Kadison-Singer from mathematical physics: An introduction

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).Comment: 12 pages, LaTeX2e "amsart" document class, grew out of a workshop at the AIM institute (with NSF support) in Palo Alto in September, 2006. v2: fine tuning. More details, clarifications, explanations, citations/ references have been added, most of the additions are motivated by suggestions coming in from KS IMA participant

Discrete reproducing kernel Hilbert spaces: Sampling and distribution of Dirac-masses

We study reproducing kernels, and associated reproducing kernel Hilbert spaces (RKHSs) $\mathscr{H}$ over infinite, discrete and countable sets $V$. In this setting we analyze in detail the distributions of the corresponding Dirac point-masses of $V$. Illustrations include certain models from neural networks: An Extreme Learning Machine (ELM) is a neural network-configuration in which a hidden layer of weights are randomly sampled, and where the object is then to compute resulting output. For RKHSs $\mathscr{H}$ of functions defined on a prescribed countable infinite discrete set $V$, we characterize those which contain the Dirac masses $\delta_{x}$ for all points $x$ in $V$. Further examples and applications where this question plays an important role are: (i) discrete Brownian motion-Hilbert spaces, i.e., discrete versions of the Cameron-Martin Hilbert space; (ii) energy-Hilbert spaces corresponding to graph-Laplacians where the set $V$ of vertices is then equipped with a resistance metric; and finally (iii) the study of Gaussian free fields.Comment: 9 figure

Unitary matrix functions, wavelet algorithms, and structural properties of wavelets

Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and subband filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/~jorgen/ .Comment: 63 pages, 10 figures/tables, LaTeX2e ("mrv9x6" document class), Contribution by Palle E. T. Jorgensen to the Tutorial Sessions, Program: ``Functional and harmonic analyses of wavelets and frames,'' 4-7 August 2004, Organizers: Judith Packer, Qiyu Sun, Wai Shing Tang. v2 adds Section 2.3.4, "Matrix completion" with reference

Representations of Cuntz algebras, loop groups and wavelets

A theorem of Glimm states that representation theory of an NGCR C*-algebra is always intractable, and the Cuntz algebra O_N is a case in point. The equivalence classes of irreducible representations under unitary equivalence cannot be captured with a Borel cross section. Nonetheless, we prove here that wavelet representations correspond to equivalence classes of irreducible representations of O_N, and they are effectively labeled by elements of the loop group, i.e., the group of measurable functions A:T-->U_N(C). These representations of O_N are constructed here from an orbit picture analysis of the infinite-dimensional loop group.Comment: 6 pages, LaTeX2e "amsproc" class; expanded version of an invited lecture given by the author at the International Congress on Mathematical Physics, July 2000 in Londo

Positive definite (p.d.) functions vs p.d. distributions

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, and the generalization to Bochner/Schwartz representations for positive definite tempered distributions on the other; in the latter case, via tempered positive measures. Via our transforms, we make precise the respective reproducing kernel Hilbert spaces (RKHSs), that of N. Aronszajn and that of L. Schwartz. Further applications are given to stationary-increment Gaussian processes

Certain representations of the Cuntz relations, and a question on wavelets decompositions

We compute the Coifman-Meyer-Wickerhauser measure $\mu$ for certain families of quadrature mirror filters (QMFs), and we establish that for a subclass of QMFs, $\mu$ contains a fractal scale. In particular, these measures $\mu$ are not in the Lebesgue class.Comment: v.2 has a new title and additional material in the introduction. Prepared using the amsproc.cls document clas

Reproducing kernels and choices of associated feature spaces, in the form of L2L^{2}-spaces

Motivated by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels $K$, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in the form $L^{2}\left(\mu\right)$; and we study the question of which measures $\mu$ are right for a particular kernel $K$. The answer to this depends on the particular application at hand. Such applications are the focus of the separate sections in the paper

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically the cascade algorithm in wavelet theory. Let $H$ be a Hilbert space, and let $\pi$ be a representation of $L^\infty(T)$ on $H$. Let $R$ be a positive operator in $L^\infty(T)$ such that $R(1)=1$, where $1$ denotes the constant function $1$. We study operators $M$ on $H$ (bounded, but non-contractive) such that $\pi(f)M=M\pi(f(z^2))$ and $M^* \pi(f)M=\pi(R^* f)$, $f \in L^\infty (T)$, where the $*$ refers to Hilbert space adjoint. We give a complete orthogonal expansion of $H$ which reduces $\pi$ such that $M$ acts as a shift on one part, and the residual part is $H^{(\infty)}=\bigcap_n[M^n H]$, where $[M^n H]$ is the closure of the range of $M^n$. The shift part is present, we show, if and only if $\ker(M^*) \neq \{0\}$. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation $\pi$, we show that, for this wavelet operator $M$, the components in the decomposition are unitarily, and canonically, equivalent to spaces $L^2(E_n) \subset L^2(R)$, where $E_n \subset R$, $n=0,1,2,...,\infty$, are measurable subsets which form a tiling of $R$; i.e., the union is $R$ up to zero measure, and pairwise intersections of different $E_n$'s have measure zero. We prove two results on the convergence of the cascade algorithm, and identify singular vectors for the starting point of the algorithm.Comment: AMS-LaTeX; 47 pages, 3 tables, 2 figures comprising 3 EPS diagram

On reproducing kernels, and analysis of measures

Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure $\mu$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $\mu$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast, when $\mu$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their It\^{o} calculus and diffusion. Examples include fractional Brownian motion, and time-change processes

Realizations and Factorizations of Positive Definite Kernels

Given a fixed sigma-finite measure space $\left(X,\mathscr{B},\nu\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $\nu$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\left(\cdot,\cdot\right)$ with the two variables from the subset of the sigma-algebra $\mathscr{B}$, sets having finite $\nu$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for $K$; and we give necessary and sufficient conditions for $K$ to have realizations and factorizations in $L^{2}\left(\nu\right)$. Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals