Ola Bratteli, Palle E.T. Jorgensen
Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale $N$
(280K, LaTeX2e amsart class, 59 pages)

ABSTRACT.  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.