Proximinality and uniformly approximable sets in L^p

2022/08/09 by Guillaume Grelier, Grelier, Guillaume, Jaime San Martı́n +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2209.02598

openalex publication_date 2022/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For any p∈[1,∞], we prove that the set of simple functions taking at most k different values is proximinal in L^p for all k≥ 1. We introduce the class of uniformly approximable subsets of L^p, which is larger than the class of uniformly integrable sets. This new class is characterized in terms of the p-variation if p∈[1,∞) and in terms of covering numbers if p=∞. We study properties of uniformly approximable sets. In particular, we prove that the convex hull of a uniformly approximable bounded set is also uniformly approximable and that this class is stable under Hölder transformations. We also prove that, for p∈ [1,∞), the unit ball of L^p is uniformly approximable if and only if L^p is finite-dimensional, while for p=∞ the unit ball is always uniformly approximable.

Related

• Uniform rectifiability and ε-approximability of harmonic functions in L^p
• Approximative compactness in Böchner spaces
• ε-Approximability of Harmonic Functions in L^p Implies Uniform Rectifiability
• Lp-estimates for the variation for singular integrals on uniformly\n rectifiable sets
• Simultaneously proximinality in L_∞(μ, X)