arXiv subjects
Publications and source records attributed to Peter Yuditskii.
We generalized the Korkin-Zolotarev theorem to the case of entire functions having the smallest $L^1$ norm on a system of intervals $E$. If $\bbC\setminus E$ is a domain of Widom type with the Direct Cauchy Theorem we give an explicit formula for the minimal deviation. Important relations between the problem and the theory of canonical systems with reflectionless resolvent functions are shown.
The main result of this work is a parametric description of the spectral surfaces of a class of periodic 5-diagonal matrices, related to the strong moment problem. This class is a self-adjoint twin of the class of CMV matrices. Jointly they form the simplest possible classes of 5-diagonal matrices.
We discuss a class of regions and conformal mappings which are useful in several problems of approximation theory, harmonic analysis and spectral theory.
We discuss several questions which remained open in our joint work with M. Sodin "Almost periodic Jacobi matrices with homogeneous spectrum, infinite-dimensional Jacobi inversion, and Hardy spaces of character--automorphic functions". In particular, we show that there exists a non-homogeneous set $E$ such that the Direct Cauchy Theorem (DCT) holds in the Widom domain $\bbC\setminus E$. On the other hand we demonstrate that the weak homogeneity condition on $E$ (introduced recently by Poltoratski and Remling) does not ensure that DCT holds in the corresponding Widom domain.
We describe polynomials of the best uniform approximation to sgn(x) on the union of two intervals in terms of special conformal mappings. This permits us to find the exact asymptotic behavior of the error of this approximation.
We find the exact upper estimate for the upper density of zeros of entire functions of exponential type whose indicator diagram is contained in a given interval.
The algebraic structure of V.P. Potapov's Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov--Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type.
For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function $\sgn(x)$ on the two symmetric intervals $[-1,-a]\cup[a,1]$ by rational functions with the only possible poles of degree $2k-1$ at the origin and of $2m-1$ at infinity. Then the following limit exists \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{m-{1/2}} (2m-1)^{k+{1/2}}=\frac 2 π(\frac{1-a^2}{2a})^{k+{1/2}} Γ(k+\frac 1 2). \end{equation}
We study the best uniform approximation by polynomials of fixed degree of the function sgn(x) on the union of two intervals symmetric with respect to the origin. We obtain precise asymptotics, with explicit constants, for the error of the best approximation as the degrees tend to infinity. Our approach is based on a new representation of the extremal polynomials in terms of certain conformal maps. We also study the best polynomial approximation of sgn(x) on [-1,1] with respect to the Hausdorff distance between the completed graphs.
We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain, which exhibit a singular component.
Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $σ$ be a positive measure with $\text{supp}(σ)=E$. Under the condition that the absolutely continuous part of $σ$ satisfies a Szegö--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $σ$. For the special case that $E$ consists of a finite number of intervals and that $σ$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szegö and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schrödinger equations.