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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
Received: 12.03.2013

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}


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