arXiv subjects

Anton Baranov

Publications and source records attributed to Anton Baranov.

49 records · Page 3Linked to original sources

arXiv2013

A model space approach to some classical inequalities for rational functions

We consider the set \mathcal{R}_{n} of rational functions of degree at most n\geq1 with no poles on the unit circle \mathbb{T} and its subclass \mathcal{R}_{n,\, r} consisting of rational functions without poles in the annulus \left\{ξ:\; r\leq|ξ|\leq\frac{1}{r}\right\}. We discuss an approach based on the model space theory which brings some integral representations for functions in \mathcal{R}_{n} and their derivatives. Using this approach we obtain L^{p}-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle \mathbb{T}.

math.FA↗

arXiv2012

Recent developments in spectral synthesis for exponential systems and for non-self-adjoint operators

We survey recent results concerning the hereditary completeness of some special systems of functions and the spectral synthesis problem for a related class of linear operators. We present a solution of the spectral synthesis problem for systems of exponentials in $L^2(-π, π)$. Analogous results are obtained for the systems of reproducing kernels in the de Branges spaces of entire functions. We also apply these results (via a functional model) to the spectral theory of rank one perturbations of compact self-adjoint operators.

math.FA↗

arXiv2012

De Branges' theorem on approximation problems of Bernstein type

The Bernstein approximation problem is to determine whether or not the space of all polynomials is dense in a given weighted $C_0$-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 $C_0$-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 $\bar{F(\bar z)}$, 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.

math.CV↗

arXiv2012

A Bernstein-type inequality for rational functions in weighted Bergman spaces

Given $n\geq1$ and $r\in[0, 1),$ we consider the set $\mathcal{R}_{n, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in $\mathcal{R}_{n, r}\:$ (as n tends to infinity and r tends to 1-) in weighted Bergman spaces with "polynomially" decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with "super-polynomially" decreasing weights.

math.FA↗

arXiv2012

A restricted shift completeness problem

We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in $L^2[0,1]$ posed by M.Carlsson and C.Sundberg.

math.CV↗

arXiv2012

Hereditary completeness for systems of exponentials and reproducing kernels

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{iλ_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional defect. This means that there is at most one (up to a constant factor) function $f$ which is orthogonal to all the summands in its formal Fourier series $\sum_n (f,\tilde e_n) e^{iλ_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to $\{e^{iλ_n t}\}$. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.

math.CV↗

arXiv2011

Systems of reproducing kernels and their biorthogonal: completeness or incompleteness?

Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems of reproducing kernels in a reproducing kernel Hilbert space $\mathcal{H}$ of analytic functions. We study the completeness problem for a class of spaces with a Riesz basis of reproducing kernels and for model subspaces $K_Θ$ of the Hardy space. We find a class of spaces where systems biorthogonal to complete systems of reproducing kernels are always complete, and show that in general this is not true. In particular we answer the question posed by N.K. Nikolski and construct a model subspace with a non-complete biorthogonal system.

math.CV↗

arXiv2010

Symbols of truncated Toeplitz operators

We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: 1) properties of truncated Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by Donald Sarason.

math.FA↗

arXiv2009

Majorization in de Branges spaces I. Representability of subspaces

In this series of papers we study subspaces of de Branges spaces of entire functions which are generated by majorization on subsets $D$ of the closed upper half-plane. The present, first, part is addressed to the question which subspaces of a given de Branges space can be represented by means of majorization. Results depend on the set $D$ where majorization is permitted. Significantly different situations are encountered when $D$ is close to the real axis or accumulates to $i\infty$.

math.CV↗

arXiv2009

Majorization in de Branges spaces II. Banach spaces generated by majorants

This is the second part in a series dealing with subspaces of de Branges spaces of entire function generated by majorization on subsets of the closed upper half-plane. In this part we investigate certain Banach spaces generated by admissible majorants. We study their interplay with the original de Branges space structure, and their geometry. In particular, we will show that, generically, they will be nonreflexive and nonseparable.

math.CV↗

arXiv2009

The Feichtinger conjecture for reproducing kernels in model subspaces

We obtain two results concerning the Feichtinger conjecture for systems of normalized reproducing kernels in the model subspace $K_Θ= H^2\ominus ΘH^2$ of the Hardy space $H^2$, where $Θ$ is an inner function. First, we verify the Feichtinger conjecture for the kernels $ \tilde k_{λ_n} = k_{λ_n}/\|k_{λ_n}\|$ under the assumption that $\sup_n |Θ(λ_n)|<1$. Secondly, we prove the Feichtinger conjecture in the case where $Θ$ is a one-component inner function, meaning that the set $\{z:|Θ(z)|<\varepsilon\}$ is connected for some $\varepsilon\in(0,1)$.

math.CV↗

arXiv2008

Weighted norm inequalities for de Branges--Rovnyak spaces and their applications

Let $\mathcal{H}(b)$ denote the de Branges--Rovnyak space associated with a function $b$ in the unit ball of $H^\infty(\mathbb{C}_+)$. We study the boundary behavior of the derivatives of functions in $\mathcal{H}(b)$ and obtain weighted norm estimates of the form $\|f^{(n)}\|_{L^2(μ)} \le C\|f\|_{\mathcal{H}(b)}$, where $f \in \mathcal{H}(b)$ and $μ$ is a Carleson-type measure on $\mathbb{C}_+\cup\mathbb{R}$. We provide several applications of these inequalities. We apply them to obtain embedding theorems for $\mathcal{H}(b)$ 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 $\{k^b_{λ_n}\}$ in $\mathcal{H}(b)$ under small perturbations of the points $λ_n$.

math.CV↗

arXiv2006

Majorants of meromorphic functions with fixed poles

Let $B$ be a meromorphic Blaschke product in the upper half-plane with zeros $z_n$ and let $K_B=H^2\ominus BH^2$ be the associated model subspace of the Hardy class. In other words, $K_B$ is the space of square summable meromorphic functions with the poles at the points $\bar z_n$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_B$ if there is a non-zero function $f\in K_B$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We study the relations between the distribution of the zeros of a Blaschke product $B$ and the class of admissible majorants for the space $K_B$.

math.CV↗