Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N

In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras O_N, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L^2(T) by (S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L^2(T). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations of O_N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.