For p ∈ (1/2, 1), the L p (ℝ)-convergence of the series \( {\displaystyle \sum_{k=1}^{\infty}\left|\mathrm{I}\mathrm{m}{\left(t-{z}_k\right)}^{-1}\right|} \) is studied, where zk are some points on the complex plane. The problem is solved completely in the case where the sequence {Re zk} has no limit points. The case where this sequence has finitely many limit points is also studied.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 416, 2013, pp. 108–116.
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Kayumov, I.R., Kayumova, A.V. Convergence of the Imaginary Parts of Simplest Fractions in L p ( ℝ ) for p < 1. J Math Sci 202, 553–559 (2014). https://doi.org/10.1007/s10958-014-2062-1
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DOI: https://doi.org/10.1007/s10958-014-2062-1