Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN

Abstract

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 ξ) (z) = m i (z) ξ 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.

Generalized Multiresolution Analyses with Given Multiplicity Functions

“…We first show that our multiplicity function m admits a low-pass filter, which is a matrix H of functions on Γ satisfying relations, introduced in [3], which generalize those of quadrature mirror filters. From H we build an isometry S H on a Hilbert space K, following an idea which goes back at least to [7], and Theorem 8 says that when the filter is low-pass, S H is a pure isometry. Then, when we apply the construction of Theorem 5 to this isometry, we obtain a direct-limit Hilbert space which has the required GMRA.…”

STATES ON THE CUNTZ ALGEBRAS AND p-ADIC RANDOM WALKS

“…In fact, modulo an action by unitary matrices 2) this is a bijection between End(B(H)) and Rep(Cuntz, H). The number of generators for the particular Cuntz algebra is called the Powers index of the endomorphism σ.…”

Positive Definite Maps, Representations and Frames

“…For details we refer the reader to [Dau92] for the scale N = 2 and to [BraJo97] for arbitrary scale N .…”