221 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

### 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

### 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

### 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

### Use of operator algebras in the analysis of measures from wavelets and iterated function systems

In this paper, we show how a class of operators used in the analysis of
measures from wavelets and iterated function systems may be understood from a
special family of representations of Cuntz algebras

### 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

### Some recent trends from research mathematics and their connections to teaching: Case studies inspired by parallel developments in science and technology

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.Comment: 9 pages, "article" document class, expanded version of an invited
presentation "Teaching of mathematics at various levels in an international
university-system, and connections to research and to current trends in
technology" at the Symposium on Mathematics Education Reform (2006 Beijing,
China) http://www.math.ohiou.edu/~shen/calculus/schedule.htm

### A family of measures associated with iterated function systems

Let $(X,d)$ be a compact metric space, and let an iterated function system
(IFS) be given on $X$, i.e., a finite set of continuous maps $\sigma_{i}$: $X\to X$, $i=0,1,..., N-1$. The maps $\sigma_{i}$ transform the measures $\mu$
on $X$ into new measures $\mu_{i}$. If the diameter of $\sigma_{i_{1}}\circ
>... \circ \sigma_{i_{k}}(X)$ tends to zero as $k\to \infty$, and if
$p_{i}>0$ satisfies $\sum_{i}p_{i}=1$, then it is known that there is a unique
Borel probability measure $\mu$ on $X$ such that \mu =\sum_{i}p_{i} \mu_{i}
\tag{*}.
In this paper, we consider the case when the $p_{i}$s are replaced with a
certain system of sequilinear functionals. This allows us to study the variable
coefficient case of (*), and moreover to understand the analog of (*) which is
needed in the theory of wavelets.Comment: 14 pages including references. Corrections made on pp.4 and 1

### Some second-order partial differential equations associated with Lie groups

In this note we survey results in recent research papers on the use of Lie
groups in the study of partial differential equations. The focus will be on
parabolic equations, and we will show how the problems at hand have solutions
that seem natural in the context of Lie groups. The research is joint with D.W.
Robinson, as well as other researchers who are listed in the references.Comment: 11 pages, 1 EPS graphic, LaTeX2e amsart document class, AMS Cyrillic
font used in bibliography. This paper is an expanded version of a lecture
given by the author at the National Research Symposium on Geometric Analysis
and Applications at the the Centre for Mathematics and its Applications at
The Australian National University in June of 200

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