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.

A Tauberian theorem for ideal statistical convergence

Leonetti P
2020

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.
Generalized density ideal; Ideal statistical convergence; Maximal ideals; Submeasures; Tauberian condition
Balcerzak, Marek; Leonetti, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2142072
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