Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-28T14:58:42.609Z Has data issue: false hasContentIssue false
  • English
  • Français

Summability methods which include the Riesz typical means. II

Published online by Cambridge University Press:  24 October 2008

B. C. Russell
B. C. Russell
Affiliation:
York University, Toronto
Get access Rights & Permissions [Opens in a new window]

Extract

By making use of a convergence-factor theorem of Bosanquet(3), Cooke((4), Theorem I) gave conditions for a regular sequence-to-sequence summability matrix B to be at least as strong as Cesàro summability (C, κ) (κ > 0), namely:

Theorem C. Let κ > 0. In order that the T-matrix B = (bρμ) shall satisfy B ⊇ (C, κ) it is necessary and sufficient that

If 0 < κ ≤ 1 then (2) alone is necessary and sufficient.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 2 , March 1971 , pp. 297 - 300
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Borwein, D.On a method of summabiity equivalent to the Cesàro method. J. London Math. Soc. 42 (1967), 339–343.CrossRefGoogle Scholar
(2)Borwein, D. and Russell, D. C.On Riesz and generalised Cesàro summabiity of arbitrary positive order. Math. Z. 99 (1967), 171–177.CrossRefGoogle Scholar
(3)Bosanquet, L. S.Note on convergence and summability factors. III. Proc. London Math. Soc. (2) 50 (1949), 482–496.Google Scholar
(4)Cooke, R. G.On T-matrices at least as efficient as (C, r) suinmability and Fourier-effective methods of summation. J. London Math. Soc. 27 (1952), 328–337.CrossRefGoogle Scholar
(5)Lorentz, G. G.Eine Bemerkung über Limitierungsverfahren die nicht schwächer als em Cesàro-Verfahren sind. Math. Z. 51 (1948), 85–91.CrossRefGoogle Scholar
(6)Maddox, I. J.Some inclusion theorems. Proc. Glasgow Math. Assoc. 6 (1964), 161–168.CrossRefGoogle Scholar
(7)Meir, A.An inclusion theorem for generalized Cesàro and Riesz means. Canad. J. Math. 20 (1968), 735–738.CrossRefGoogle Scholar
(8)Russell, D. C.Note on inclusion theorems for infinite matrices. J. London Math. Soc. 33 (1958), 50–62.CrossRefGoogle Scholar
(9)Russell, D. C.On generalized Cesàro means of integral order. Tôhoku Math. J. (2) 17 (1965), 410–442. Corrigenda, 18 (1966), 454–455.CrossRefGoogle Scholar
(10)Russell, D. C.Inclusion theorems for section-bounded matrix transformations. Math. Z. 113 (1970), 255–265.CrossRefGoogle Scholar
(11)Russell, D. C.Summability methods which include the Riesz typical means. I. Proc. Cambridge Philos. Soc. 69 (1970), 99–106.CrossRefGoogle Scholar
(12)Russell, D. C. Note on convergence factors, II. Indian J. Math. (to appear).Google Scholar