Thinnable ideals and invariance of cluster points

Leonetti, Paolo

doi:10.1216/rmj-2018-48-6-1951

Cited by 13 publications

(14 citation statements)

References 21 publications

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“…which we refer to as the lower Pólya density on H. Among other things, p has found a number of remarkable applications in analysis and economic theory (see e.g. [19,22,26,33]), but what is perhaps more interesting in the frame of the present work is that p is an upper density in the sense of our definitions. This follows from Proposition 13 and the fact that p is the pointwise limit of the real net of the upper α-densities on H (see [21,Theorem 4.3]).…”

Section: Structural Resultsmentioning

confidence: 81%

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On the notions of upper and lower density

Leonetti¹,

Tringali²

2019

Proceedings of the Edinburgh Mathematical Society

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Let P(N) be the power set of N. We say that a function µ ⋆ : P(N) → R is an upper density if, for all X, Y ⊆ N and h, k ∈ N + , the following hold: (f1) µ ⋆ (N) = 1; (f2)We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya, and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)-(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that µ ⋆ (X) ≤ 1 for every X ⊆ N.Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

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Section: Structural Resultsmentioning

confidence: 81%

“…which we refer to as the lower Pólya density on H. Among other things, p ⋆ has found a number of remarkable applications in analysis and economic theory (see, e.g., [34], [23], [20], and [27]),…”

Section: Structural Resultsmentioning

confidence: 99%

On the notions of upper and lower density

Leonetti¹,

Tringali²

2019

Proceedings of the Edinburgh Mathematical Society

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“…is comeager, cf. also [27] for the case I = Z and [22] for a measure theoretic analogue. We will extend this result to all meager ideals.…”

Section: I-cluster Pointsmentioning

confidence: 99%

The Baire Category of Subsequences and Permutations which preserve Limit Points

Balcerzak

Leonetti

2020

Results Math

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Let $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ N . We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ I -cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221–229]. As an application, if x is a sequence with values in a first countable compact space which is $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.

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“…In particular, Γ x (I) is compact for all real bounded sequences x, cf. [2,13,14] for basic facts and characterizations of I-cluster points.…”

Section: Introductionmentioning

confidence: 99%

Characterizations of the ideal core

Leonetti

2019

Journal of Mathematical Analysis and Applications

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Given an ideal I on ω and a sequence x in a topological vector space, we let the I-core of x be the least closed convex set containing {x n : n / ∈ I} for all I ∈ I. We show two characterizations of the I-core. This implies that the I-core of a bounded sequence in R k is simply the convex hull of its I-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and e-convergence of double sequences.2010 Mathematics Subject Classification. Primary: 40A35. Secondary: 54A20, 40A05.

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Thinnable ideals and invariance of cluster points

Cited by 13 publications

References 21 publications

On the notions of upper and lower density

On the notions of upper and lower density

The Baire Category of Subsequences and Permutations which preserve Limit Points

Characterizations of the ideal core

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