EXISTENCE AND ACCURACY FOR MATRIX REFINEMENT EQUATIONS

A refinement equation is a functional equation of the form f(x) = Sigm a(k=0)(N) c(k) f(2x - k). It is the starting point for the constructio n of wavelets and for subdivision schemes in. approximation theory. Th e recent theory of multiwavelets relies on a matrix refinement equatio n, where the c(k) are r x r matrices and the solution f is vector-valu ed, f(x) = (f(1)(x) ,..., f(r)(x))(t). In this paper we sketch some re cent results on the existence and uniqueness of solutions to matrix: r efinement equations, and on the conditions needed for exact reconstruc tion of polynomials 1,x ,..., x(p-1) from integer translates of f(1) , ..., f(r).