Lebesgue-type inequalities in greedy approximation
Stephen Dilworth, Gustavo Garrigos, Eugenio Hernandez, Denka, Kutzarova, Vladimir Temlyakov
TL;DR
This paper establishes improved Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm in smooth Banach spaces, providing optimal bounds in sequence spaces and for the multivariate Haar system in Lp.
Contribution
It introduces a new property for dictionaries and derives sharper bounds for greedy algorithms in Banach spaces, extending previous results.
Findings
Improved bounds for WCGA in uniformly smooth Banach spaces.
Optimal bounds for sequence spaces with Littlewood-Paley norm.
Validation of bounds for multivariate Haar system in Lp, 1<p<2.
Abstract
We present new results regarding Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces. We improve earlier bounds in Temlyakov (Forum Math Sigma 2014), for dictionaries satisfying a new property introduced here. We apply these results to derive optimal bounds in two natural examples of sequence spaces. In particular, optimality is obtained in the case of the multivariate Haar system in Lp with 1<p<2, under the Littlewood-Paley norm.
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Taxonomy
TopicsMathematical Approximation and Integration · Approximation Theory and Sequence Spaces · Advanced Banach Space Theory