Sobolev Regularity for Refinement Equations via Ergodic Theory

The refinement equation f(x) = ∑N k=0 ck f(2x− k) plays a key role in wavelet theory and in subdivision schemes in approximation theory. This paper explores the relationship of the refinement equation to the mapping τ(x) = 2x mod 1. A simple necessary condition for the existence of an integrable solution to the refinement equation is obtained by considering the periodic cycles of τ . Another simple necessary condition for the existence of an integrable solution satisfying (1+ |γ|) f̂(γ) ∈ L(R) is obtained by considering the ergodic property of τ . In particular, for p = 2 this is a necessary condition for f to lie in the Sobolev space H.