On the Approximate Weak Chebyshev Greedy Algorithm in uniformly smooth Banach spaces

We study greedy approximation in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA), introduced and studied in [6], is defined for any Banach space X and a dictionary D, and provides nonlinear n-term approximation with respect to D. In this paper we study the Approximate Weak...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 436; no. 1; pp. 288 - 304
Main Author: Dereventsov, A.V.
Format: Journal Article
Language:English
Published: Elsevier Inc 01.04.2016
Subjects:
Approximate Weak Chebyshev Greedy Algorithm
Banach space
Greedy algorithm
Nonlinear approximation
Weak Chebyshev Greedy Algorithm
Greedy algorithm
Weak Chebyshev Greedy Algorithm
Approximate Weak Chebyshev Greedy Algorithm
Banach space
Nonlinear approximation
ISSN:0022-247X, 1096-0813
Online Access:Get full text
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Summary:We study greedy approximation in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA), introduced and studied in [6], is defined for any Banach space X and a dictionary D, and provides nonlinear n-term approximation with respect to D. In this paper we study the Approximate Weak Chebyshev Greedy Algorithm (AWCGA) – a modification of the WCGA that was studied in [7]. In the AWCGA we are allowed to calculate n-term approximation with a perturbation in computing the norming functional and a relative error in calculating the approximant. Such permission is natural for the numerical applications and simplifies realization of the algorithm. We obtain conditions that are necessary and sufficient for the convergence of the AWCGA for any element of X. In particular, we show that if perturbations and errors are from ℓ1 space then the conditions for the convergence of the AWCGA are the same as for the WCGA. For specifically chosen perturbations and errors we estimate the rate of convergence for any element f from the closure of the convex hull of D and demonstrate that in special cases the AWCGA performs as well as the WCGA.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.12.006