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Direct Theorems on Methods of Summability

Published online by Cambridge University Press:  20 November 2018

G. G. Lorentz
G. G. Lorentz*
Affiliation:
University of Toronto
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1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular methods 1.1(1) or at least by all such methods subject to some simple additional conditions. Questions of this kind have been treated by the author (Lorentz [2], [5]) and, from another point of view, by R. P. Agnew [2] [3] ; in this paper we wish to discuss the problem systematically.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 4 , 01 August 1949 , pp. 305 - 319
Copyright
Copyright © Canadian Mathematical Society 1949

References

Agnew, R. P., [1] “Analytic extension by Hausdorff methods,” Trans. Amer. Math. Soc, vol. 52 (1942), 217-237.Google Scholar
Agnew, R. P., [2] “Convergence fields of methods of summability,” Ann. of Math., vol. (2) 46 (1945), 93-101.Google Scholar
Agnew, R. P., [3] “A simple sufficient condition that a method of summability be stronger than convergence,” Bull. Amer. Math. Soc, vol. 52 (1946), 128–132.Google Scholar
Amerio, L., [1] Un metodo di sommazione per le serie di potenze,” Ann. Scuola norm, sup., Pisa, Set. fis. mat., vol. (2) 8 (1939), 167-180.Google Scholar
Bernstein, S., [1] “Sur la convergence de certaines suites des polynomes,” J. Math. Pures Appl., vol. (9) 15 (1935), 345–358.Google Scholar
Boas, R. P., Jr., [1] “Tauberian theorems for (C, 1) summability,” Duke Math. J., vol. 4 (1938), 227-230.Google Scholar
Karamata, J., [1] “Eine weitere Umkehrung des Cesàroschen Limitierungsverfahrens,” Glas Srpske Akad., vol. 163 (80) (1935), 59-70.Google Scholar
Lorentz, G. G., [1] “Zur Theorie der Polynome von S. Bernstein,” Rec. Math., n.s. vol. 2 (1937), 543-556.Google Scholar
Lorentz, G. G., [2] “Beziehungen zwischen den Umkehrsätzen der Limitierungstheorie,” Bericht der Math. Tagung, Tübingen, (1946), 97-99.Google Scholar
Lorentz, G. G., [3] “Über Limitierungsverfahren, die von einem Stieltjes-Integral abhängen,” Acta Math., vol. 79 (1947), 255-272.Google Scholar
Lorentz, G. G., [4] “Tauberian theorems and Tauberian conditions,” Trans. Amer. Math. Soc, vol. 63 (1948), 226–234.Google Scholar
Lorentz, G. G., [5] “A contribution to the theory of divergent series,” Acta-Math., vol. 80 (1948), 167– 190.Google Scholar
Orlicz, W., [1] “Zur allgemeinen Limitierungstheorie,” Tôhoku Math. J., vol. 26 (1926), 233-237.Google Scholar
Ananda Rau, K., [1] “On the convergence and summability of Dirichlet's series,” Proc. London Math. Soc. lis., vol. 34 (1932), 414-440.Google Scholar
Rey-Pastor, J., [1] “Un método de sumacion de series,” Rend. Cire. Mat. Palermo, vol. 55 (1931), 450-455.Google Scholar
Schur, I., [1] “Über lineare Transformationen in der Théorie der unendlichen Reihen,” J. Reine Angew. Math., vol. 151 (1921), 79-111.Google Scholar