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Register a new userA Tauberian theorem for ideal statistical convergence
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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 $$ textstyle left{n in mathbf{N}: frac{1}{n}|{k le n: x_k otin U}| ge varepsilonright} in mathcal{I} $$ for all neighborhoods $U$ of $ell$ and all $varepsilon>0$. 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 ${Asubseteq mathbf{N}: sum_{a in A}1/a<infty}$ or the density zero ideal ${Asubseteq mathbf{N}: lim_{nto infty} frac{1}{n}|Acap [1,n]|=0}$ 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.
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