Abstract

We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L_{n}^{mathrm{ch}} for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L_{n}^{mathrm{ch}} with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained.

Highlights

  • Nonlinear n-term approximations with respect to biorthogonal systems such as the trigonometric system and wavelet bases are frequently used in image or signal processing, PDE solvers and statistic learning

  • The fundamental question of a nonlinear approximation is how to construct good algorithms to realize the best n-term approximation. It turns out the Thresholding Greedy Algorithm (TGA), which was proposed by Konyagin and Temlyakov in [2], in some sense is a suitable method for nonlinear n-term approximation

  • Greedy Algorithm (CTGA), which is an enhancement of TGA

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Summary

Introduction

Nonlinear n-term approximations with respect to biorthogonal systems such as the trigonometric system and wavelet bases are frequently used in image or signal processing, PDE solvers and statistic learning (see [1]). Theorem 1.3 If is a K-quasi-greedy basis in a Banach space (over K = R or C), for all n ≥ 1, μdn 2K The upper bound of Lcnh(T d) follows from the known results of Ln(T d) and the proof of the lower bound of Lcnh(T d) relies on a theorem on the lower estimate of Lcnh( ) for a general basis in a Banach space X.

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