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Riemann rearrangement theorem for some types of convergence. (English) Zbl 1210.40001

The authors reexamine the Riemann Rearrangement Theorem for two types of generalized convergence. Namely, for a series \(\sum x_n\) of reals the following two sets are considered: \(RS_{st}(\sum x_n)\) consisting of all sums of statistically convergent rearrangements of \(\sum x_n\), and \(RS_{2}(\sum x_n)\) consisting of all elements of the form \(\lim_{n \to \infty} \sum_{k = 1}^{2n}x_{\pi(k)}\). It is shown that for a statistically convergent series whose statistical sum equals \(a\) exactly four possibilities can happen: (1) \(RS_{st}(\sum x_n) = \{a\}\); (2) \(RS_{st}(\sum x_n) = \{a + \lambda\mathbb Z\} \cup \{- \infty, +\infty\}\) for some \(\lambda \in \mathbb R\); (3) \(RS_{st}(\sum x_n) = \overline{\mathbb R}\); (2) \(RS_{st}(\sum x_n) = \{- \infty, a, +\infty\}\).
Surprisingly the description for \(RS_{2}(\sum x_n)\) is involved and highly non-trivial. Namely, if \(\lim_{n \to \infty} \sum_{k = 1}^{2n}x_k = a\), then \(RS_{2}(\sum x_n)\) is either (1) \(a + \{c_1 z_1 + \dots + c_l z_l: c_i \in\mathbb Z\), \(\sum_1^l c_k \in 2\mathbb Z\), \(z_i \in E\}\), where \(E\) is a separated set, or (2) the whole \(\mathbb R\), or (3) just \(\{a\}\).
A generalization of the first result to some other types of filter convergence can be found in [A. Leonov, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 826, No. 58, 134–140 (2008; Zbl 1164.40301)]. For the corresponding results for Cesàro convergence see [F. Bagemihl and P. Erdős, Acta Math. 92, 35–53 (1954; Zbl 0056.28202)] and [S. Mazur, S. Arch. Towarz. Nauk. Lwow 4, 411–424 (1929)]; for other matrix summation methods see [G. G. Lorentz and K. Zeller, Acta Math. 100, 149–169 (1958; Zbl 0085.04703)].

MSC:

40A05 Convergence and divergence of series and sequences
40A35 Ideal and statistical convergence
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References:

[1] Bagemihl, F.; Erdös, P., Rearrangements of \(C_1\)-summable series, Acta Math., 92, 35-53 (1954) · Zbl 0056.28202
[2] Connor, J.; Ganichev, M.; Kadets, V., A characterization of Banach spaces with separable duals via weak statistical convergence, J. Math. Anal. Appl., 244, 251-261 (2000) · Zbl 0982.46007
[3] Fast, H., Sur la convergence statistique, Colloq. Math., 2, 241-244 (1951) · Zbl 0044.33605
[4] Hadwiger, H., Über das Umordnungsproblem im Hilbertschen Raum, Math. Z., 46, 70-79 (1940) · JFM 66.0537.02
[5] Kadets, M.; Kadets, V., Series in Banach Spaces: Conditional and Unconditional Convergence, Oper. Theory Adv. Appl., vol. 94 (1997), Birkhäuser: Birkhäuser Basel, p. 36 · Zbl 0876.46009
[6] Lorentz, G. G.; Zeller, K., Series rearrangements and analytic sets, Acta Math., 100, 149-169 (1958) · Zbl 0085.04703
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