Haar wavelet characterization of dyadic Lipschitz regularity

Abstract

We obtain a necessary and sufficient condition on the Haar coefficients of a real function f defined on R+ for the Lipschitz α regularity of f with respect to the ultrametric δ(x, y) = inf{|I| : x, y ∈ I; I ∈ D}, where D is the family of all dyadic intervals in R+ and α is positive. Precisely, f ∈ Lipδ (α) if and only if |⟨fhj k ⟩| ≤ C2−(α+1/2)j for some constant C, every j ∈ Z and every k = 0, 1, 2, . . . Here, as usual, h j k (x) = 2j/2h(2jx − k) and h(x) = X[0,1/2)(x) − X[1/2,1)(x).