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Direct Theorems on Methods of Summability II

Published online by Cambridge University Press:  20 November 2018

G. G. Lorentz
G. G. Lorentz*
Affiliation:
The University of Toronto
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1.1. This paper is a continuation of the papers of the author [14], [15]. We begin by recapitulating the main definitions. If {n,} is an increasing sequence of positive integers, the value of the characteristic or the counting function ω(n) of {nv} is, for any , the number of n satisfying the inequality . Suppose that A is a linear method of summation corresponding to the transformation

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 3 , 1951 , pp. 236 - 256
Copyright
Copyright © Canadian Mathematical Society 1951

References

[1] Agnew, R. P., On equivalence of methods of evaluation of sequences, Tôhoku Math. J., vol. 35 (1932), 244–252.Google Scholar
[2] Agnew, R. P., Analytic extension of Hausdorff methods, Trans. Amer. Math. Soc, vol. 52 (1942), 217–237.Google Scholar
[3] Cooke, R. G., On mutual consistency and regular T-limits, Proc. London Math. Soc, vol. (2), 41 (1936), 113–125.Google Scholar
[4] Garabedian, H. L., Hille, E. and Wall, H. S., Formulations of the Hausdorff inclusion problem, Duke Math. J., vol. 8 (1941), 193–213.Google Scholar
[5] Hardy, G. H., Divergent series, Oxford Univ. Press (1949).Google Scholar
[6] Hardy, G. H. and Riesz, M., The general theory of Dirichlefs series, Cambridge (1915).Google Scholar
[7] Hill, J. D., Summability of sequences of Q-s and l-s,Ann. of Math., vol. (2) 46 (1945), 556–562.Google Scholar
[8] Ingham, A. E., Note on the converse of Abel-s theorem, Proc. London Math. Soc, vol. (2) 23 (1924), 326–336.Google Scholar
[9] Kakutani, Shizo, Weak convergence in uniformly convex spaces, Tôhoku Math. J., vol. 45 (1938), 188–193.Google Scholar
[10] Knopp, K., Über das Eulersche Summierungsverfahren, (II), Math. Zeitschr., vol. 18 (1923), 125–156.Google Scholar
[11] Knopp, K. and Lorentz, G. G., Beitrage zur absoluten Limitierung, Archiv der Math., vol. 2 (1949), 10–16.Google Scholar
[12] Lorentz, G. G., Beziehungen zwischen den Umkehrsdtzen der Limitierungstheorie, Bericht der Math. Tagung Tubingen, (1946), 97–99.Google Scholar
[13] Lorentz, G. G., Über Limiterungsverfahren, die von einem Stieltjes-Integral abhangen, Acta Math., vol. 79 (1947), 255–272.Google Scholar
[14] Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., vol. 80 (1948), 167–190.Google Scholar
[15] Lorentz, G. G., Direct theorems on methods of summability, Can. J. of Math., vol. 1 (1949), 305–319.Google Scholar
[16] Rado, R., Some elementary Tauberian theorems (I), Quart. J. Math. (Oxford ser.), vol. 9 (1938), 274–282.Google Scholar
[17] Zygmund, A., On certain integrals, Trans. Amer. Math. Soc, vol. 55 (1944), 170–204.Google Scholar