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On convergence subsystems of orthonormal systems. (English) Zbl 0811.42009

Although not every orthonormal system is a convergence system, Marcinkiewicz and Menshov proved that every orthonormal system \(\{\phi_ n\}\) contains a convergence subsystem \(\{\phi_{n_ k}\}\). B. S. Kashin [Usp. Mat. Nauk 40, No. 2(242), 181-182 (1985; Zbl 0591.42017)] obtained an upper estimate of the growth rate of the indices \(n_ k\). In this paper, the author obtains a lower estimate. If \(R_ k\geq k\) and \(R_ k= o(k\log_ 2 k)\), as \(k\to\infty\), then there is an orthonormal system \(\{\phi_ n\}\) such that none of its subsystems \(\{\phi_{n_ k}\}\) is a convergence system when \(n_ k\leq R_ k\) for \(k= 1,2,\dots\;\).

MSC:

42C15 General harmonic expansions, frames

Citations:

Zbl 0591.42017
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References:

[1] D.E. Menshov, Sur les séries des fonctions orthogonales I.Fund. Math. 4 (1923), 82–105.
[2] J. Marcinkiewicz, Sur la convergence des séries orthogonales.Studia Math. 6 (1936), 39–45. · JFM 62.0284.01
[3] D.E. Menshov, Sur la convergence et la sommation des séries des fonctions orthogonales.Bull. Soc. Math. de France 64 (1936), 147–170.
[4] G. Bennet, Lectures on matrix transformation ofl p spaces.Notes in Banach spaces, pp. 39–80,Univ. Texas Press, Austin, Tex., 1980.
[5] B.S. Kashin The choice of convergence subsystem from the given orthonormal system. (Russian)Uspehi Mat. Nauk 40 (1985), No. 2 (242), 181–182. English translation:Russ. Math. Surv. 40 (1985), 215–216.
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[7] H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen.Math. Ann. 87(1922), 112–138.
[8] B.S. Kashin, A.A. Saakyan, Orthogonal series. (Russian), ”Nauka”,Moscow, 1984 English translation: Translations of Mathematical Monographs, 75,American Math. Society, Providence, RI, 1989. · Zbl 0632.42017
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