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We deal with Riesz-type families (see Proc. ) of summability methods A α for converging functions and sequences. The methods A α in a Riesztype family depend on a continuous parameter α, and are connected through certain generalized integral Nörlund methods. By extending and applying the results of Stadtmüller and Tali (Anal. Math., 2003, 29, 227-242), we compare speeds of convergence in a Riesz-type family. As expected, the speed of convergence cannot increase if we switch from one summability method to a stronger one. Comparative estimations for speeds are found. In particular, the families of integral Riesz methods, generalized integral Nörlund methods, and Abel-and Borel-type summability methods are considered. Numerical examples are given.

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.

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The paper deals with certain families {A?}(?>?0) of summability methods. Strong and statistical convergences in Ces?ro- and Euler-Knopp-type families {A?} are investigated. Convergence of a sequence x = (xn) with respect to the different strong summability methods [A?+1]t (with positive exponents t = (tn)) in a family {A?} is compared, and characterized with the help of statistical convergence. A convexity theorem for comparison of three strong summability methods [A?+1]t, [A?+1]t and [A?+1]t (? > ? > ? > ?0) in a Ces?ro-type family {A?} is proved. This theorem can be seen as a generalization of some convexity theorems known earlier. Interrelations between strong convergence and certain statistical convergence are also studied and described with the help of theorems proved here. All the results can be applied to the families of generalized N?rlund methods (N, p?n, qn).