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 Mat. Sb. (N.S.), 1972, Volume 89(131), Number 3(11), Pages 355–365 (Mi msb3238)

On the convergence of series of weakly multiplicative systems of functions

V. F. Gaposhkin

Abstract: A system of measurable functions $\{\varphi_k\}$ defined on a measurable space is called weakly multiplicative if it satisfies the relations
$$\int_X\varphi_{k_1}\varphi_{k_2}…\varphi_{k_p} d\mu=0\quad(\forall p\geqslant2, k_1<k_2<…<k_p).$$

In this paper the convergence in the metric of $L_p$ and a.e. is investigated for series of weakly multiplicative system of functions. One of the results is: {\it If $\{\varphi_k\}$ is weakly multiplicative and $\sup_k\|\varphi_k\|_p\leqslant M$ for some $p>2,$ then any series $\sum c_k\varphi_k$ with coefficients in $l_2$ converges unconditionally a.e. and in $L_p$}. For $p=2n$, instead of weak multiplicativity it is sufficient to require the condition $\int_X\varphi_{k_1}…\varphi_{k_{2n}} d\mu=0$ ($\forall k_1<…<k_{2n}$).
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Sbornik, 1972, 18:3, 361–372

Bibliographic databases:

UDC: 517.522
MSC: Primary 42A60; Secondary 60G45, 60G50

Citation: V. F. Gaposhkin, “On the convergence of series of weakly multiplicative systems of functions”, Mat. Sb. (N.S.), 89(131):3(11) (1972), 355–365; Math. USSR-Sb., 18:3 (1972), 361–372

Citation in format AMSBIB
\Bibitem{Gap72} \by V.~F.~Gaposhkin \paper On the convergence of series of weakly multiplicative systems of functions \jour Mat. Sb. (N.S.) \yr 1972 \vol 89(131) \issue 3(11) \pages 355--365 \mathnet{http://mi.mathnet.ru/msb3238} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=334315} \zmath{https://zbmath.org/?q=an:0249.42013} \transl \jour Math. USSR-Sb. \yr 1972 \vol 18 \issue 3 \pages 361--372 \crossref{https://doi.org/10.1070/SM1972v018n03ABEH001818} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. S. Krantsberg, “On divergent Fourier series in orthogonal systems”, Math. USSR-Sb., 22:4 (1974), 547–560
2. M. Longnecker, R. J. Serfling, “Moment inequalities for S n under general dependence restrictions, with applications”, Z Wahrscheinlichkeitstheorie verw Gebiete, 43:1 (1978), 1
3. P. A. Yaskov, “On asymptotic constancy of diagonal elements of a random orthogonal projection”, Russian Math. Surveys, 69:4 (2014), 755–756
4. A. I. Rubinshtein, “Ob odnom mnozhestve slabo multiplikativnykh sistem”, Matem. zametki, 105:3 (2019), 471–475
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