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 ...
详细信息
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 4 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. (C) 2015 Elsevier Inc. All rights reserved.
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 [19] for dictionaries satisfying a new propert...
详细信息
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 [19] 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 L-p with 1 < p <= 2, under the Littlewood-Paley norm. (C) 2020 Elsevier Inc. All rights reserved.
暂无评论