Lebesgue constants for Chebyshev thresholding greedy algorithms

Chunfang Shao,Peixin Ye

doi:10.1186/s13660-018-1694-y

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

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.

Results

Conclusion