Comparison of speeds of convergence in Riesz-type families of summability methods

Abstract

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.

Strong summability methods in a Riesz-type family

“…5 = 0. In papers [14] and [15] the speeds Ay and Xß of convergence x = x{u) with respect to methods Ay and Aß {ß > y) are compared in a Riesz-type family (see [14], Theorem 1 and [15], Theorem 2). Speed Ay = A is supposed to be a given speed and Xß = Xß {u) is defined by the relations i^^.…”

“…In the present paper the authors continue their studies on Riesz-type families started in [14] and [15]. The main idea of the paper is to define the family of strong summability methods [Aa+i]it on the basis of a given Riesz-type family {A«} (a /^^ 0:0) and to describe it with the help of different inclusion theorems.…”

“…We need the following proposition (see [10] and [14]). In the present paper the authors continue their studies on Riesz-type families started in [14] and [15].…”

“…where Cyß is the integral method defined by the function Next we introduce some examples of Riesz-type famihes (see, e.g., [15]). The methods (ß, a,^") satisfy (1.4) and (1.5) with My ^ = l/r(/3-7) (see [13]).…”

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]).…”