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- Title
On Convergence of Series of Simple Partial Fractions in $$ Lp\left(\mathrm{\mathbb{R}}\right) $$.
- Authors
Dodonov, A.
- Abstract
We show that the necessary condition for the convergence of the series of simple partial fractions $$ {\displaystyle \sum_{k=1}^{\infty }{\left(z-{z}_k\right)}^{-1}} $$ in $$ Lp\left(\mathrm{\mathbb{R}}\right) $$, 1 < p < ∞, is the convergence of the series $$ {\displaystyle \sum_{k=1}^{\infty }{\left|{z}_k\right|}^{-1/q}1{\mathrm{n}}^{-1-\varepsilon}\left(\left|{z}_k\right|+1\right)} $$, ε > 0. In the case 1 < p < 2, we obtain a convergence criterion in terms of the imaginary parts of poles under the condition that all the poles z = x + iy belong to the angle | z| ≤ α| y| with a fixed α > 0.
- Subjects
STOCHASTIC convergence; MATHEMATICAL series; PARTIAL fractions; MATHEMATICAL analysis; NUMERICAL analysis
- Publication
Journal of Mathematical Sciences, 2015, Vol 210, Issue 5, p648
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-015-2583-2