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P. Yuditskii

Publications and source records attributed to P. Yuditskii.

At least 19 recordsLinked to original sources
arXiv2020

Pointwise Remez inequality

The standard well-known Remez inequality gives an upper estimate of the values of polynomials on $[-1,1]$ if they are bounded by $1$ on a subset of $[-1,1]$ of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded $1$ on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik-Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.

math.CA↗
arXiv2019

Martin Functions of Fuchsian Groups and Character Automorphic Subspaces of the Hardy Space in the Upper Half Plane

We establish exact conditions for non triviality of all subspaces of the standard Hardy space in the upper half plane, that consist of character automorphic functions with respect to the action of a discrete subgroup of $SL_2(\mathbb R)$. Such spaces are natural objects in the context of the spectral theory of almost periodic differential operators and in asymptotics of approximations by entire functions. A naive idea: it should be completely parallel to the celebrated Widom characterization for Hardy spaces on Riemann surfaces with a minor modification, namely, one has to substitute the Green function of the domain with the Martin function. Basically, this is correct, but...

math.CV↗
arXiv2018

Sharp Remez inequality

Let an algebraic polynomial $P_n(ζ)$ of degree $n$ be such that $|P_n(ζ)|\le 1$ for $ζ\in E\subset\mathbb{T}$ and $|E|\ge 2π-s$. We prove the sharp Remez inequality $$ \sup_{ζ\in\mathbb{T}}|P_n(ζ)|\le \mathfrak{T}_{n}\left(\sec \frac{s} 4\right),$$ where $\mathfrak{T}_{n}$ is the Chebyshev polynomial of degree $n$. The equality holds if and only if $$ P_n(e^{iz})=e^{i(nz/2+c_1)}\mathfrak{T}_n\left(\sec\frac s 4\cos \frac {z-c_0} 2\right), \quad c_0,c_1\in\mathbb{R}. $$ This gives the solution of the long-standing problem on the sharp constant in the Remez inequality for trigonometric polynomials.

math.CA↗
arXiv2014

Killip-Simon problem and Jacobi flow on GSMP matrices

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys completely the absolutely continuos (a.c.) spectrum of the initial operator A (von Neumann). However, if A is the discrete free 1-D Schrödinger operator and B is an arbitrary Jacobi matrix (of Hilbert-Schmidt class) the a.c. spectrum remains perfectly the same, that is, the interval [-2,2]. Moreover, Killip and Simon described explicitly the spectral properties for such A+B. Jointly with Damanik they generalized this result to the case of perturbations of periodic Jacobi matrices in the non-degenerated case. Recall that the spectrum of a periodic Jacobi matrix is a system of intervals of a very specific nature. Christiansen, Simon and Zinchenko posed in a review dedicated to F. Gesztesy (2013) the following question: "is there an extension of the Damanik-Killip-Simon theorem to the general finite system of intervals case?" In this paper we solve this problem completely. Our method deals with the Jacobi flow on GSMP matrices. GSMP matrices are probably a new object in the spectral theory. They form a certain Generalization of matrices related to the Strong Moment Problem, the latter ones are a very close relative of Jacobi and CMV matrices. The Jacobi flow on them is also a probably new member of the rich family of integrable systems. Finally, related to Jacobi matrices of Killip-Simon class, analytic vector bundles and their curvature play a certain role in our construction and, at least on the level of ideology, this role is quite essential.

math.SP↗
arXiv2012

Kotani-Last problem and Hardy spaces on surfaces of Widom type

It is a small theory of non almost periodic ergodic families of Jacobi matrices with pure (however) absolutely continuous spectrum. And the reason why this effect may happen: under our "axioms" we found an analytic condition on the resolvent set that is responsible for (exactly equivalent to) this effect.

math-ph↗
arXiv2010

Scattering theory for CMV matrices: uniqueness, Helson--Szegő and Strong SzegŐ theorems

We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition is given in terms of the boundedness of a transformation operator associated to the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.

math.SP↗
arXiv2010

The scattering problem in Ryckman's class of Jacobi matrices

We give a complete solution of the scattering problem for Jacobi matrices from a class which was recently introduced by E. Ryckman. We characterize the scattering data for this class and illustrate the inverse scattering on some simple examples.

math.CV↗
arXiv2010

On Complex (non analytic) Chebyshev Polynomials in $\bbC^2$

We consider the problem of finding a best uniform approximation to the standard monomial on the unit ball in $\bbC^2$ by polynomials of lower degree with complex coefficients. We reduce the problem to a one-dimensional weighted minimization problem on an interval. In a sense, the corresponding extremal polynomials are uniform counterparts of the classical orthogonal Jacobi polynomials. They can be represented by means of special conformal mappings on the so-called comb-like domains. In these terms, the value of the minimal deviation and the representation for a polynomial of best approximation for the original problem are given. Furthermore, we derive asymptotics for the minimal deviation.

math.CA↗
arXiv2009

Asymptotics of the best polynomial approximation of $|x|^p$ and of the best Laurent polynomial approximation of $\sgn(x)$ on two symmetric intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of $|x|^p$ on two symmetric intervals. Note, that in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of $\sgn(x)$ on $[-1,-a]\cup[a,1]$ by Laurent polynomials.

math.CA↗
arXiv2008

Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) singular measure. Our main point of interest is the solution of the inverse scattering problem (the heart of the Faddeev--Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restriction of this correspondence (from a matrix to the scattering function) is one to one. In this paper we show that the main questions on inverse scattering can be solved with the help of three important classical results: Adamyan-Arov-Krein (AAK) Theory, Helson-Szego Theorem and Strong Szego Limit Theorem. Each of these theorem states the equivalence of certain conditions. Actually, to each theorem we add one more equivalent condition related to the CMV inverse scattering problem.

math.SP↗
arXiv2008

Remarks on Nehari's problem, matrix $A_2$ condition, and weighted bounded mean oscillation

We consider Nehari's problem in the case of non-uniqueness of solution. The solution set is then parametrized by the unit ball of $H^{\infty}$ by means of so-called {\em regular generators} -- bounded holomorphic functions $ϕ$. The definition of {\em regularity} is given below, but let us mention now that 1) the following assumption on modulus of $ϕ$ is sufficient for {\em regularity}: $\frac{1}{1-|ϕ|^2}\in L^1(\mathbb{T})$; 2) there is no necessary and sufficient condition of {\em regularity} on bounded holomorphic $ϕ$ in terms of $|ϕ|$ on $\mathbb{T}$, \cite{Kh1}. This makes reasonable the attempt to find a weaker sufficient condition on $|ϕ|$ than the condition in 1). This is done here. Also we are discussing certain new necessary and sufficient conditions of {\em regularity} in terms of bounded mean (weighted) oscillations of $ϕ$. They involve the matrix $A_2$ condition from \cite{TV}.

math-ph↗
arXiv2007

Reflectionless measures with a point mass and singular continuous component

We construct mesures supported on a compact subset E of the real line having zero principal value of their Cauchy integral a.e. on E with respect to Lebesgue measure and having singular components. E is sufficiently regular (Widom property is satisfied) but not homogeneous as for homogeneous spectrum such construction is impossible. This impossibility played an important role in characterizing almost periodic Jacobi matrices with homogeneous spectrum (Sodin-Yuditskii).

math-ph↗
arXiv2007

CMV matrices with asymptotically constant coefficients. Szegö over Blaschke class, Scattering Theory

We develop a modern extended scattering theory for CMV matrices with asymptotically constant Verblunsky coefficients. We demonstrate that an orthonormal system in a certain "weighted'' Hilbert space, which we call the Fadeev-Marchenko (FM) space, behaves asymptotically as the system in the standard (free) case. The duality between the two types of Hardy subspaces in it plays the key role in the proof of all asymptotics involved. We show that the traditional (Faddeev-Marchenko) condition is too restrictive to define the class of CMV matrices for which there exists a unique scattering representation. The main results are: 1) Szegö-Blaschke class: the class of twosided CMV matrices acting in $l^2$, whose spectral density satisfies the Szegö condition and whose point spectrum the Blaschke condition, corresponds precisely to the class where the scattering problem can be posed and solved. That is, to a given CMV matrix of this class, one can associate the scattering data and related to them the FM space. The CMV matrix corresponds to the multiplication operator in this space, and the orthonormal basis in it (corresponding to the standard basis in $l^2$) behaves asymptotically as the basis associated with the free system. 2) $A_2$-Carleson class: from the point of view of the scattering problem, the most natural class of CMV matrices is that one in which a) the scattering data determine the matrix uniquely and b) the associated Gelfand- Levitan- Marchenko transformation operators are bounded. Necessary and sufficient conditions for this class can be given in terms of an $A_2$ kind condition for the density of the absolutely continuous spectrum and a Carleson kind condition for the discrete spectrum. Similar close to the optimal conditions are given directly in terms of the scattering data.

math.SP↗
arXiv2007

On Scattering for CMV Matrices

Adamjan-Arov (Lax--Phillips) model space is considered as a scattering representation space for a CMV matrix in context of an extended Marchenko--Faddeev scattering theory. That is, there exists a basis in which the multiplication by independent variable is a CMV matrix. This basis as well as Verblunski coefficients are computed explicitly in terms of Nehari interpolation. Asymptotically the Verblynski coefficients go to zero. Moreover, relations between the basis and wandering subspaces are established. Transformation from scattering representation to spectral representation is given.

math.SP↗
arXiv2006

Finite difference operators with a finite--band spectrum

We study the correspondence between almost periodic difference operators and algebraic curves (spectral surfaces). An especial role plays the parametrization of the spectral curves in terms of, so called, branching divisors. The multiplication operator by the covering map with respect to the natural basis in the Hardy space on the surface is the $2d+1$--diagonal matrix; the $d$--root of the product of the Green functions (counting their multiplicities) with respect to all infinite points on the surface is the symbol of the shift operator. We demonstrate an application of our general construction to a particular covering, which generate widely discussed almost periodic CMV matrices. We discuss an important theme: covering of one spectral surface by another one and related to this operation transformations on the set of multidiagonal operators (so called Renormalization Equations). We proof several new results dealing with Renormalization Equations for periodic Jacobi matrices (polynomial coverings) and the case of a rational double covering.

math.SP↗
arXiv2005

Inverse scattering problem for a special class of canonical systems and non-linear Fourier integral. Part I: asymptotics of eigenfunctions

An original approach to the inverse scattering for Jacobi matrices was suggested in a recent paper by Volberg-Yuditskii. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that generalizes the case of Sturm-Liouville (Schrödinger) operator.

math-ph↗
arXiv2005

On a new asymptotic problem in the scattering setting

In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting.

math.SP↗