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Anton Baranov - One of the best experts on this subject based on the ideXlab platform.
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Krein-type theorems and orDered structure for Cauchy-De Branges spaces
arXiv: Complex Variables, 2018Co-Authors: Evgeny Abakumov, Anton Baranov, Yurii BelovAbstract:We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of De Branges' OrDering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.
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De Branges spaces and Fock spaces
arXiv: Complex Variables, 2018Co-Authors: Anton Baranov, Hélène Bommier-hatoAbstract:Relations between two classes of Hilbert spaces of entire functions, De Branges spaces and Fock-type spaces with non-radial weights, are studied. It is shown that any De Branges space can be realized as a Fock-type space with equivalent area norm, and several constructions of a representing weight are suggested. For some special classes of weights (e.g., weights Depending on the imaginary part only) the corresponding De Branges spaces are explicitly Described.
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De Branges spaces and Fock spaces
Complex Variables and Elliptic Equations, 2017Co-Authors: Anton Baranov, Hélène Bommier-hatoAbstract:Relations between two classes of Hilbert spaces of entire functions, De Branges spaces and Fock-type spaces with nonradial weights, are studied. It is shown that any De Branges space can be realize...
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Fock type spaces with Riesz bases of reproducing kernels and De Branges spaces
Studia Mathematica, 2017Co-Authors: Anton Baranov, Yurii Belov, Alexander BorichevAbstract:We Describe the radial Fock type spaces which possess Riesz bases of normalized reproducing kernels and which are (are not) isomorphic to De Branges spaces in terms of the weight functions.
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Riesz bases of reproducing kernels in Fock type spaces and De Branges spaces
arXiv: Complex Variables, 2015Co-Authors: Anton Baranov, Yurii Belov, Alexander BorichevAbstract:We Describe the radial Fock type spaces which possess Riesz bases of normalized reproducing kernels and which are (are not) isomorphic to De Branges spaces in terms of the weight functions.
Harald Woracek - One of the best experts on this subject based on the ideXlab platform.
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Perturbation of chains of De Branges spaces
Journal d'Analyse Mathématique, 2018Co-Authors: Harald WoracekAbstract:We investigate the structure of the set of De Branges spaces of entire functions which are contained in a space L2(μ). Thereby, we follow a perturbation approach. The main result is a growth DepenDent stability theorem. Namely, assume that measures μ1 and μ2 are close to each other in a sense quantified relative to a proximate orDer. ConsiDer the sections of corresponding chains of De Branges spaces C1 and C2 which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate orDer. Then either these sections coinciDe or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.
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De Branges Spaces and Growth Aspects
Operator Theory, 2014Co-Authors: Harald WoracekAbstract:The subject of this survey is to review the basics of Louis De Branges’ theory of Hilbert spaces of entire functions, and to present results bringing together the notions of De Branges spaces on one hand and growth functions (proximate orDers) on the other hand. After a few introductory words, the paper starts off with a short companion on De Branges theory (section “A Short Companion on Hilbert Spaces of Entire Functions”) where much of the terminology and cornerstones of the theory are presented. Then growth functions are very briefly introduced (section “Growth Functions”). The following two sections of the survey are Devoted to growth properties. First (section “General Theorems Relating De Branges Spaces and Growth”), some general theorems, where the growth of elements of a De Branges space is discussed in relation with generating Hermite–Biehler functions and associated canonical systems, and results on growth of subspaces of a given space are presented. Second (section “Some Examples”), some more concrete examples which appear “in nature,” and where growth of different rates is exhibited. It should be said explicitly that this survey is of course far from being exhaustive. For example, since the main purpose is to study growth properties of spaces of entire functions, all what relates to spectral measures (inclusion in L2-spaces, etc.) is omitted from the presentation.
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Symmetry in De Branges almost Pontryagin spaces
Integral Equations and Operator Theory, 2013Co-Authors: Henrik Winkler, Harald WoracekAbstract:In many examples of De Branges spaces symmetry appears naturally. Presence of symmetry gives rise to a Decomposition of the space into two parts, the ‘even’ and the ‘odd’ part, which themselves can be regarDed as De Branges spaces. The converse question is to DeciDe whether a given space is the ‘even’ part or the ‘odd’ part of some symmetric space, and, if yes, to Describe the totality of all such symmetric spaces. We consiDer this question in an inDefinite (almost Pontryagin space) setting, and give a complete answer. Interestingly, it turns out that the answers for the ‘even’ and ‘odd’ cases read quite differently; the latter is significantly more complex.
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De Branges’ theorem on approximation problems of Bernstein type
Journal of the Institute of Mathematics of Jussieu, 2013Co-Authors: Anton Baranov, Harald WoracekAbstract:The Bernstein approximation problem is to Determine whether or not the space of all polynomials is Dense in a given weighted C0-space on the real line. A theorem of L. De Branges characterizes non–Density by existence of an entire function of Krein class being related with the weight in a certain way. An analogous result holds true for weighted sup–norm approximation by entire functions of exponential type at most τ and bounDed on the real axis (τ > 0 fixed). We consiDer approximation in weighted C0-spaces by functions belonging to a prescribed subspace of entire functions which is solely assumed to be invariant unDer division of zeros and passing from F (z) to F (z), and establish the precise analogue of De Branges’ theorem. For the proof we follow the lines of De Branges’ original proof, and employ some results of L. Pitt.
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Existence of zerofree functions N-associated to a De Branges Pontryagin space
Monatshefte für Mathematik, 2010Co-Authors: Harald WoracekAbstract:In the theory of De Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we Deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a De Branges Pontryagin space. Let a De Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (inDefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., Dealing with admissible majorants in De Branges spaces or the continuation problem for hermitian inDefinite functions.
Emmanuel Fricain - One of the best experts on this subject based on the ideXlab platform.
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composition operators on De Branges rovnyak spaces
Results in Mathematics, 2019Co-Authors: Emmanuel Fricain, Muath Karaki, Javad MashreghiAbstract:We study the compactness of the composition operator on De Branges–Rovnyak spaces. Inspired by a paper by Lyubarskii–Malinnikova on moDel spaces, we give some necessary and some sufficient conditions for compactness. In the paper of Lyubarskii-Malinnikova, the key point is some Bernstein inequality on moDel spaces due to Cohn (and based on a Deep inequality of Axler–Chang–Sarason involving the Hardy–Littlewood maximal function). We generalize the result of Cohn to some subspace of a De Branges–Rovnyak space (in many cases Dense) and then get a sufficient condition (analogue to Lyubarskii–Malinnikova’s condition) for compactness of the composition operator on that subspace.
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constructive approximation in De Branges rovnyak spaces
Constructive Approximation, 2016Co-Authors: Omar Elfallah, Emmanuel Fricain, Karim Kellay, Javad Mashreghi, Thomas RansfordAbstract:In most classical holomorphic function spaces on the unit disk in which the polynomials are Dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the De Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are Dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive siDe, we prove the following approximation theorem for Toeplitz operators on general De Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are Dense in \(\mathcal{H}(b)\).
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Constructive approximation in De Branges-Rovnyak spaces
arXiv: Functional Analysis, 2015Co-Authors: Omar El-fallah, Emmanuel Fricain, Karim Kellay, Javad Mashreghi, Ransford TomAbstract:In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates $f\_r(z):=f(rz)~(r \textless{} 1)$. We show that this is \emph{not} the case for the De Branges--Rovnyak spaces $\cH(b)$. More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function $f\in\cH(b)$ such that $\lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty$. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are Dense in $\cH(b)$. We give the first constructive proof of this fact.
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WEIGHTED NORM INEQUALITIES FOR De Branges-ROVNYAK SPACES AND THEIR APPLICATIONS
American Journal of Mathematics, 2010Co-Authors: Anton Baranov, Emmanuel Fricain, Javad MashreghiAbstract:Let H(b) Denote the De Branges-Rovnyak space associated with a function b in the unit ball of H 1 (C+). We study the boundary behavior of the Derivatives of functions in H(b) and obtain weighted norm estimates of the form k f (n) k L2(µ) ≤ Ck f k H(b), where f ∈ H(b) and µ is a Carleson-type measure on C+ ∪ R. We proviDe several applications of these inequalities. We apply them to obtain embedding theorems for H(b) spaces. These results extend Cohn and Volberg-Treil embedding theorems for the moDel (star-invariant) subspaces which are special classes of De Branges-Rovnyak spaces. We also exploit the inequalities for the Derivatives to study stability of Riesz bases of reproducing kernels {k bn } in H(b) unDer small perturbations of the pointsn.
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On certain Riesz families in vector-valued De Branges-Rovnyak spaces
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Nicolas Chevrot, Emmanuel Fricain, Dan TimotinAbstract:Abstract We obtain criteria for the Riesz basis property for families of reproducing kernels in vector-valued De Branges–Rovnyak spaces H ( b ) . In particular, it is shown that in several situations the property implies a special form for the function b . We also study the completeness of a related family.
Thomas Ransford - One of the best experts on this subject based on the ideXlab platform.
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outer functions and divergence in De Branges rovnyak spaces
arXiv: Complex Variables, 2019Co-Authors: Javad Mashreghi, Thomas RansfordAbstract:In most classical holomorphic function spaces on the unit disk in which the polynomials are Dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a De Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials are Dense, and a function $f\in{\mathcal H}(b)$ such that $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.
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constructive approximation in De Branges rovnyak spaces
Constructive Approximation, 2016Co-Authors: Omar Elfallah, Emmanuel Fricain, Karim Kellay, Javad Mashreghi, Thomas RansfordAbstract:In most classical holomorphic function spaces on the unit disk in which the polynomials are Dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the De Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are Dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive siDe, we prove the following approximation theorem for Toeplitz operators on general De Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are Dense in \(\mathcal{H}(b)\).
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Dirichlet spaces with superharmonic weights and De Branges-Rovnyak spaces
Complex Analysis and Operator Theory, 2015Co-Authors: Omar El-fallah, Karim Kellay, Javad Mashreghi, Hubert Klaja, Thomas RansfordAbstract:Wec onsiDer Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be iDentified as De Branges–Rovnyak spaces. As an application, we obtain the dilation inequality $D_\omega(f_r)≤ \frac{2r}{1+r} D_\omega(f) $ $(0\leq r
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which De Branges rovnyak spaces are dirichlet spaces and vice versa
Journal of Functional Analysis, 2013Co-Authors: Constantin Costara, Thomas RansfordAbstract:Abstract We investigate for which pairs b , μ the De Branges–Rovnyak space H b is equal to the generalized Dirichlet space D μ .
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De Branges rovnyak spaces and dirichlet spaces
Journal of Functional Analysis, 2010Co-Authors: Nicolas Chevrot, Dominique Guillot, Thomas RansfordAbstract:Sarason has shown that the local Dirichlet spaces Dλ may be consiDered as manifestations of De Branges–Rovnyak spaces H(b), and has used this iDentification to give a new proof that the spaces Dλ are star-shaped. We investigate which other Dirichlet spaces D(μ) arise as De Branges–Rovnyak spaces, and which other De Branges–Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces insiDe Dλ.
Nicolas Chevrot - One of the best experts on this subject based on the ideXlab platform.
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De Branges rovnyak spaces and dirichlet spaces
Journal of Functional Analysis, 2010Co-Authors: Nicolas Chevrot, Dominique Guillot, Thomas RansfordAbstract:Sarason has shown that the local Dirichlet spaces Dλ may be consiDered as manifestations of De Branges–Rovnyak spaces H(b), and has used this iDentification to give a new proof that the spaces Dλ are star-shaped. We investigate which other Dirichlet spaces D(μ) arise as De Branges–Rovnyak spaces, and which other De Branges–Rovnyak spaces H(b) are star-shaped. We also prove a transfer principle which represents H(b)-spaces insiDe Dλ.
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On certain Riesz families in vector-valued De Branges-Rovnyak spaces
Journal of Mathematical Analysis and Applications, 2009Co-Authors: Nicolas Chevrot, Emmanuel Fricain, Dan TimotinAbstract:Abstract We obtain criteria for the Riesz basis property for families of reproducing kernels in vector-valued De Branges–Rovnyak spaces H ( b ) . In particular, it is shown that in several situations the property implies a special form for the function b . We also study the completeness of a related family.