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

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 $\phi(N)\approx \log(\log N)$.