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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

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.

UDC: 517.538.52+517.444

Received: 12.03.2013


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

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