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

Abstract

this chapter will resume our study of the geometric approach via Hilbert space to separation of variables/tensor product stressed in the last two chapters; and we will note a number of applications of this idea. The approach further serves to clarify a number of themes involving combinatorics of the recursive bases studied throughout the book. The separate themes are as follows.

• (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 L²(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 L²(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 ℓ²); and it is precisely in these ℓ² spaces where our Cuntz relations are realized.

If one finds a difficulty in a calculation which is otherwise quite convincing, one should not push the difficulty away; one should rather try to make it the centre of the whole thing. —Werner Heisenberg