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Inclusion relations for Riesz typical means

Published online by Cambridge University Press:  24 October 2008

A. Jakimovski  and
J. Tzimbalario
A. Jakimovski
Affiliation:
Department of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
J. Tzimbalario
Affiliation:
Department of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
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Abstract

Necessary and sufficient conditions for sequence-to-sequence or sequence-to-function summability method to include (R, λ, α), when 1 < α ≤ 2, are given. Also, for suitably restricted sequences λ, necessary and sufficient conditions for a series-to-sequence or series-to-function summability method to include (R, λ, α) for 1 < α ≤ 2 are given. These results are obtained by showing that a certain sequence {δj} (j ≥ 0) is a Schauder-basis in Rλα(N) for each α, 1 < α ≤ 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 3 , November 1972 , pp. 417 - 423
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Borwein, D.On the abscissae of summability of a Dirichlet series. J. London Math. Soc. 30 (1955), 68–71.CrossRefGoogle Scholar
(2)Borwein, D. and Russell, D. C.On Riesz and generalized Cesàro summability of arbitrary positive order. Math. Z. 99 (1967), 171–177.CrossRefGoogle Scholar
(3)Hardy, G. H. and Riesz, M. The general theory of Dirichlet's series. Cambridge Tract No. 18; 1915, 1952.Google Scholar
(4)Kuttner, B.Some theorems on the relation between Riesz and Abel typical means. Proc. Cambridge Philos. Soc. 57 (1961), 61–75.CrossRefGoogle Scholar
(5)Maddox, I. J.Some inclusion theorems. Proc. Glasgow Math. Assoc. 6 (1964), 161–168.Google Scholar
(6)Meir, A.An inclusion theorem for generalized Cesàro and Riesz means. Canad. J. Math. 20 (1968), 735–738.CrossRefGoogle Scholar
(7)Peyerimhoff, A.Konvergeuz-und Summierbarkeitsfaktoren. Math. Z. 55 (1951), 23–54.Google Scholar
(8)Rangachari, M. S.On some generalizations of Riemann summability. Math. Z. 88 (1965), 166–188.CrossRefGoogle Scholar
Addendum, Rangachari, M. S.On some generalizations of Riemann summability. Math. Z. 91 (1966), 344–347.CrossRefGoogle Scholar
(9)Russell, D. C.On Riesz and Riemann summabiity. Trans. Amer. Math. Soc. 104 (1962), 383–391.CrossRefGoogle Scholar
(10)Russell, D. C.Summability methods which include the Riesz typical means. I. Proc. Cambridge Philos. Soc. 69 (1971), 99–106.Google Scholar
(11)Russell, D. C.Summability methods which include the Riesz typical means. II. Proc. Cambridge Philos. Soc. 69 (1971), 297–300.CrossRefGoogle Scholar
(12)Russell, D. C.Note on convergence factors. Tôhoku Math. J. 18 (1966), 414–428.CrossRefGoogle Scholar