392 research outputs found

### 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

### 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 $L^{2}$-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

### 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

### 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

### Closed subspaces which are attractors for representations of the Cuntz algebras

We analyze the structure of co-invariant subspaces for representations of the
Cuntz algebras O_N for N = 2,3,..., N < infinity, with special attention to the
representations which are associated to orthonormal and tight-frame wavelets in
L^2(R) corresponding to scale number N.Comment: 32 pages, LaTeX2e "birkart" document class; accepted for publication
in the Proceedings of the 2002 IWOTA conference at Virginia Tech in
Blacksburg, VA. v4 revision: changes and corrections to Theorem 4.4 and
Corollary 7.1. Also Theorem 4.4 is relabeled "Proposition 4.4", and
clarifying remarks are adde

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