Comparison of Speeds of Convergence in Riesz‐type Families of Summability Methods. Ii

Abstract

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 A β 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, 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.

Characterization of some matrix classes involving some sets with speed

“…The necessary and sufficient conditions for A 2 .m ; m /, A 2 .c ; c /, and A 2 .c ; m /, were first introduced by Kangro ( [5], [6], [7]). The estimation and the comparison of speeds of convergence of series and sequences, based on Kangro's concepts of convergence, boundedness, and summability with speed, have also been studied byŠeletski and Tali ( [9], [10]), Stadtmüller and Tali ( [11]), and Tammeraid ([12], [13], [14], [15]). For more results on matrix transforms of m and c , one can refer to ( [5], [7], [8]).…”

Tauberian Remainder Theorems for the Weighted Mean Method of Summability

“…A number of authors including Kangro [4], Tammeraid [11] andŠeletski and Tali [10] proved Tauberian remainder theorems for some summability methods using summability with given rapidity. Meronen and Tammaraid [5] obtained a Tauberian condition under which λ-boundedness of {ξ n } transformed by the generalized Euler-Knopp method implies λ-boundedness of {ξ n }.…”