Metric entropies of sets in abstract Wiener space

Xicheng Zhang

doi:10.1016/j.bulsci.2005.02.002

Metric entropies of sets in abstract Wiener space

Xicheng Zhang^*

^*Corresponding author for this work

Huazhong University of Science and Technology

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

Let (X, H μ) be an abstract Wiener space, E (ε, K) denote the metric entropy of a set K ⊂ X. If K is not a slim set, then we prove that 0 < lim inf ε→0 ε² E(ε). In particular, if lim inf_3→0 ε²E (ε, K) = 0, then K is a slim set. Moreover, if K is compact and contained in the closure of B₀^H (R) in X, where B₀^H := {h ∈ H: ∥h∥_H < R} is a ball in H, then lim supε→0 ε² E(ε, K) < ∞.

Original language	English
Pages (from-to)	559-566
Number of pages	8
Journal	Bulletin des Sciences Mathematiques
Volume	129
Issue number	7
DOIs	https://doi.org/10.1016/j.bulsci.2005.02.002
Publication status	Published - Aug 2005
Externally published	Yes

Keywords

Capacity
Metric entropy
Slim set
Small ball

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Zhang, X. (2005). Metric entropies of sets in abstract Wiener space. Bulletin des Sciences Mathematiques, 129(7), 559-566. https://doi.org/10.1016/j.bulsci.2005.02.002

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PY - 2005/8

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N2 - Let (X, H μ) be an abstract Wiener space, E (ε, K) denote the metric entropy of a set K ⊂ X. If K is not a slim set, then we prove that 0 < lim inf ε→0 ε2 E(ε). In particular, if lim inf3→0 ε2E (ε, K) = 0, then K is a slim set. Moreover, if K is compact and contained in the closure of B0H (R) in X, where B0H := {h ∈ H: ∥h∥H < R} is a ball in H, then lim supε→0 ε2 E(ε, K) < ∞.

AB - Let (X, H μ) be an abstract Wiener space, E (ε, K) denote the metric entropy of a set K ⊂ X. If K is not a slim set, then we prove that 0 < lim inf ε→0 ε2 E(ε). In particular, if lim inf3→0 ε2E (ε, K) = 0, then K is a slim set. Moreover, if K is compact and contained in the closure of B0H (R) in X, where B0H := {h ∈ H: ∥h∥H < R} is a ball in H, then lim supε→0 ε2 E(ε, K) < ∞.

KW - Capacity

KW - Metric entropy

KW - Slim set

KW - Small ball

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JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

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ER -

Metric entropies of sets in abstract Wiener space

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Keywords

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