354 research outputs found
On zero sets in the Dirichlet space
We study the zeros sets of functions in the Dirichlet space. Using Carleson
formula for Dirichlet integral, we obtain some new families of zero sets. We
also show that any closed subset of E \subset \TT with logarithmic capacity
zero is the accumulation points of the zeros of a function in the Dirichlet
space. The zeros satisfy a growth restriction which depends on .Comment: Journal of Geometric Analysis (2011
On the structure of covariant phase observables
We study the mathematical structure of covariant phase observables. Such an
observable can alternatively be expressed as a phase matrix, as a sequence of
unit vectors, as a sequence of phase states, or as an equivalent class of
covariant trace-preserving operations. Covariant generalized operator measures
are defined by structure matrices which form a W*-algebra with phase matrices
as its subset. The properties of the Radon-Nikodym derivatives of phase
probability measures are studied.Comment: 11 page
Local dynamics for fibered holomorphic transformations
Fibered holomorphic dynamics are skew-product transformations over an
irrational rotation, whose fibers are holomorphic functions. In this paper we
study such a dynamics on a neighborhood of an invariant curve. We obtain some
results analogous to the results in the non fibered case
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Divided Differences & Restriction Operator on Paley-Wiener Spaces for Carleson Sequences
For a sequence of complex numbers we consider the restriction
operator defined on Paley-Wiener spaces
(). Lyubarskii and Seip gave necessary and sufficient conditions on
for to be an isomorphism between and a
certain weighted space. The Carleson condition appears to be necessary.
We extend their result to Carleson sequences (finite unions of disjoint
Carleson sequences). More precisely, we give necessary and sufficient
conditions for to be an isomorphism between and
an appropriate sequence space involving divided differences
Interpolation and harmonic majorants in big Hardy-Orlicz spaces
Free interpolation in Hardy spaces is caracterized by the well-known Carleson
condition. The result extends to Hardy-Orlicz spaces contained in the scale of
classical Hardy spaces , . For the Smirnov and the Nevanlinna
classes, interpolating sequences have been characterized in a recent paper in
terms of the existence of harmonic majorants (quasi-bounded in the case of the
Smirnov class). Since the Smirnov class can be regarded as the union over all
Hardy-Orlicz spaces associated with a so-called strongly convex function, it is
natural to ask how the condition changes from the Carleson condition in
classical Hardy spaces to harmonic majorants in the Smirnov class. The aim of
this paper is to narrow down this gap from the Smirnov class to ``big''
Hardy-Orlicz spaces. More precisely, we characterize interpolating sequences
for a class of Hardy-Orlicz spaces that carry an algebraic structure and that
are strictly bigger than . It turns out that the
interpolating sequences are again characterized by the existence of
quasi-bounded majorants, but now the weights of the majorants have to be in
suitable Orlicz spaces. The existence of harmonic majorants in such Orlicz
spaces will also be discussed in the general situation. We finish the paper
with an example of a separated Blaschke sequence that is interpolating for
certain Hardy-Orlicz spaces without being interpolating for slightly smaller
ones.Comment: 19 pages, 2 figure
Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis
We study the behaviour of the Standard map critical function in a
neighbourhood of a fixed resonance, that is the scaling law at the fixed
resonance. We prove that for the fundamental resonance the scaling law is
linear. We show numerical evidence that for the other resonances , , and and relatively prime, the scaling law follows a
power--law with exponent .Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit
Are Devaney hairs fast escaping?
Beginning with Devaney, several authors have studied transcendental entire
functions for which every point in the escaping set can be connected to
infinity by a curve in the escaping set. Such curves are often called Devaney
hairs. We show that, in many cases, every point in such a curve, apart from
possibly a finite endpoint of the curve, belongs to the fast escaping set. We
also give an example of a Devaney hair which lies in a logarithmic tract of a
transcendental entire function and contains no fast escaping points.Comment: 22 pages, 1 figur
Operator theory and function theory in Drury-Arveson space and its quotients
The Drury-Arveson space , also known as symmetric Fock space or the
-shift space, is a Hilbert function space that has a natural -tuple of
operators acting on it, which gives it the structure of a Hilbert module. This
survey aims to introduce the Drury-Arveson space, to give a panoramic view of
the main operator theoretic and function theoretic aspects of this space, and
to describe the universal role that it plays in multivariable operator theory
and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page
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