2024 GREEDY APPROXIMATION ALGORITHMS FOR SPARSE COLLECTIONS
Guillermo Rey
Author Affiliations +
Publ. Mat. 68(1): 251-265 (2024). DOI: 10.5565/PUBLMAT6812411

Abstract

We describe a greedy algorithm that approximates the Carleson constant of a collection of general sets. The approximation has a logarithmic loss in a general setting, but is optimal up to a constant with only mild geometric assumptions. The constructive nature of the algorithm gives additional information about the almost disjoint structure of sparse collections.

As applications, we give three results for collections of axis-parallel rectangles in every dimension. The first is a constructive proof of the equivalence between Carleson and sparse collections, first shown by Hänninen. The second is a structure theorem proving that every finite collection ε can be partitioned into O(N) sparse subfamilies, where N is the Carleson constant of ε. We also give examples showing that such a decomposition is impossible when the geometric assumptions are dropped. The third application is a characterization of the Carleson constant involving only L1, estimates.

Acknowledgements

The author would like to thank the referees for their thoughtful comments and many helpful suggestions.

Supported by Grant MICIN/AEI/PID2019-105599GB-I00.

Citation

Download Citation

Guillermo Rey. "GREEDY APPROXIMATION ALGORITHMS FOR SPARSE COLLECTIONS." Publ. Mat. 68 (1) 251 - 265, 2024. https://doi.org/10.5565/PUBLMAT6812411

Information

Received: 23 February 2022; Accepted: 20 July 2022; Published: 2024
First available in Project Euclid: 25 December 2023

MathSciNet: MR4682731
Digital Object Identifier: 10.5565/PUBLMAT6812411

Subjects:
Primary: 42B25

Keywords: Carleson sequence , maximal function , Multiparameter , sparse collection

Rights: Copyright © 2024 Universitat Autònoma de Barcelona, Departament de Matemàtiques

JOURNAL ARTICLE
15 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.68 • No. 1 • 2024
Back to Top