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 | References | Similar Articles | Additional Information
Abstract: The problem whether a real sequence has a dilution which is
summable to a number
is transformed by means of two sequences measuring the oscillation of
about
. (If it does not oscillate, the condition, known, is that
is a limit point of
.) For the
th consecutive block of
on one side of
is the minimum of their distances from
the sum of distances. Then there must exist positive numbers
such that
. The necessary condition and the sufficient condition coincide for very smooth sequences at
.
- [1] V. Drobot, On the dilution of series, Ann. Polon. Math. (to appear). MR 0382906 (52:3788)
- [2] D. Gaier, Limitierung Gestreckter Folgen, Publ. Ramanujan Inst. No. 1 (1968/69), 223-234. MR 42 #3465. MR 0268568 (42:3465)
- [3] G. H. Hardy, Divergent series, Clarendon Press, Oxford, 1949. MR 11, 25. MR 0030620 (11:25a)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1974-0350250-2
Article copyright:
© Copyright 1974
American Mathematical Society