We consider the smoothness of solutions of a system of refinement equations
written in the form
phi = Sigma(alpha is an element of z) a(alpha)phi(2.-alpha),
where the vector of functions phi = (phi(1),...,phi(r))(T) is in (L-p( R))(
r) and a is a finitely supported sequence of r x r matrices called the refi
nement mask. We use the generalized Lipschitz space Lip*(nu, L-p(R)), nu >0
, to measure smoothness of a given function.
Our method is to relate the optimal smoothness, nu(p)(phi), to the p-norm j
oint spectral radius of the block matrices A(epsilon), epsilon = 0,1, given
by A(epsilon) = (a(epsilon + 2 alpha - beta))(alpha,beta), when restricted
to a certain finite dimensional common invariant subspace V. Denoting the
p-norm joint spectral radius by rho(p)(A(0)\(V), A(1)\(V)), we show that nu
(p)(phi) greater than or equal to 1/p - log(2)rho(p)(A(0)\(V), A(1)\(V)) wi
th equality when the shifts of phi(1),...,phi(r) are stable and the invaria
nt subspace is generated by certain vectors induced by difference operators
of sufficiently high order. This allows an effective use of matrix theory.
Also the computational implementation of our method is simple.
When p = 2, the optimal smoothness is also given in terms of the spectral r
adius of the transition matrix associated with the refinement mask.
To illustrate the theory, we give a detailed analysis of two examples where
the optimal smoothness can be given explicitly. We also apply our methods
to the smoothness analysis of multiple wavelets. These examples clearly dem
onstrate the applicability and practical power of our approach.