Hostname: page-component-7479d7b7d-jwnkl Total loading time: 0 Render date: 2024-07-13T09:20:33.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Theorems on Strong Riesz Summability Factors

Published online by Cambridge University Press:  20 January 2009

B. L. R. Shawyer
B. L. R. Shawyer
Affiliation:
The University, Nottingham
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Throughout this note it will be assumed that

(i) unless otherwise stated, k; is a positive integer,

(ii) a ≧ 0; m > − 1,

(iii) q ≧ 1; q′ is the conjugate number to q, and is defined by

(iv) the functions φ(w), ψ(w) are defined in [0, ∞) with absolutely continuous kth derivatives in every interval [a, W],

(v) φ(w) is non-negative and unboundedly increasing,

(vi) λ = {λn} is an unboundedly increasing sequence with λ1 > 0.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 15 , Issue 1 , June 1966 , pp. 19 - 27
Copyright
Copyright © Edinburgh Mathematical Society 1966

References

REFERENCES

(1)Ahmed, Z. U.On absolute Riesz summability factors, Rend. Circ. Math. Palermo, II, XI (1962), 91–104.CrossRefGoogle Scholar
(2)Borwein, D.A theorem on Riesz summability, J. London Math. Soc. 31 (1956), 319–324.CrossRefGoogle Scholar
(3)Borwein, D. and Shawyer, B.L. R.On Riesz summability factors, Proc. Glasgow Math. Assoc. 5 (1962), 188–196.CrossRefGoogle Scholar
(4)Borwein, D. and Shawyer, B. L. R.On absolute Riesz summability factors, J. London Math. Soc. 39 (1964), 455–465.CrossRefGoogle Scholar
(5)Borwein, D. and Shawyer, B. L. R.On strong Riesz summability factors, London Math. Soc. 40 (1965), 111–126.CrossRefGoogle Scholar
(6)Dikshtt, G. D.On the absolute summability factors of infinite series, Indian J. Math. 1 (1958), 33–40.Google Scholar
(7)Dikshit, G. D.On the absolute summability factors of infinite series, II, Proc. Nat. Inst. Sci. India A 26 (1960), 86–94.Google Scholar
(8)Glatfeld, M.On strong Rieszian summability, Proc. Glasgow Math. Assoc. 3 (1957), 123–131.CrossRefGoogle Scholar
(9)Guha, U. C.Convergence factors for Riesz summability, J. London Math. Soc. 31(1956), 311–319.CrossRefGoogle Scholar
(10)Hardy, G. H.Orders of Infinity (Cambridge Tract No. 12).Google Scholar
(11)Hsiang, Fu ChengOn the strong Riesz summability of an infinite series, Indian J. Math. 4 (1962), 47–52.Google Scholar
(12)Srivastava, P.On strong Rieszian summability of infinite series, Proc. Nat. Inst. Sci. India A 23 (1957), 58–71.Google Scholar
(13)Srivastava, P.On the second theorem of consistency for strong Riesz summability, Indian J. Math. 1 (1958), 1–16.Google Scholar
(14)srivastava, P.On the second theorem of consistency for strong Riesz summability (IT), Indian J. Math. 2 (1960), 97–117.Google Scholar
(15)Srivastava, P.Theorems on strong Riesz summability, Quarterly J. Math. (2) 11 (1960), 229–240.CrossRefGoogle Scholar
(16)de la Vallee Poussin, C-J.Cours d'analyse infinitesimale (Louvain, Paris).Google Scholar