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 \(k= 1,2,\dots\;\).


42C15 General harmonic expansions, frames


Zbl 0591.42017
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