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 Zap. Nauchn. Sem. POMI, 2013, Volume 416, Pages 108–116 (Mi znsl5697)

Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$

I. R. Kayumov, A. V. Kayumova

Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: For $p\in(1/2,1)$, the $L_p(\mathbb R)$-convergence of the series $\sum_{k=1}^\infty|\operatorname{Im}(t-z_k)^{-1}|$ is studied, where the $z_k$ are some points on the complex plane. The problem is solved completely in the case where the sequence $\{\operatorname{Re}z_k\}$ has no limit points. Also, the case where this sequence has finitely many limit points is studied.

Key words and phrases: simplest fractions, Hardy inequality, $L_p$-convergence.

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English version:
Journal of Mathematical Sciences (New York), 2014, 202:4, 553–559

UDC: 517.538.52+517.444

Citation: I. R. Kayumov, A. V. Kayumova, “Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$”, Investigations on linear operators and function theory. Part 41, Zap. Nauchn. Sem. POMI, 416, POMI, St. Petersburg, 2013, 108–116; J. Math. Sci. (N. Y.), 202:4 (2014), 553–559

Citation in format AMSBIB
\Bibitem{KayKay13} \by I.~R.~Kayumov, A.~V.~Kayumova \paper Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$ \inbook Investigations on linear operators and function theory. Part~41 \serial Zap. Nauchn. Sem. POMI \yr 2013 \vol 416 \pages 108--116 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl5697} \transl \jour J. Math. Sci. (N. Y.) \yr 2014 \vol 202 \issue 4 \pages 553--559 \crossref{https://doi.org/10.1007/s10958-014-2062-1} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84922076687}