Abstract
Given an ideal I on the positive integers, a real sequence (xn) is said to be I-statistically convergent to ℓ provided that n∈N:[Formula presented]|{k≤n:xk∉U}|≥ε∈Ifor all neighborhoods U of ℓ and all ε>0. First, we show that I-statistical convergence coincides with J-convergence, for some unique ideal J=J(I). In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if I is the summable ideal {A⊆N:∑a∈A1∕a<∞} or the density zero ideal {A⊆N:limn→∞[Formula presented]|A∩[1,n]|=0} then 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 I is maximal.
Original language | English |
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Pages (from-to) | 83-95 |
Number of pages | 13 |
Journal | Indagationes Mathematicae |
Volume | 31 |
Issue number | 1 |
DOIs | |
Publication status | Published - Jan 2020 |
Keywords
- Generalized density ideal
- Ideal statistical convergence
- Maximal ideals
- Submeasures
- Tauberian condition
ASJC Scopus subject areas
- Mathematics(all)