We show that there are Hilbert spaces constructed from the Hausdorff measures H s on the real line R with 0 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C ⊂ [0, 1], the Hilbert space is a separable subspace of L 2 (R, (dx) s ) where s = log 3 (2). While we develop the general theory of multiresolutions in fractal Hilbert spaces, the emphasis is on the case of scale 3 which covers the traditional Cantor set C. Introducingwe first describe the subspace in L 2 (R, (dx) s ) which has the following family as an orthonormal basis (ONB):Since the affine iteration systems of Cantor type arise from a certain algorithm in R d which leaves gaps at each step, our wavelet bases are in a sense gap-filling constructions.

Abstract. We study representations of the Cuntz algebras ON . While, for fixed N , the set of equivalence classes of representations of ON is known not to have a Borel cross section, there are various subclasses of representations which can be classified. We study monic representations of ON , that have a cyclic vector for the canonical abelian subalgebra. We show that ON has a certain universal representation which contains all positive monic representations. A large class of examples of monic representations is based on Markov measures. We classify them and as a consequence we obtain that different parameters yield mutually singular Markov measure, extending the classical result of Kakutani. The monic representations based on the Kakutani measures are exactly the ones that have a one-dimensional cyclic S * i -invariant space.

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson's canonical measure µ, and we ask when µ is a spectral measure, i.e., when the Hilbert space L 2 (µ) has an orthonormal basis (ONB) of exponentials {e λ | λ ∈ Λ}. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3.

Let R R be an expanding matrix with integer entries, and let B , L B,L be finite integer digit sets so that ( R , B , L ) (R,B,L) form a Hadamard triple on R d {\mathbb {R}}^d in the sense that the matrix 1 | det R | [ e 2 π i ⟨ R − 1 b , ℓ ⟩ ] ℓ ∈ L , b ∈ B \begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*} is unitary. We prove that the associated fractal self-affine measure μ = μ ( R , B ) \mu = \mu (R,B) obtained by an infinite convolution of atomic measures μ ( R , B ) = δ R − 1 B ∗ δ R − 2 B ∗ δ R − 3 B ∗ ⋯ \begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in L 2 ( μ ) L^2(\mu ) . This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

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Abstract. We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B, L in R d of the same cardinality which generate complex Hadamard matrices.Our harmonic analysis for these iterated function systems (IFS) (X, µ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator R W acting on functions on X, and a corresponding class H of continuous R Wharmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L 2 (µ). For affine IFSs we establish orthogonal bases in L 2 (µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d .

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