The Weak Chebyshev Greedy Algorithm (WCGA) in $L^p (\log L)^\alpha$ spaces

TL;DR

This paper extends Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm in Banach spaces, applying it to Zygmund spaces and analyzing optimal recovery of sparse signals with specific iteration bounds.

Contribution

It generalizes previous results to broader smoothness conditions and provides sharp bounds for sparse signal recovery in Zygmund spaces using the WCGA.

Findings

Optimal recovery occurs with iterations $oxed{ ext{O}(N^{ ext{max}\{1,2/p' ight ext{}}} ( ext{log} N)^{| ext{α}| p'})$

Bounds are sharp for $p \\leq 2$ in Zygmund spaces

For the trigonometric system in $L^2( ext{log} L)^{ ext{α}}$, $oxed{ ext{approximate log(log N)}}$ iterations suffice

Abstract

We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $\mathbb{X}$. First, we generalize a result of Temlyakov to cover situations in which the modulus of smoothness and the so called A3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces $\mathbb{X}=L^p(\log L)^\alpha$, with $1<p<\infty$ and $\alpha\in\mathbb{R}$, and show that, when the Haar system is used, then optimal recovery of $N$-sparse signals occurs when the number of iterations is $\phi(N)=O(N^{\max\{1,2/p'\}} \,(\log N)^{|\alpha| p'})$. Moreover, this quantity is sharp when $p\leq 2$. Finally, an expression for $\phi(N)$ in the case of the trigonometric system is also given, which in the special case of $L^2(\log L)^\alpha$, with $\alpha>0$, takes the form $\phi(N)\approx \log(\log N)$.