Wavelet analysis of refinement equations

The Besov regularity of a compactly supported refinement equation solution is determined by the spectral radius of a linear operator acting on $\ell ^p (\mathbb{Z})$. The proof of this is obtained by using a wavelet basis. Exact criteria for Holder and Sobolev regularity follow immediately. Continuity, differentiability, and integrability can also be characterized.The results are applied to examples from the theory of orthonormal and biorthogonal wavelets and subdivision schemes for curve design.