437 research outputs found
Unitary groups and spectral sets
We study spectral theory for bounded Borel subsets of \br and in particular
finite unions of intervals. For Hilbert space, we take of the union of
the intervals. This yields a boundary value problem arising from the minimal
operator \Ds = \frac1{2\pi i}\frac{d}{dx} with domain consisting of
functions vanishing at the endpoints. We offer a detailed interplay
between geometric configurations of unions of intervals and a spectral theory
for the corresponding selfadjoint extensions of \Ds and for the associated
unitary groups of local translations. While motivated by scattering theory and
quantum graphs, our present focus is on the Fuglede-spectral pair problem.
Stated more generally, this problem asks for a determination of those bounded
Borel sets in \br^k such that has an orthogonal basis
of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex
exponentials restricted to .
In the general case, we characterize Borel sets having this spectral
property in terms of a unitary representation of (\br, +) acting by local
translations. The case of is of special interest, hence the
interval-configurations. We give a characterization of those geometric
interval-configurations which allow Fourier spectra directly in terms of the
selfadjoint extensions of the minimal operator \Ds. This allows for a direct
and explicit interplay between geometry and spectra. As an application, we
offer a new look at the Universal Tiling Conjecture and show that the
spectral-implies-tile part of the Fuglede conjecture is equivalent to it and
can be reduced to a variant of the Fuglede conjecture for unions of integer
intervals.Comment: We improved the paper and partition it into several independent part
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.Comment: 44 pages, LaTeX2e ("jotart" document class); v2: A few opening
paragraphs were added to the paper; an addition where a bit of the history is
explained, and where some more relevant papers are cited. Corrected a
typographical error in Proposition 8.1. v3: A few minor additions: More
motivation and explanations in the Intro; Remark 3.3 is new; and eleven
relevant references/citations are added; v4: corrected and updated
bibliography; v5: more bibliography updates and change of LaTeX document
clas
Monic representations of the Cuntz algebra and Markov measures
We study representations of the Cuntz algebras \O_N. While, for fixed ,
the set of equivalence classes of representations of \O_N is known not to
have a Borel cross section, there are various subclasses of representations
which can be classified. We study monic representations of \O_N, that have a
cyclic vector for the canonical abelian subalgebra. We show that \O_N has a
certain universal representation which contains all positive monic
representations. A large class of examples of monic representations is based on
Markov measures. We classify them and as a consequence we obtain that different
parameters yield mutually singular Markov measure, extending the classical
result of Kakutani. The monic representations based on the Kakutani measures
are exactly the ones that have a one-dimensional cyclic -invariant
space
Fourier duality for fractal measures with affine scales
For a family of fractal measures, we find an explicit Fourier duality. The
measures in the pair have compact support in \br^d, and they both have the
same matrix scaling. But the two use different translation vectors, one by a
subset in \br^d, and the other by a related subset . Among other
things, we show that there is then a pair of infinite discrete sets
and in \br^d such that the -Fourier exponentials are
orthogonal in , and the -Fourier exponentials are
orthogonal in . These sets of orthogonal "frequencies" are
typically lacunary, and they will be obtained by scaling in the large. The
nature of our duality is explored below both in higher dimensions and for
examples on the real line.
Our duality pairs do not always yield orthonormal Fourier bases in the
respective -Hilbert spaces, but depending on the geometry of certain
finite orbits, we show that they do in some cases. We further show that there
are new and surprising scaling symmetries of relevance for the ergodic theory
of these affine fractal measures.Comment: v
- …