Minimality of the data in wavelet filters

Orthogonal wavelets. or wavelet Frames. for L-2(R) are associated with quad rature mirror filters (QMF). a set of complex numbers which relate the dyad ic scaling of functions on R to the L-translates. In this paper. We show th at generically. the data in the QMF-systems of wavelets are minimal, in the sense that the data cannot be nontrivially reduced. The minimality propert y is given a geometric formulation in the Hilbert space l(2)(Z). and it is then shown that minimality corresponds to irreducibility of a wavelet repre sentation of the algebra l(2); and so our result is that this family of rep resentations: of l(2) On the Hilbert space l(2)(Z) is irreducible for a gen eric set of values of the parameters which label the wavelet representation s. (C) 2001 Academic Press.