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 {A_α} (α > α₀ ) of summability methods A_α . Note that a typical Riesz‐type family is the family formed by Riesz methods A_α = (R, α), α > 0. In [9] the comparative estimates for speeds of convergence for two methods A_γ and A_β in a Riesz‐type family {A_α} were proved on the base of an inclusion theorem. In the present paper these estimates are improved by comparing speeds of three methods A_γ, A_β and A_δ 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.

Keywords:

• speed of convergence
• Tauberian remainder theorem
• Riesz‐type family of summability methods
• Riesz methods
• generalized integral Nörlund methods
• Borel‐type methods

Notes

This research was supported by the Estonian Science Foundation (grants No. 7033 and No. 7319).