Orthogonal wavelets, or wavelet frames, for L^2(R) are associated with quadrature mirror filters (QMF). The latter constitute a set of complex numbers which relate the dyadic scaling of functions on R to the Z-translates, and which satisfy the QMF-axioms. In this paper, we show that generically, the data in the QMF-systems of wavelets is minimal, in the sense that it cannot be nontrivially reduced. The minimality property 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 representation of the algebra O_2; and so our result is that this family of representations of O_2 on the Hilbert space l^2(Z) is irreducible for a generic set of values of the parameters which label the wavelet representations.

LaTeX2e amsart class; 69 pages, 2 tables, 4 figures, 12 pages of plots (total 162 EPS graphics); full-resolution EPS graphics available at ftp://ftp.math.uiowa.edu/pub/jorgen/MinimalityWavelet Changes: correction in Theorem 5.9 and its proof; some added implications and clarifications; slightly more condensed, with an expanded introduction and more graphics in a smaller format. Accepted for publication in Advances in Mathematics