Generalized self-similarity applied to Matrix Refinement Equations

C. CABRELLI; C. HEIL; U. MOLTER

ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK

WILEY-V C H VERLAG GMBH

Lugar: Weinheim; Año: 1996 vol. 76 p. 493 - 494

0044-2267

We show how some recent results in self-similarity can be applied to the study of solutions of Matrix Refinement Equations (MRE). MRE are refinement equations of the form f(x) = ∑k=0 N Ckf(2x - k), where f is a multivalued function and the coefficients Ck are matrices. Solutions of these equations (refinable functions) are the building blocks in the construction of multiwavelets. We show that MRE are a particular case of generalized self-similarity equations of the form f(x) = script O sign(x, φ1(x, f ○ g1(x), . . . , φτ(x, f ○ gτ(x))), which have recently been studied. We see how analyzing the MRE in this context allows us to obtain conditions for existence and smoothness of the solutions in a very simple way.