arXiv subjects
Publications and source records attributed to A. Volberg.
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.
Our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed by an arbitrary measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.
First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $ϕ, ϕ(0)=1$, such that $\|(Bϕ)'\| \leq C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotic of orthogonal polynomials.
In this work we build a certain machine that allows to construct almost periodic Jacobi matrices with singularly continuous spectrum a prescribed p-adic hull.
We consider the Jacobi matrix generated by a balanced measure of hyperbolic polynomial map. The conjecture of Bellissard says that this matrix should have an extremely strong periodicity property. We show how this conjecture is related to a certain noncommutative version of Bowen--Ruelle theory, and how the two weight Hilbert transform naturally appears in this context.
We prove a partial result concerning the long-standing problem on limit periodicity of the Jacobi matrix associated with the balanced measure on the Julia set of an expending polynomial. Besides this, connections of the problem with the Faybusovich--Gekhtman flow and many other objects (the Hilbert transform, the Schwarz derivative, the Ruelle and Laplace operators) that, we sure, are of an independent interest, are discussed.
This work is in a stream initiated by a paper of Killip and Simon [Ann. of Math. (2003)]. Using methods of Functional Analysis and the classical Szegö Theorem we prove sum rule identities in a very general form. Then, we apply the result to obtain new asymptotics for orthonormal polynomials.
Consider a class of functions of one real variable with the following uniqueness property: if a function f(x) from the class vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for f(x) outside a small exceptional set. Such estimates are well-known and useful for polynomials and analytic functions. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions.
The goal of this paper is to attract attention of the reader to a dimension-free geometric inequality that can be proved using the classical needle decomposition. This inequality allows us to derive sharp dimension-free estimates for the distribution of values of polynomials in n-dimensional convex bodies. Such estimates, in their turn, lead to a surprising result about the distribution of zeroes of random analytic functions; informally speaking, we show that for simple families of analytic functions, there exists a "typical" distribution of zeroes such that the portion of the family occupied by the functions whose distribution of zeroes deviates from that typical one by some fixed amount, is about Const exp{-size of the deviation}. The paper is essentially self-contained. When choosing the style, we tried to make it an enjoyable reading for both a senior undergraduate student and an expert. As to the question "What is new in the paper?", we beleive that the answer to it is a function of two variables, the first being "what is written" and the second being "who is reading". Since we have no knowledge of the value of the second variable, we can only give the range of answers with the first variable fixed. For the targeted audience it will be the standard range [Nothing, Everything] (with both endpoints included).
This is a companion to our previous paper. Here, we derive local dimension-free estimates for volumes of sub- and super-level sets of analytic functions of several variables.
Solving inverse scattering problem for a discrete Sturm-Liouville operator with the fast decreasing potential one gets reflection coefficients $s_\pm$ and invertible operators $I+H_{s_\pm}$, where $ H_{s_\pm}$ is the Hankel operator related to the symbol $s_\pm$. The Marchenko-Fadeev theorem (in the continuous case) and the Guseinov theorem (in the discrete case), guarantees the uniqueness of solution of the inverse scattering problem. In this article we asks the following natural question --- can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators $I+H_{s_\pm}$ are invertible? Can one claim that uniqueness implies invertibility or vise versa? Moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials. So we merege here two mostly developed cases of inverse problem for Sturm-Liouville operators: the inverse problem with (almost) periodic potential and the inverse problem with the fast decreasing potential.