175 research outputs found
De Branges spaces and Krein's theory of entire operators
This work presents a contemporary treatment of Krein's entire operators with
deficiency indices and de Branges' Hilbert spaces of entire functions.
Each of these theories played a central role in the research of both renown
mathematicians. Remarkably, entire operators and de Branges spaces are
intimately connected and the interplay between them has had an impact in both
spectral theory and the theory of functions. This work exhibits the
interrelation between Krein's and de Branges' theories by means of a functional
model and discusses recent developments, giving illustrations of the main
objects and applications to the spectral theory of difference and differential
operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors
were corrected. The bibliography style was change
de Branges-Rovnyak spaces: basics and theory
For a contractive analytic operator-valued function on the unit disk
, de Branges and Rovnyak associate a Hilbert space of analytic
functions and related extension space
consisting of pairs of analytic functions on the unit disk . This
survey describes three equivalent formulations (the original geometric de
Branges-Rovnyak definition, the Toeplitz operator characterization, and the
characterization as a reproducing kernel Hilbert space) of the de
Branges-Rovnyak space , as well as its role as the underlying
Hilbert space for the modeling of completely non-isometric Hilbert-space
contraction operators. Also examined is the extension of these ideas to handle
the modeling of the more general class of completely nonunitary contraction
operators, where the more general two-component de Branges-Rovnyak model space
and associated overlapping spaces play key roles. Connections
with other function theory problems and applications are also discussed. More
recent applications to a variety of subsequent applications are given in a
companion survey article
The Zero-Removing Property and Lagrange-Type Interpolation Series
The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros
Applications of M.G. Krein's Theory of Regular Symmetric Operators to Sampling Theory
The classical Kramer sampling theorem establishes general conditions that
allow the reconstruction of functions by mean of orthogonal sampling formulae.
One major task in sampling theory is to find concrete, non trivial realizations
of this theorem. In this paper we provide a new approach to this subject on the
basis of the M. G. Krein's theory of representation of simple regular symmetric
operators having deficiency indices (1,1). We show that the resulting sampling
formulae have the form of Lagrange interpolation series. We also characterize
the space of functions reconstructible by our sampling formulae. Our
construction allows a rigorous treatment of certain ideas proposed recently in
quantum gravity.Comment: 15 pages; v2: minor changes in abstract, addition of PACS numbers,
changes in some keywords, some few changes in the introduction, correction of
the proof of the last theorem, and addition of some comments at the end of
the fourth sectio
Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces
The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established
On the class SI of J-contractive functions intertwining solutions of linear differential equations
In the PhD thesis of the second author under the supervision of the third
author was defined the class SI of J-contractive functions, depending on a
parameter and arising as transfer functions of overdetermined conservative 2D
systems invariant in one direction. In this paper we extend and solve in the
class SI, a number of problems originally set for the class SC of functions
contractive in the open right-half plane, and unitary on the imaginary line
with respect to some preassigned signature matrix J. The problems we consider
include the Schur algorithm, the partial realization problem and the
Nevanlinna-Pick interpolation problem. The arguments rely on a correspondence
between elements in a given subclass of SI and elements in SC. Another
important tool in the arguments is a new result pertaining to the classical
tangential Schur algorithm.Comment: 46 page
Generalized KdV Equation for Fluid Dynamics and Quantum Algebras
We generalize the non-linear one-dimensional equation of a fluid layer for
any depth and length as an infinite order differential equation for the steady
waves. This equation can be written as a q-differential one, with its general
solution written as a power series expansion with coefficients satisfying a
nonlinear recurrence relation. In the limit of long and shallow water (shallow
channels) we reobtain the well known Korteweg-de-Vries equation together with
its single-soliton solution.Comment: 17 pages, Latex, PACS: 47.20.Ky, 43.25.Rq, 47.35.+i, 03.40.Kf,
43.25.Fe, 02.20.Tw, MSC: 16W30, 17B37, 81R50, 35Q51, 34B15, 34L30, 76E3
Theoretical Insights into the Use of Structural Similarity Index In Generative Models and Inferential Autoencoders
Generative models and inferential autoencoders mostly make use of
norm in their optimization objectives. In order to generate perceptually better
images, this short paper theoretically discusses how to use Structural
Similarity Index (SSIM) in generative models and inferential autoencoders. We
first review SSIM, SSIM distance metrics, and SSIM kernel. We show that the
SSIM kernel is a universal kernel and thus can be used in unconditional and
conditional generated moment matching networks. Then, we explain how to use
SSIM distance in variational and adversarial autoencoders and unconditional and
conditional Generative Adversarial Networks (GANs). Finally, we propose to use
SSIM distance rather than norm in least squares GAN.Comment: Accepted (to appear) in International Conference on Image Analysis
and Recognition (ICIAR) 2020, Springe
The class of n-entire operators
We introduce a classification of simple, regular, closed symmetric operators
with deficiency indices (1,1) according to a geometric criterion that extends
the classical notions of entire operators and entire operators in the
generalized sense due to M. G. Krein. We show that these classes of operators
have several distinctive properties, some of them related to the spectra of
their canonical selfadjoint extensions. In particular, we provide necessary and
sufficient conditions on the spectra of two canonical selfadjoint extensions of
an operator for it to belong to one of our classes. Our discussion is based on
some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2.
References added. Examples added. arXiv admin note: text overlap with
arXiv:1104.476
Schur functions and their realizations in the slice hyperholomorphic setting
we start the study of Schur analysis in the quaternionic setting using the
theory of slice hyperholomorphic functions. The novelty of our approach is that
slice hyperholomorphic functions allows to write realizations in terms of a
suitable resolvent, the so called S-resolvent operator and to extend several
results that hold in the complex case to the quaternionic case. We discuss
reproducing kernels, positive definite functions in this setting and we show
how they can be obtained in our setting using the extension operator and the
slice regular product. We define Schur multipliers, and find their co-isometric
realization in terms of the associated de Branges-Rovnyak space
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