## On the choice of a subsystem of convergence with logarithmic density from an arbitrary orthonormal system.(Russian)Zbl 0668.42012

A system of functions $$(f_ n(x))^{\infty}_{n=1}$$ (x$$\in (0,1))$$ is called a convergence system if $$\sum c_ nf_ n(x)$$ convergence almost everywhere in (0,1) whenever$$\sum c^ 2_ n<\infty$$. A classical result of Marcinkiewicz and Menshov asserts that every orthonormal sequence $$(f_ n(x))^{\infty}_{n=1}$$ contains a convergence subsystem $$(f_{n_ k})^{\infty}_{n=1}$$. G. Bennett [Notes In Banach spaces, Semin. and Courses, Austin/Tex. 1975-79, 39-80 (1980; Zbl 0457.46009)] conjectured that there is a sequence $$(r_ n)^{\infty}_{n=1}$$ of positive reals such that for every $$(f_ n(x))^{\infty}_{n=1}$$ in the Marcinkiewicz-Menshov theorem $$n_ k$$ may be chosen satisfying the condition $$\lim_{k\to \infty}n_ k/r_ k=0.$$ This conjecture was proved by B. S. Kashin [Usp. Mat. Nauk 40, No.2(242), 181-182 (1985; Zbl 0591.42017)]. In the same paper B. S. Kashin asked, is it true that we may pose $$r_ k=k^{1+a}$$ for any $$a>0?$$ The author proved that for every $$c>1$$ we may choose $$r_ k=c^ k$$. He also obtained analogous result for $$S_ p$$-systems.
Reviewer: M.Ostrovskij

### MSC:

 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 40A30 Convergence and divergence of series and sequences of functions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

### Keywords:

convergence system; $$S_ p$$-systems

### Citations:

Zbl 0457.46009; Zbl 0591.42017
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