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Convergence and summability factors in a sequence (II)

Part of:Direct theorems on summability

Published online by Cambridge University Press:  26 February 2010

L. S. Bosanquet
L. S. Bosanquet
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WC1E 6BT
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Extract

1.1. In an earlier paper [6] the author proved

Theorem A. If α ≥ 0, β ≥ 0 (α, β integers), p > –1, p–r > –1 then necessary and sufficient conditions for a sequence (εn) to be such that

are

and

If α ≥ 1 (α an integer) and (ii) holds, then (i)bis equivalent to

MSC classification

Secondary:40D15: Convergence factors and summability factors

Information

Type
Research Article
Information
Mathematika , Volume 30 , Issue 2 , December 1983, pp. 255 - 273
Copyright
Copyright © University College London 1983

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References

1.Andersen, A. F.. Studier over Cesàro's summabititetsmetode (Copenhagen Thesis, 1921).Google Scholar
2.Andersen, A. F.. Comparison theorems in the theory of Cesàro summability. Proc. London Math. Soc. (2), 27 (1928), 39–71.CrossRefGoogle Scholar
3.Andersen, A. F.. On the extensions within the theory of Cesaro summability of a classical convergence theorem of Dedekind. Proc. London Math. Soc. (3), 8 (1958), 1–52.CrossRefGoogle Scholar
4.Bosanquet, L. S.. Note on the Bohr-Hardy theorem. J. London Math. Soc. (1), 17 (1942), 166–173.CrossRefGoogle Scholar
5.Bosanquet, L. S.. Note on convergence and summability factors (III). Proc. London Math. Soc. (2), 50 (1949), 482–496.Google Scholar
6.Bosanquet, L. S.. On convergence and summability factors in a sequence. Mathematika, 1 (1954), 24–44.CrossRefGoogle Scholar
7.Bosanquet, L. S.. On the order of magnitude of fractional differences. The Golden Jubilee Commemoration Volume, Calcutta Math. Soc, (1958-1959), 161–172.Google Scholar
8.Bosanquet, L. S.. An inequality for sequence transformations. Mathematika, 13 (1966), 26–41.CrossRefGoogle Scholar
9.Bosanquet, L. S. and Tatchell, J. B.. A note on summability factors. Mathematika, 4 (1957), 25–40.CrossRefGoogle Scholar
10.Chow, H. C.. Note on convergence and summability factors. J. London Math. Soc. (1), 29 (1954), 459–476.CrossRefGoogle Scholar
11.Hardy, G. H.. Divergent series (Oxford University Press, 1949).Google Scholar
12.Knopp, K.. Beweis eines von I. Schur in die Theorie der C-Summierbarkeit aufgestellten Satzes. J.für Math., 187 (1949), 70–74.Google Scholar
13.Kuttner, B.. On differences of fractional order. Proc. London Math. Soc. (3), 7 (1957), 453–466.CrossRefGoogle Scholar