A new factorization technique of the matrix mask of univariate re nable functionsGerlind

A univariate compactly supported reenable function can always be factored into B k f, with B k the B-spline of order k, f a compactly supported distribution, and k the approximation orders provided by the underlying shift-invariant space S(). Factorizations of univariate reenable vectors were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the rst precise deenition of that process. The main goal of this article is the introduction of a special factorization algorithm of reenable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector is shown to bèalmost' in the form B k F, with F still compactly supported and reenable, and k the approximation order of S((): `almost' in the sense that and B k F diier at most in one entry. The algorithm guarantees F to retain the possible favorable properties of , such as the stability of the shifts of and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to whose shifts are not stable, as well as to reenable vectors which are not compactly supported. The usefulness of this speciic factorization for the study of the smoothness of FSI wavelets (known also as`multiwavelets' and`multiple wavelets') is explained. The analysis invokes in an essential way the theory of nitely generated shift-invariant (FSI) spaces, and, in particular, the tool of superfunction theory.