Regularity of refinable function vectors

The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm and the existence and regularity of compactly supported solutions of vector refinement equations.

Abstract

We study the existence and regularity of compactly supported solutions φ = (φv) v= /r− of vector refinement equations. The space spanned by the translates of φv can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.