# Pairs of representations of the Cuntz algebras \( \mathcal{O}_n \), and their application to multiresolutions

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## Abstract

- (1)
The tensor-product idea yields an explicit representation of the unitary operator

*U*which we use to model scale-similarity in our constructions both for standard wavelets and for fractals; see especially Lemma 9.3.2. - (2)
Using tensor products we show that there are

*two*representations of the Cuntz relations involved in modeling basis constructions on scale-similarity (such as was pioneered first for the standard wavelet bases in*L*^{2}(ℝ^{d}). As already noted, our use of multiresolutions naturally entails representation of the Cuntz relations; representations which come directly from subband filters. But in addition, we find a second family of representations, one which reveals symmetry under certain permutations; see Section 9.3. The resulting second class of representations of the Cuntz relations has in fact already been studied earlier (for different purposes) under the name “permutative representations.” - (3)
The tensor-product idea further shows that the representation-theoretic approach is as useful for new basis constructions in function spaces on

*fractals*as it is for the standard wavelet bases in*L*^{2}(ℝ^{d}). In Chapter 4 we introduced natural Hilbert spaces on affine fractals, and in Theorems 9.6.1 and 9.6.2 below we show how these spaces admit bases of generalized wavelets (i.e., “fractal wavelets”) and of wavelet packets. - (4)
Our use of tensor products helps organize the possibilities for the wavelet packets introduced in Chapter 7. In Theorem 9.4.3 we show that when a system of (admissible) digital filters is given, then the feasible choices of wavelet packets are in one-to-one correspondence with certain discrete pavings, or tilings.

- (5)
Finally, in Section 9.5 we outline how our use of tensor products clarifies the

*discrete wavelet transform*. Recall that a choice of a multiresolution (in the generalized sense outlined in Chapter 7) automatically entails a discrete wavelet transform. When a resolution subspace is selected, this allows us to process data which is organized in an associated sequence space (i.e., in an ℓ^{2}); and it is precisely in these ℓ^{2}spaces where our Cuntz relations are realized.

## Keywords

Hilbert Space Representation Duality Wavelet Packet Tensor Factorization Permutative Representation## Preview

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