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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.)
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