arXiv subjects

Dorin Ervin Dutkay

Publications and source records attributed to Dorin Ervin Dutkay.

At least 73 records · Page 4Linked to original sources

arXiv2008

Spectral duality for a class of unbounded operators

We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical approximation of infinite models with suitable sequences of finite models which in turn allow (relatively) easy computations. Let $X$ be an infinite set and let $\H$ be a Hilbert space of functions on $X$ with inner product $\ip{\cdot}{\cdot}=\ip{\cdot}{\cdot}_{\H}$. We will be assuming that the Dirac masses $δ_x$, for $x\in X$, are contained in $\H$. And we then define an associated operator $Δ$ in $\H$ given by $$(Δv)(x):=\ip{δ_x}{v}_{\H}.$$ Similarly, for every finite subset $F\subset X$, we get an operator $Δ_F$. If $F_1\subset F_2\subset...$ is an ascending sequence of finite subsets such that $\cup_{k\in\bn}F_k=X$, we are interested in the following two problems: (a) obtaining an approximation formula $$\lim_{k\to\infty}Δ_{F_k}=Δ;$$ and (b) establish a computational spectral analysis for the truncated operators $Δ_F$ in (a).

math.FA↗

arXiv2008

Spectral Theory for Discrete Lapacians

We give the spectral representation for a class of selfadjoint discrete graph Laplacians $Δ$, with $Δ$ depending on a chosen graph $G$ and a conductance function $c$ defined on the edges of $G$. We show that the spectral representations for $Δ$ fall in two model classes, (1) tree-graphs with $N$-adic branching laws, and (2) lattice graphs. We show that the spectral theory of the first class may be computed with the use of rank-one perturbations of the real part of the unilateral shift, while the second is analogously built up with the use of the bilateral shift. We further analyze the effect on spectra of the conductance function $c$: How the spectral representation of $Δ$ depends on $c$. Using $Δ_G$, we introduce a resistance metric, and we show that it embeds isometrically into an energy Hilbert space. We introduce an associated random walk and we calculate return probabilities, and a path counting number.

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arXiv2008

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in $\{0, 1, 2, 3\}$) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis.

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arXiv2008

Quasiperiodic Spectra and Orthogonality for Iterated Function System Measures

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a ``small perturbation'' of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs $X$ admit spectral duality. When $X$ is given, we identify geometric conditions on $X$ for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices.

math.CA↗

arXiv2007

Orthonormal dilations of Parseval wavelets

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.

math.FA↗

arXiv2007

Fourier series on fractals: a parallel with wavelet theory

We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(μ)$ with measures $μ$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics. Motivated by applications, we draw parallels between analysis of fractal measures on the one hand, and the geometry of wavelets on the other. We are motivated by spectral theory for commuting partial differential operators and related duality notions. While stated initially for bounded and open regions in $\br^d$, they have since found reformulations in the theory of fractals and wavelets. We include a historical sketch with questions from early operator theory.

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arXiv2007

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.

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arXiv2007

A duality approach to representations of Baumslag-Solitar groups

We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations $u t = t^2 u$ where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an isometry in $\mathcal H$. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the dilated operators may be the unitary operators which define a dyadic wavelet multiresolutions of $L^2(\br)$ with the subspace $\mathcal H$ serving as the corresponding resolution subspace. That is, the initialized resolution which is generated by the wavelet scaling function(s). In the dilated Hilbert space, the Baumslag-Solitar relations then take the more familiar form $u t u^{-1} = t^2$. We offer an operator theoretic framework including the standard construction; and we show how the representations of certain discrete semidirect group products serve to classify the possibilities. For this we analyze and compare several types of unitary representations of these semidirect products: the induced representations in Mackey's theory, the wavelet representations on $L^2(\br)$, the irreducible representation on the dual, the finite dimensional representations, and the the regular representation.

math.OA↗

arXiv2007

The algebra of harmonic functions for a matrix-valued transfer operator

We analyze matrix-valued transfer operators. We prove that the fixed points of transfer operators form a finite dimensional $C^*$-algebra. For matrix weights satisfying a low-pass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algortihms.

math.FA↗

arXiv2006

Analysis of orthogonality and of orbits in affine iterated function systems

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 mu, and we ask when mu is a spectral measure, i.e., when the Hilbert space L^2(mu) has an orthonormal basis (ONB) of exponentials. 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.

math.CA↗

arXiv2006

Oversampling generates super-wavelets

We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space $L^2(\br)\oplus...\oplus L^2(\br)$.

math.FA↗

arXiv2006

Martingales, endomorphisms, and covariant systems of operators in Hilbert space

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in symbolic dynamics. In the theory of wavelets, in the study of subshifts, in the analysis of Julia sets of rational maps of a complex variable, and, more generally, in the study of dynamical systems, we are faced with the problem of building a unitary operator from a mapping r in a compact metric space X. The space X may be a torus, or the state space of subshift dynamical systems, or a Julia set. While our motivation derives from some wavelet problems, we have in mind other applications as well; and the issues involving covariant operator systems may be of independent interest.

math.CA↗

arXiv2006

Hilbert spaces built on a similarity and on dynamical renormalization

We develop a Hilbert space framework for a number of general multi-scale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the more familiar approach to wavelet theory which starts with the two-to-one endomorphism $r: z \mapsto z^2$ in the one-torus $\bt$, a wavelet filter, and an associated transfer operator. This leads to a scaling function and a corresponding closed subspace $V_0$ in the Hilbert space $L^2(\br)$. Using the dyadic scaling on the line $\br$, one has a nested family of closed subspaces $V_n$, $n \in \bz$, with trivial intersection, and with dense union in $L^2(\br)$. More generally, we achieve the same outcome, but in different Hilbert spaces, for a class of non-linear problems. In fact, we see that the geometry of scales of subspaces in Hilbert space is ubiquitous in the analysis of multiscale problems, e.g., martingales, complex iteration dynamical systems, graph-iterated function systems of affine type, and subshifts in symbolic dynamics. We develop a general framework for these examples which starts with a fixed endomorphism $r$ (i.e., generalizing $r(z) = z^2$) in a compact metric space $X$. It is assumed that $r : X\to X$ is onto, and finite-to-one.

math.DS↗

arXiv2005

Disintegration of projective measures

In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain desintegration. This is a desintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.

math.CA↗

arXiv2005

Low-pass filters and representations of the Baumslag Solitar group

We analyze representations of solvable Baumslag Solitar group that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

math.CA↗

arXiv2005

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.

math.FA↗

arXiv2005

The spectrum of the wavelet Galerkin operator

We give a complete description of spectrum of the wavelet Galerkin operator $$R_{m_0,m_0}f(z)=\frac{1}{N}\sum_{w^N=z}|m_0|^2(w)f(w),\quad(z\in\m athbb{T})$$ associated to a a low-pass filter $m_0$ and a scale $N$, in the Banach spaces $C(\mathbb{T})$ and $L^{p}(\mathbb{T})$, $1\leq p\leq\infty$.

math.FA↗

arXiv2005

Positive definite maps, representations and frames

We present a unitary approach to the construction of representations and intertwining operators. We apply it to the $C^*$-algebras, groups, Gabor type unitary systems and wavelets. We give an application of our method to the theory of frames, and we prove a general dilation theorem which is in turn applied to specific cases, and we obtain in this way a dilation theorem for wavelets.

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