# Applied Summability Methods by M. Mursaleen

By M. Mursaleen

This brief monograph is the 1st booklet to concentration solely at the research of summability equipment, that have turn into energetic components of analysis lately. The e-book presents simple definitions of series areas, matrix differences, average matrices and a few targeted matrices, making the cloth available to mathematicians who're new to the topic. one of the middle goods lined are the evidence of the top quantity Theorem utilizing Lambert's summability and Wiener's Tauberian theorem, a few effects on summability exams for singular issues of an analytic functionality, and analytic continuation via Lototski summability. nearly summability is brought to turn out Korovkin-type approximation theorems and the final chapters characteristic statistical summability, statistical approximation, and a few purposes of summability equipment in fastened aspect theorems.

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**Example text**

Z/ > 1, [46]. z/ D z 1. z 1/ with a suitable real , we obtain a domain of summability any given half plane, the boundary of which is a straight line passing through z D 1. / D fx W x 1g. 5 is a generalization of Okada’s theorem. Here we have the additional result about the uniform summability in compact subsets, which has proved for special domains D in [78] (see also [26, p. 189]). r/ for z D re i , where is continuous for 0 Ä r < 1. 0 Ä ˛ < 2 / is continuous. ank /. Since E 1 X jank Xk j D EjXk j kD1 1 X jank j Ä MEjXk j; kD1 P the series 1 kD0 ank Xk converges absolutely with probability one.

41]). 1). d. Let 1 X 1 e m nDmC1 n=m Xns : P Then Ym C Zm ! 0, as m ! 1, Ym and Zm are independent and symmetric. P Therefore it follows easily from the Levy’s inequality [57, p. 247] that Zm ! 0. Y1 ; : : : ; Ym / are independent and Ym C Zm ! , Zm ! 1/ obtain by Lemma 3 of [23] that Ym ! e. m Xm / ! 0, we have by Lemma 3 of [10] that Xms =m ! e. By the BorelCantelli lemma, this implies that EjX1 j < 1. As established before, we then have Xn ! 1. 2. Chow [22] has shown that unlike the Cesàro and Abel methods which require EjX1 j < 1 for summability, the Euler and Borel methods require EX12 < 1 for summability.

1C 1 / ; and the final estimate will converge when summed on n provided that ˛ < = 2. 4. 5 ([81, Lemma 3]). If Bn , then for every > 0; 1 X nD0 1 D 0; EjX j1C < 1, and maxk2N jank j Ä ˇ "ˇ ˇX0 ˇ ˇ ˇ P ˇ ank Xk ˇ ˇ ˇ # < 1; k where X0 X ank Xk D ank Xk ; ŒkWjank Xk j