Abstract
Orthogonal wavelets, or wavelet frames, for L2(R) are associated with quadrature mirror filters (QMF), a set of complex numbers which relate the dyadic scaling of functions on R to the Z-translates. In this paper, we show that generically, the data in the QMF-systems of wavelets are minimal, in the sense that the data cannot be nontrivially reduced. The minimality property is given a geometric formulation in the Hilbert space ℓ2(Z), and it is then shown that minimality corresponds to irreducibility of a wavelet representation of the algebra O2; and so our result is that this family of representations of O2 on the Hilbert space ℓ2(Z) is irreducible for a generic set of values of the parameters which label the wavelet representations.
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