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Matrix Summability of Geometrically Dominated Series

Published online by Cambridge University Press:  20 November 2018

G. H. Fricke  and
J. A. Fridy
G. H. Fricke
Affiliation:
Wright State University, Dayton, Ohio
J. A. Fridy
Affiliation:
Kent State University, Kent, Ohio
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 3 , 01 June 1987 , pp. 568 - 582
Copyright
Copyright © Canadian Mathematical Society 1987

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