A Tauberian theorem for ideal statistical convergence

Abstract

Given an ideal $\mathcal{I}$ on the positive integers, a real sequence $(x_n)$ is said to be $\mathcal{I}$-statistically convergent to $\ell$ provided that $$\left\{n\in \mathbf{N}:\frac{1}{n}|\{k\le n:x_k\notin U\}|\ge \epsilon \right\}\in \mathcal{I}$$ for all neighborhoods $U$ of $\ell$ and all $\epsilon >0$. First, we show that $\mathcal{I}$-statistical convergence coincides with $\mathcal{J}$-convergence, for some unique ideal $\mathcal{J}=\mathcal{J}(\mathcal{I})$. In addition, $\mathcal{J}$ is Borel [analytic, coanalytic, respectively] whenever $\mathcal{I}$ is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if $\mathcal{I}$ is the summable ideal $\{A\subseteq \mathbf{N}:{\sum }_{a\in A}1/a<\infty \}$ or the density zero ideal $\{A\subseteq \mathbf{N}:{\lim }_{n\to \infty }\frac{1}{n}|A\cap [1,n]|=0\}$ then $\mathcal{I}$-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if $\mathcal{I}$ is maximal.

15 pages, comments are welcome