Let $(x_k)$, for $k\in \mathbb{N}\cup \{0\}$  be a sequence of real or complex numbers and set $(EC)_{n}^{1}=\frac{1}{2^n}\sum_{j=0}^{n}{\binom{n}{j}\frac{1}{j+1}\sum_{v=0}^{j}{x_v}},$ $n\in \mathbb{N}\cup \{0\}.$  We present necessary and sufficient conditions, under which $st-\lim_{}{x_k}= L$ follows from $st-\lim_{}{(EC)_{n}^{1}} = L,$ where L is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

The main goal of this article is to investigate the integration theory of Mcshane type for functions with values in ordered spaces. Afected by the work of Boccuto, Riecan, and Vrábelová with Kurzweil-Henstock integration we studied the same problems for another important type of integration on such space as Mcshane ones.In this paper we present in other way the definition of Mcshane integral on the Riesz space using a very important lemma of famous Fremlin. In the second section we reconstruct allmost all the propeties of Mchane integral given in [5], [6] and these ones become a little more stronger. We arrive some new results compared with Henstock-Kurzweil ones. In the third section we define the strong version of Mcshane integral and give the neccesary and sufficent condition of this concept. In the fourth section we prove the fundamental theorems of Calculus for the D‫-ܯ‬integral and in the fifth we extended an application of this integration to Walsh series .Mathematics Subject Classification: 28B15, 28B05, 28A39, 42C10, 42C25, 46G10

No abstract

No abstract

No abstract