Some equations relating multiwavelets and multiscaling functions

The local trace function introduced in \cite{Dut} is used to derive equations that relate multiwavelets and multiscaling functions in the context of a generalized multiresolution analysis, without appealing to filters. A construction of normalized tight frame wavelets is given. Particular instances of the construction include normalized tight frame and orthonormal wavelet sets

Harmonic Analysis of Signed Ruelle Transfer Operators

Motivated by wavelet analysis, we prove that there is a one-to-one correspondence between the following data: Solutions to $R(h)=h$ where $R$ is a certain non-positive Ruelle transfer operator; Operators that intertwine a certain class of representations of the $C^*$-algebra $\mathfrak{A}_N$ on two unitary generators $U$, $V$ subject to the relation $$UVU^{-1}=V^N$$ This correspondence enables us to give a criterion for the biorthogonality of a pair of scaling functions and calculate all solutions of the equation $R(h)=h$ in some concrete cases.Comment: 24 page

Probability and Fourier duality for affine iterated function systems

Let $d$ be a positive integer, and let $\mu$ be a finite measure on \br^d. In this paper we ask when it is possible to find a subset $\Lambda$ in \br^d such that the corresponding complex exponential functions $e_\lambda$ indexed by $\Lambda$ are orthogonal and total in $L^2(\mu)$. If this happens, we say that $(\mu, \Lambda)$ is a spectral pair. This is a Fourier duality, and the $x$-variable for the $L^2(\mu)$-functions is one side in the duality, while the points in $\Lambda$ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures $\mu$ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in \br^d; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems

Covariant representations for matrix-valued transfer operators

Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-to-one correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on $\mathbb{R}^n$ if the weight satisfies some low-pass condition.Comment: new version, motivation adde

Weighted Fourier frames on fractal measures

We generalize an idea of Picioroaga and Weber to construct Paseval frames of weighted exponential functions for self-affine measures