## 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

Zbl 0591.42017
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### References:

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