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On the consistency and relative strength of regular summability methods

Published online by Cambridge University Press:  24 October 2008

J. Copping
J. Copping
Affiliation:
University of Exeter
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Extract

Statement of results. Let A and B be two matrix summability methods, and S a given set of sequences. We shall say that B is S-stronger than A if B sums every sequence which belongs to S and is summed by A. If each of A, B is S-stronger than the other, A and B will be called S-equivalent. If B is S-stronger than A, but A is not S-stronger than B, we say that B is strictly S-stronger than A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 3 , July 1966 , pp. 421 - 428
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Baker, J. W. and Petersen, G. M.Inclusion of sets of regular summability matrices. Proc. Cambridge Philos. Soc. 60 (1964), 705–712.Google Scholar
(2)Baker, J. W. and Petersen, G. M.Inclusion of sets of regular summability matrices. II. Proc. Cambridge Philos. Soc. 61 (1965), 381–394.Google Scholar
(3)Brudno, A. L.Summation of bounded sequences. Mat. Sb. 16 (1945), 191–247 (in Russian).Google Scholar
(4)Copping, J.Conditions for a K-matrix to evaluate some bounded divergent sequences. J. London Math. Soc. 32 (1957), 217–227.CrossRefGoogle Scholar
(5)Darevsky, V. M.On intrinsically perfect methods of summation. Bull. Acad. Sci. U.R.S.S., Ser. Math. ( = Izv. Akad. Nauk SSSR), 10 (1946), 97–104.Google Scholar
(6)Hardy, G. H.An extension of a theorem on oscillating series. Proc. London Math. Soc. 12 (1913), 174–180.Google Scholar
(7)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(8)Mazur, S. and Orlicz, W.Sur les méthodes linéaires de sommation. Comptes Rendus, 196 (1933), 32–34.Google Scholar
(9)Mazur, S. and Orlicz, W.On linear methods of summability. Studia Math. 14 (1954), 129–160.Google Scholar
(10)Petersen, G. M.Summability methods and bounded sequences. J. London Math. Soc. 31 (1956), 324–326.CrossRefGoogle Scholar
(11)Petersen, G. M.Consistency of summation matrices for unbounded sequences. Quart. J. Math. Oxford Ser. 14 (1963), 161–169.CrossRefGoogle Scholar
(12)Pitt, H. R.Tauberian theorems. Tata Institute Monograph No. 2 (Oxford–Bombay, 1958).Google Scholar
(13)Jurkat, W. and Peyerimhoff, A.Mittelwertsätze und vergleichssätze für matrixtrans-formationen. Math. Z. 56 (1952), 152–178.Google Scholar