 00441 Ola Bratteli, Palle E.T. Jorgensen
 Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale $N$
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Nov 8, 00

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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 scale2 theory of wavelets with the scaleN 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.
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