Albert Cohen, Ingrid Daubechies, Gerlind Plonka
Regularity of refinable function vectors
Preprint series: Preprints aus dem Fachbereich Mathematik, Universität Rostock

MSC:

39B62 Systems of functional equations

42A05 Trigonometric polynomials, inequalities, extremal problems

Abstract: We study the existence and regularity of compactly supported
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.
Keywords: Refinable function vectors, Refinement mask, Fourier transform, Spectral radius