**MSC:**- 39B62 Systems of functional equations
- 42A05 Trigonometric polynomials, inequalities, extremal problems

solutions $\phi = (\phi_{\nu})_{\nu=0}^{r-1}$ of vector

refinement equations.

The space spanned by the translates of $\phi_{\nu}$ can only

provide approximation order if the refinement mask

$\mbox{\boldmath$P$}$ has certain particular factorization

properties. We show, how the factorization of

$\mbox{\boldmath$P$}$ can lead to decay of

$|\hat{\phi}_{\nu}(u)|$ as $|u| \rightarrow \infty$.

The results on decay are used in order to prove uniqueness

of solutions and convergence of the cascade algorithm.