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Proceedings of the American Mathematical Society

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On dilution and Cesàro summation


Author: John R. Isbell
Journal: Proc. Amer. Math. Soc. 45 (1974), 397-400
MSC: Primary 40G05
DOI: https://doi.org/10.1090/S0002-9939-1974-0350250-2
MathSciNet review: 0350250
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Abstract: The problem whether a real sequence $ ({s_i})$ has a dilution which is $ (C,1)$ summable to a number $ s$ is transformed by means of two sequences measuring the oscillation of $ ({s_i})$ about $ s$. (If it does not oscillate, the condition, known, is that $ s$ is a limit point of $ ({s_i})$.) For the $ j$th consecutive block of $ {s_i}$ on one side of $ s,{\alpha _j}$ is the minimum of their distances from $ s,{\beta _j}$ the sum of distances. Then there must exist positive numbers $ {p_j}$ such that $ {\beta _j} + {p_j}{\alpha _j} = o({p_1} + \cdots + {p_{j - 1}})$. The necessary condition and the sufficient condition coincide for very smooth sequences at $ {\alpha _i}\log {\beta _i} = o(i)$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0350250-2
Article copyright: © Copyright 1974 American Mathematical Society

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