TY - JOUR
T1 - A Tauberian theorem for ideal statistical convergence
AU - Balcerzak, Marek
AU - Leonetti, Paolo
PY - 2020/1
Y1 - 2020/1
N2 - 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.
AB - 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.
KW - Generalized density ideal
KW - Ideal statistical convergence
KW - Maximal ideals
KW - Submeasures
KW - Tauberian condition
UR - http://www.scopus.com/inward/record.url?scp=85074502018&partnerID=8YFLogxK
U2 - 10.1016/j.indag.2019.10.001
DO - 10.1016/j.indag.2019.10.001
M3 - Article
AN - SCOPUS:85074502018
SN - 0019-3577
VL - 31
SP - 83
EP - 95
JO - Indagationes Mathematicae
JF - Indagationes Mathematicae
IS - 1
ER -