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Summation and absolute summation of numeric sequences by Riesz methods. (English. Russian original) Zbl 0714.40008

Sov. Math. 33, No. 12, 93-95 (1989); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1989, No. 12(331), 76-78 (1989).
Let \(T=(a_{nk})\) be an infinite matrix of real numbers. A sequence \(s=\{s_ n\}\) of real numbers is said to be T-limitable to \(\sigma\) (T- lim \(s_ n=\sigma)\) if \(\lim_{n\to \infty}\sigma_ n=\sigma\), where \(\sigma_ n=\sum^{\infty}_{k=0}a_{nk}s_ k\) \((n=0,1,...)\). It is well-known that if \(s=\{s_ n\}\) is an arbitrary divergent sequence and x an arbitrary number then there is a matrix t such that \(T-\lim s_ n=x.\) If we restrict ourselves only to non-negative matrices \(T=(a_{nk})\) (i.e. if \(a_{nk}\geq 0\) for all n,k), then for each divergent sequence \(s=\{s_ n\}\) and each x with \(\lim_{n\to \infty}\inf s_ n\leq x\leq \lim_{n\to \infty}\sup s_ n\) there exists a non-negative matrix t such that T-lim \(s_ n=x\). The author investigates similar questions for Riesz matrices given by increasing sequences \(\{\lambda_ n\}\) of positive real numbers with \(\lambda_ n\to +\infty\). Here we have \(\sigma_ n=(1/\lambda_{n+1})\sum^{n}_{k=0}(\lambda_{k+1}-\lambda_ k)s_ k\) \((n=0,1,...)\). The author shows that the set of all divergent sequences can be divided into two classes \(S_ 1\), \(S_ 2\) so that if \(s=\{s_ n\}\in S_ 1\) then for each x, \(\lim_{n\to \infty}\inf s_ n\leq x\leq \lim_{n\to \infty}\sup s_ n\) there is a Riesz method T with T-lim \(s_ n=x\) and if \(s=\{s_ n\}\in S_ 2\) then there exists no Riesz method T by which the sequence s would be T-limitable to any finite real number.
Reviewer: T.Šalať

MSC:

40G05 Cesàro, Euler, Nörlund and Hausdorff methods
40G99 Special methods of summability
40C05 Matrix methods for summability

Keywords:

Riesz matrices
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