Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-17T23:27:52.083Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Remarks Concerning (f, dn) and [F, dn] Summability Methods

Published online by Cambridge University Press:  20 November 2018

C. F. Koch
C. F. Koch*
Affiliation:
Southern Illinois University, Carbondale, Illinois
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.

In this paper we note a simple connection between the (ƒ, dn) method of summability (defined by Smith (8)) and a composition of [F, dn] (defined by Jakimovski (4)) and Sonnenschein methods (9; 10). This connection is then used to supply some sufficient conditions for the regularity of (ƒ, dn) methods by using known regularity conditions for various [F, dn] and Sonnenschein methods. Finally, the connection is further exploited to obtain information about the Lebesgue constants for a certain class of [F, dn] methods by investigating related (ƒ, dn) methods.

2. Definitions and the regularity theorem. The (ƒ, dn) method of summability is defined by Smith (8) essentially as follows. Letƒ(z) be a nonconstant entire function satisfying ƒ (l) = 1 and let {dn} be a sequence of complex numbers satisfying dt ≠ – 1, di ≠ – ƒ(0) (i ≧ 1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1361 - 1365
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Agnew, R. P., Comparison of products of methods of summability, Trans. Amer. Math. Soc. 43 (1938), 327–343.Google Scholar
2. Bajsanski, B., Sur une classe générale de procédés de sommations du type d'Euler-Borel, Publ. Inst. Math. (Beograd) X (1956), 131-152.Google Scholar
3. Cowling, V. F. and Miracle, C. L., Some results for the generalized Lototsky transform, Can. J. Math. 14 (1962), 418–435.Google Scholar
4. Jakimovski, A., A generalization of the Lototsky method of summability, Michigan Math. J. 6 (1959), 277–290.Google Scholar
5. Lorch, L. and Newman, D. J., On the [F, dn] summation of Fourier series, Comm. Pure Appl. Math. 15 (1962), 109–118.Google Scholar
6. Miracle, C. L., Some regular [F, dn] matrices with complex elements, Can. J. Math. 15 (1963), 503–535.Google Scholar
7. Sledd, W. T., Regularity methods for Karamata matrices, J. London Math. Soc. 38 (1963), 105–107.Google Scholar
8. Smith, Gaston, On the (f, dn) method of summability, Can. J. Math. 17 (1965), 506–526.Google Scholar
9. Sonnenschein, J., Sur les séries divergentes, Thèses, Bruxelles, 1946.Google Scholar
10. Sonnenschein, J., Sur les séries divergentes, Bull. Acad. Roy. Belg. 35 (1949), 594–601.Google Scholar