2011/05/17 by Dorin Ervin Dutkay, Dutkay, Dorin Ervin, Palle E. T. Jørgensen +3
Mathematics · Physics and Astronomy · #31C35 #47A67 #65T60 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1105.3442
openalex publication_date 2011/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random-walks, as well as classes of harmonic functions. Our setting entails representations built from certain finite-to-one endomorphisms r in compact metric spaces X, and we study their dilations to automorphisms in induced solenoids. Our wavelet representations are covariant systems formed from the dilated automorphisms. They depend on assigned measures μ on X. It is known that when the data (X, r, μ) are given the associated wavelet representation is typically reducible. By introducing wavelet filters associated to (X, r) we build random walks in X, path-space measures, harmonic functions, and an associated Martin boundary. We construct measures on the solenoid (X_∞, r_∞), built from (X, r). We show that r_∞ induces unitary operators U on Hilbert space \H and representations π of the algebra L^∞(X) such that the pair (U, r_∞), together with the corresponding representation π forms a cross-product in the sense of C^*-algebras. We note that the traditional wavelet representations fall within this wider framework of (\H, U, π) covariant crossed products.