On dilution and Cesàro summation
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- by John R. Isbell PDF
- Proc. Amer. Math. Soc. 45 (1974), 397-400 Request permission
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 ${\alpha _i}\log {\beta _i} = o(i)$.References
- Vladimir Drobot, On the dilution of series, Ann. Polon. Math. 30 (1975), no. 3, 323–331. MR 382906, DOI 10.4064/ap-30-3-323-331
- D. Gaier, Limitierung gestreckter Folgen, Publ. Ramanujan Inst. 1 (1968/69), 223–234 (German). MR 268568
- G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
Additional Information
- © Copyright 1974 American Mathematical Society
- 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