Refinable functions and cascade algorithms play a fundamental role in wavelet analysis, which is useful in many applications. In this paper we shall study several properties of refinable functions, cascade algorithms and wavelets, associated with Hölder continuous masks, in the weighted subspaces L 2,p,γ (R) of L 2 (R), where 1 p ∞, γ 0 and f ∈ L 2,p,γ (R) means