Stability and Independence of the Shifts of Finitely Many Refinable Functions

Typical constructions of wavelets depend on the stability of the shifts of an underlying re nable function. Unfortunately, several desirable properties are not available with compactly supported orthogonal wavelets, e.g. symmetry and piecewise polynomial structure. Presently, multiwavelets seem to o er a satisfactory alternative. The study of multiwavelets involves the consideration of the properties of several (simultaneously) re nable functions. In Section 2 of this paper, we characterize stability and linear independence of the shifts of a nite re nable function set in terms of the re nement mask. Several illustrative examples are provided. The characterizations given in Section 2 actually require that the re nable functions be minimal in some sense. This notion of minimality is made clear in Section 3, where we provide su cient conditions on the mask to ensure minimality. The conditions are shown to be also necessary under further assumptions on the re nement mask. An example is provided illustrating how the software package MAPLE can be used to investigate at least the case of two simultaneously re nable functions.