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

  • C. F. Koch (a1)

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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).

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References

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1. Agnew, R. P., Comparison of products of methods of summability, Trans. Amer. Math. Soc. 43 (1938), 327343.
2. Bajsanski, B., Sur une classe générale de procédés de sommations du type d'Euler-Borel, Publ. Inst. Math. (Beograd) X (1956), 131-152.
3. Cowling, V. F. and Miracle, C. L., Some results for the generalized Lototsky transform, Can. J. Math. 14 (1962), 418435.
4. Jakimovski, A., A generalization of the Lototsky method of summability, Michigan Math. J. 6 (1959), 277290.
5. Lorch, L. and Newman, D. J., On the [F, dn] summation of Fourier series, Comm. Pure Appl. Math. 15 (1962), 109118.
6. Miracle, C. L., Some regular [F, dn] matrices with complex elements, Can. J. Math. 15 (1963), 503535.
7. Sledd, W. T., Regularity methods for Karamata matrices, J. London Math. Soc. 38 (1963), 105107.
8. Smith, Gaston, On the (f, dn) method of summability, Can. J. Math. 17 (1965), 506526.
9. Sonnenschein, J., Sur les séries divergentes, Thèses, Bruxelles, 1946.
10. Sonnenschein, J., Sur les séries divergentes, Bull. Acad. Roy. Belg. 35 (1949), 594601.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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