The WCGA in Lp(logL)α Spaces - Garrigós, Gustavo (Constructive approximation, 2025)
We present some new results concerning Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces 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 A 3 parameter are not necessarily power functions. Secondly, we apply this new theorem to the Zygmund spaces X = L p ( log L ) α , with 1 < p < ∞ and 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 ϕ ( N ) = O ( N max { 1 , 2 / p ′ } ( log N ) | α | p ′ ) . Moreover, this quantity is sharp when p ≤ 2 . Finally, an expression for ϕ ( N ) in the case of the trigonometric system is also given.