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A Note on Strong Riesz Summability

Published online by Cambridge University Press:  20 November 2018

B. Thorpe
B. Thorpe*
Affiliation:
Department of Pure Mathematics, The University, Birmingham. B15 2TT, England
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Abstract

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This note proves that if 1 ≤ p < ∞ and 1 − 1/p < k < 2 − 1/p then the space of sequences strongly Riesz summable [R, λ, k]p to 0 has AK. Using general results of Jakimovski and Russell it is then possible to deduce a best possible limitation condition and a convergence factor result for [R, λ, k]p.

Keywords

40F0540H05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 3 , 01 September 1982 , pp. 263 - 272
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Borwein, D., Linear Functionals connected with strong Cesàro summability, Journal London Math. Soc, 40 (1965), 628-634.Google Scholar
2. Bosanquet, L. S., Some extensions of M. Riesz's mean value theorem, India. J. Math. 9 (1967), 65-90.Google Scholar
3. Glatfeld, M., On strong Rieszian summability, Proc. Glasgow Math. Assoc. 3 (1957), 123-131.Google Scholar
4. Hardy, G. H. and Riesz, M., The general theory of Dirichlefs series, Cambridg. Tract. No.18, (1915).Google Scholar
5. Jakimovski, A. and Russell, D. C., Best order conditions in linear spaces, Studia Math. 51 (1976), 101-120.Google Scholar
6. Jakimovski, A. and Tzimbalario, J., Inclusion relations for absolute Riesz means and a conjecture of Maddox, Proc. London Math. Soc. (3) 30 (1975), 366-384.Google Scholar
7. Kratz, W. and Shawyer, B. L. R., A strong-Riesz mean-value theorem with applications to summability factor theorems, J. reine u. angew. Math. 298 (1978) 16-31.Google Scholar
8. Kuttner, B. and Thorpe, B., A note on strong Cesàro summability, to appear in Acta Mathematica (Hungary).Google Scholar
9. Peyerimhoff, A., Konvergenz und Summierbarkeitsfaktoren Math. Z. 55 (1951), 23-54.Google Scholar
10. Shawyer, B. L. R., On the strong-Riesz mean-value theorem, J. reine u. angew. Math. 306 (1979) 208-211.Google Scholar
11. Srivastava, P., On strong Rieszian summability of infinite series, Proc. Nat. Inst. Sci. India (A) 23 (1957), 58-71.Google Scholar
12. Zeller, K., Abschnittskonvergenz in FK-Raumen, Math. Z. 55 (1951), 55-70.Google Scholar