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

Published online by Cambridge University Press:  20 November 2018

C. F. Koch
C. F. Koch*
Affiliation:
Southern Illinois University, Carbondale, Illinois
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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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1361 - 1365
Copyright
Copyright © Canadian Mathematical Society 1969

References

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