Extended abstract theory of universal series and applications

In this article, we extend the recently developed abstract theory of universal series to include averaged sums of the form \frac1f(n)?_j=0ⁿ a_j x_j{\frac{1}{\phi(n)}\sum_{j=0}^{n} a_j x_j} for a given fixed sequence of vectors (x _j ) in a topological vector space X over a field \mathbbK{\mathbb{K}} of real or complex scalars, where (f(n)){(\phi(n))} is a sequence of non-zero field scalars. We give necessary and sufficient conditions for the existence of a sequence of coefficients (a _j ) which make the above sequence of averaged sums dense in X. When applied, the extended theory gives new analogues to well known classical theorems including those of Seleznev, Fekete and Menchoff.