arXiv subjects
Publications and source records attributed to Palle E. T. Jorgensen.
By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding.
We study measures on $\mathbb{R}^d$ which are induced by a class of infinite and recursive iterations in symbolic dynamics. Beginning with a finite set of data, we analyze prescribed recursive iteration systems, each involving subdivisions. The construction includes measures arising from affine and contractive iterated function systems with and without overlap (IFSs), i.e., limit measures $μ$ induced by a finite family of affine mappings in $\mathbb{R}^d$ (the focus of our paper), as well as equilibrium measures in complex dynamics. By a systematic analysis of the Fourier transform of the measure $μ$ at hand (frequency domain), we identify asymptotic laws, spectral types, dichotomy, and chaos laws. In particular we show that the cases when $μ$ is singular carry a gradation, ranging from Cantor-like fractal measures to measures exhibiting chaos, i.e., a situation when small changes in the initial data produce large fluctuations in the outcome, or rather, the iteration limit (in this case the measures). Our method depends on asymptotic estimates on the Fourier transform of $μ$ for paths at infinity in $\mathbb{R}^d$. We show how properties of $μ$ depend on perturbations of the initial data, e.g., variations in a prescribed finite set of affine mappings in $\mathbb{R}^d$, in parameters of a rational function in one complex variable (Julia sets and equilibrium measures), or in the entries of a given infinite positive definite matrix.
While finite non-commutative operator systems lie at the foundation of quantum measurement, they are also tools for understanding geometric iterations as used in the theory of iterated function systems (IFSs) and in wavelet analysis. Key is a certain splitting of the total Hilbert space and its recursive iterations to further iterated subdivisions. This paper explores some implications for associated probability measures (in the classical sense of measure theory), specifically their fractal components. We identify a fractal scale $s$ in a family of Borel probability measures $μ$ on the unit interval which arises independently in quantum information theory and in wavelet analysis. The scales $s$ we find satisfy $s\in \mathbb{R}_{+}$ and $s\not =1$, some $s <1$ and some $s>1$. We identify these scales $s$ by considering the asymptotic properties of $μ(J) /| J| ^{s}$ where $J$ are dyadic subintervals, and $| J| \to0$.
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.
We identify multiresolution subspaces giving rise via Hankel transforms to Bessel functions. They emerge as orthogonal systems derived from geometric Hilbert-space considerations, the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of Hilbert spaces. We study the theory of representations of the $C^{\ast}$-algebra $% O_{ν+1}$ arising from this multiresolution analysis. A connection with Markov chains and representations of $O_{ν+1}$ is found. Projection valued measures arising from the multiresolution analysis give rise to a Markov trace for quantum groups $SO_q$.
Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L^2(0,1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar measures (mu). That is, we consider recursive and orthogonal decompositions for the Hilbert space L^2(mu) where mu is some self-similar measure on [0,1]. Up to two specific reflection symmetries, our scheme produces infinite families of orthonormal bases in L^2(0,1). Our approach is as versatile as the more traditional spline constructions. But while singly generated spline bases typically do not produce orthonormal bases, each of our present algorithms does.
In this paper we extend previous work on IFSs without overlap. Our method involves systems of operators generalizing the more familiar Cuntz relations from operator algebra theory, and from subband filter operators in signal processing.
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.
We give an informal overview of the Kadison-Singer extension problem with emphasis on its initial connections to Dirac's formulation of quantum mechanics. Let H be an infinite dimensional separable Hilbert space, and B(H) the algebra of all bounded operators in H. In the language of operator algebras, the Kadison-Singer problem asks whether or not for a given MASA D in B(H), every pure state on D has a unique extension to a pure state on B(H). In other words, are these pure-state extensions unique? It was shown recently by Pete Casazza and co-workers that this problem is closely connected to central open problems in other parts of mathematics (harmonic analysis, combinatorics (via Anderson pavings), Banach space theory, frame theory), and applications (signal processing, internet coding, coding theory, and more).
We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in $\br^d$, and the ``IFS'' refers to such a finite system of transformations, or functions. The iteration limits are pairs $(X, μ)$ where $X$ is a compact subset of $\br^d$, (the support of $μ$) and the measure $μ$ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space $L^2(X,μ)$; and (2) the interplay between the geometry of $(X, μ)$ on the one side, and the spectral data entailed by possible Fourier bases.
We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed relatively constant over a long period of time, they must be varied in accordance with the needs and the demands of students, and they must constantly updated keeping an eye to current research and to modern international trends in technology. Our thoughts and suggestions on the use of these trends in teaching have been tried out by the author, and they are now in textbooks, some by the author.
This paper introduces Fourier duality for a class of affine iterated function systems (IFS) T_i. These systems are determined by a finite family of contractive affine maps in R^d. Our Fourier duality applies to the resulting probability measure mu which is fixed by (T_i). When the IFS is given, the support of the associated mu is a compact set X in R^d, typically a fractal. Our Fourier duality refers to the Hilbert space L^2(X, mu): We show that under a certain unitarity condition involving a pair of affine iterated function systems (T_i) and (S_j) it is possible to recursively construct a Fourier bases in the Hilbert space L^2(X, mu) with the Fourier basis for one depending on the other.
We introduce a harmonic analysis for a class of affine iteration models in $\br^d$. Using Hilbert-space geometry, we develop a new duality notion for affine and contractive iterated function systems (IFSs) and we construct some identities for the Fourier transform of the measure corresponding to infinite Bernoulli convolutions.
We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space $L^2(\br)\oplus...\oplus L^2(\br)$.
We consider frames F in a given Hilbert space, and we show that every F may be obtained in a constructive way from a reproducing kernel and an orthonormal basis in an ambient Hilbert space. The construction is operator-theoretic, building on a geometric formula for the analysis operator defined from F. Our focus is on the infinite-dimensional case where a priori estimates play a central role, and we extend a number of results which are known so far only in finite dimensions. We further show how this approach may be used both in constructing useful frames in analysis and in applications, and in understanding their geometry and their symmetries.
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.
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.
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.