Amer. Math. Soc. 76 (1968). MR 0203351 (34:3204)
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Keywords: Uniformly bounded orthonormal system, orthogonal completion
Article copyright: © Copyright 1990 American Mathematical Society