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.