The WCGA in $$L^p(\log L)^{\alpha }$$ Spaces

Abstract

AbstractWe present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces $${{\mathbb {X}}}$$ X . First, we generalize Temlyakov’s theorem (Temlyakov in Forum Math Sigma 2(12):26, 2014) to cover situations in which the modulus of smoothness and the $${\texttt {A3}}$$ A 3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces $${{\mathbb {X}}}=L^p(\log L)^{\alpha }$$ X = L p ( log L ) α , with $$1<p<\infty $$ 1 < p < ∞ and $${\alpha }\in {{\mathbb {R}}}$$ α ∈ R , and show that, when the Haar system is used, then exact recovery of N-sparse signals occurs when the number of iterations is $$\phi (N)=O(N^{\max \{1,2/p'\}} \,(\log N)^{|{\alpha }| p'})$$ ϕ ( N ) = O ( N max { 1 , 2 / p ′ } ( log N ) | α | p ′ ) . Moreover, this quantity is sharp when $$p\le 2$$ p ≤ 2 . Finally, an expression for $$\phi (N)$$ ϕ ( N ) in the case of the trigonometric system is also given.