Comparison of speeds of convergence in Riesz‐type families of summability methods. II
Abstract
Certain summability methods for functions and sequences are compared by their speeds of convergence. The authors are extending their results published in paper [9] for Riesz‐type families {Aα} (α > α0 ) of summability methods Aα . Note that a typical Riesz‐type family is the family formed by Riesz methods Aα = (R, α), α > 0. In [9] the comparative estimates for speeds of convergence for two methods Aγ and Aβ in a Riesz‐type family {Aα}were proved on the base of an inclusion theorem. In the present paper these estimates are improved by comparing speeds of three methods Aγ, Aβ and Aδ on the base of a Tauberian theorem. As a result, a Tauberian remainder theorem is proved. Numerical examples given in [9] are extended to the present paper as applications of the Tauberian remainder theorem proved here.
First published online: 09 Jun 2011
Keywords:
speed of convergence, Tauberian remainder theorem, Riesz‐type family of summability methods, Riesz methods, generalized integral Nörlund methods, Borel‐type methodsHow to Cite
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Copyright (c) 2010 The Author(s). Published by Vilnius Gediminas Technical University.
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Copyright (c) 2010 The Author(s). Published by Vilnius Gediminas Technical University.
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This work is licensed under a Creative Commons Attribution 4.0 International License.