Geodesic
Matrix Summability of Geometrically Dominated Series
Canadian journal of mathematics, Tome 39 (1987) no. 3, pp. 568-582

Voir la notice de l'article provenant de la source Cambridge University Press

One of the common uses of summability theory is found in its applications to power series. A partial listing such as [l]-[5], [13], and [15]-[17] might serve to remind us of the many instances of summability theory applied to power series. In some studies, the summability transformation is applied to the sequence of partial sums of the power series, while in others it is applied to the general term akzk as a series-to-series transformation. In [2] and [5] the transformation is applied to the coefficient sequence of the Taylor series. In the present study we investigate matrix transformations that are applied to the sequence {akzk ), but we are not concerned with the usual preservation of convergence or sums. At a point within its disc of convergence, a power series exhibits more than ordinary convergence; it converges very rapidly, being dominated by a convergent geometric series. 
Fricke, G. H.; Fridy, J. A. Matrix Summability of Geometrically Dominated Series. Canadian journal of mathematics, Tome 39 (1987) no. 3, pp. 568-582. doi: 10.4153/CJM-1987-026-9
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[1] 1. Agnew, R. P., Summability of power series, Amer. Math. Monthly 53 (1946), 251–259. Google Scholar

[2] 2. Brown, H. I., Entire methods of summation, Comp. Math. 21 (1969), 35–42. Google Scholar

[3] 3. Cowling, V. F., Summability and analytic continuation, Proc. Amer. Math. Soc. 1 (1950), 536–542. Google Scholar

[4] 4. Dienes, P., The Taylor series (Dover, New York, 1957). Google Scholar

[5] 5. Fricke, G. and Powell, R. E., A theorem on entire methods of summation, Comp. Math. 22 (1970), 253–259. Google Scholar

[6] 6. Fridy, J. A., Mercerian-type theorems for absolute summability, Port. Math. 33 (1974), 141–145. Google Scholar

[7] 7. Fridy, J. A., Absolute summability matrices that are stronger than the identity mapping, Proc. Amer. Math. Soc. 47 (1975), 112–118. Google Scholar

[8] 8. Fridy, J. A., Abel transformations into l1 Can. Math. Bull. 25 (1982), 421–427. Google Scholar

[9] 9. Fridy, J. A. and Roberts, K. L., Some Tauberian theorems for Euler and Borel summability, Internat. J. Math. & Math. Sci. 3 (1980), 731–738. Google Scholar

[10] 10. Hardy, G. H., Divergent series (Clarendon Press, Oxford, 1949). Google Scholar

[11] 11. Knopp, K., Theory and application of infinite series (Blackie & Son, London, 1928). Google Scholar

[12] 12. Knopp, K. and Lorentz, G. G., Beiträge zur absolut en Limitierungen, Arch. Math. 2 (1949), 10–16. Google Scholar

[13] 13. Okada, Y., Über die Annaherung and analytischer Funktionen, Math. Zeit. 23 (1925), 62–71. Google Scholar

[14] 14. Powell, R. E. and Shah, S. M., Summability theory and its applications (Van Nostrand Reinhold, London, 1972). Google Scholar

[15] 15. Riesz, M., Sur les séries de Dirichlet et les entières, Comp. Rend. 149 (1909), 909–912. Google Scholar

[16] 16. Riesz, M., Über einen Satz des Herrn Fatou, Crelle J. 14 (1911), 89–99. Google Scholar

[17] 17. Tomm, Ludwig, A regular summability method which sums the geometric series to its proper value in the whole complex plane, Can. Math. Bull. 26 (1983), 179–188. Google Scholar

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