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On translativity of absolute weighted mean methods. (English) Zbl 0663.40003

The author studies the translativity of the absolute summability methods \(M_ q\). Given a sequence \((q_ n)\) for which \(Q_ n=q_ 0+q_ 1+...+q_ n\neq 0\) for each n, we say that the sequence \((s_ n)\) is \(M_ q\)-summable to S if \(t_ n=1/Q_ n\sum^{n}_{k=0}q_ kS_ k\) converges to S; if moreover \((t_ n)\) is of bounded variation, then we say that \((S_ n)\) is absolutely summable \(M_ q\) or summable \(| M_ q|\). A method of summability is called translative to the left if the limitability of \((S_ 0,S_ 1,...)\) implies that of \((0,S_ 1,S_ 2,...)\); A is translative to the right if the converse holds. When both conditions hold we say that A is translative. Here one gives necessary and sufficient conditions for the translativity to the left or to the right, and finally for the translativity of \(| M_ q|\)- summability.
The last two sections are devoted to produce examples and counter- examples of particular translative or not \(| M_ q|\)-methods. For the translativity to the right of regular \(M_ q\)-methods one can see H. L. Garabedian and W. C. Randels [Duke J. Math. 4, 529-533 (1938; Zbl 0019.20902)] while the translativity to the left of regular \(M_ q\)-methods has been studied by the author [J. Indian Math. Soc., New Ser. Math. 11, 1444-1457 (1980; Zbl 0455.40012)].
Reviewer: F.Barbieri

MSC:

40F05 Absolute and strong summability
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