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Rate of convergence of Thresholding Greedy Algorithms

V. N. Temlyakov

TL;DR

This work analyzes the rate of convergence of the Thresholding Greedy Algorithm (TGA) for basis representations in uniformly smooth Banach spaces, with a focus on L_p spaces. It establishes a unified framework where the approximation error is controlled by the product of the signal norm and its A1-norm, and derives sharp, α-dependent rates γ_m(α,p,Ψ) across greedy, unconditional, and quasi-greedy bases. Notably, the trigonometric system and Haar basis attain optimal bounds, with precise asymptotics in different ranges of p, and a practical criterion UB(a,K) links basis structure to the achievable rate. The paper also shows robustness of these results under replacing the A1-norm with weaker variants, broadening applicability to sparse approximation problems in function spaces and informing greedy-based algorithms in analysis and signal processing.

Abstract

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. We obtain some results for greedy bases, unconditional bases, and quasi-greedy bases. In particular, we prove that our bounds for the trigonometric basis and for the Haar basis are optimal.

Rate of convergence of Thresholding Greedy Algorithms

TL;DR

This work analyzes the rate of convergence of the Thresholding Greedy Algorithm (TGA) for basis representations in uniformly smooth Banach spaces, with a focus on L_p spaces. It establishes a unified framework where the approximation error is controlled by the product of the signal norm and its A1-norm, and derives sharp, α-dependent rates γ_m(α,p,Ψ) across greedy, unconditional, and quasi-greedy bases. Notably, the trigonometric system and Haar basis attain optimal bounds, with precise asymptotics in different ranges of p, and a practical criterion UB(a,K) links basis structure to the achievable rate. The paper also shows robustness of these results under replacing the A1-norm with weaker variants, broadening applicability to sparse approximation problems in function spaces and informing greedy-based algorithms in analysis and signal processing.

Abstract

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of and the -norm of . We obtain some results for greedy bases, unconditional bases, and quasi-greedy bases. In particular, we prove that our bounds for the trigonometric basis and for the Haar basis are optimal.
Paper Structure (7 sections, 26 theorems, 118 equations)

This paper contains 7 sections, 26 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Trigonometric system
  3. Unconditional and quasi-greedy bases
  4. Unconditional bases
  5. Quasi-greedy bases
  6. A criterion for good rate of approximation
  7. Replacing the $\|f\|_{A_1(\Psi)}$ by a weaker norm

Key Result

Proposition 1.1

Let $X$ be a uniformly smooth Banach space with $\rho(u,X) \le \gamma u^q$, $1<q\le 2$. Then for any dictionary ${\mathcal{D}}$ and any $\alpha \in [0,1]$ we have for all $f\in X$ such that $\|f\|_{A_1({\mathcal{D}})} <\infty$, $f\neq 0$

Theorems & Definitions (47)

  • Proposition 1.1
  • Corollary 1.1
  • Definition 1.1
  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.1
  • proof
  • Theorem 2.3
  • ...and 37 more