Schur flow for orthogonal polynomials on the unit circle and its integrable discretization

A one-parameter deformation of the measure of orthogonality for orthogonal polynomials on the unit circle is considered. The corresponding dynamics of the Schur parameters of the orthogonal polynomials is shown to be characterized by the complex semi-discrete modified KdV equation, namely, the Schur flow. A discrete analogue of the Miura transformation is found. An integrable discretization of the Schur flow enables us to compute a Padé approximation of the Carathéodory functions, or equivalently, to compute a Perron–Carathéodory continued fraction in a polynomial time.     

On Ranges and Moore�CPenrose Inverses Related to Matrix Carath��odory and Schur Functions

By Cayley transformation, there is an interplay between matrix-valued Carathéodory and Schur functions in the unit disk. One of the main aims of the paper is to study this interplay with a view to the fact that the matrix function given by the values of a Carathéodory function via Moore–Penrose inverses is also a Carathéodory function. In doing so, we also analyze ranges and null spaces of the values of the matrix function in question and get some results of independence concerning the concretely chosen point of the domain.     

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk

It is shown that the following conditions are equivalent for the generalized Schur class functions at a boundary point t₀ ∈ ??: 1) Carathéodory–Julia type condition of order n; 2) agreeing of asymptotics of the original function from inside and of its continuation by reflection from outside of the unit disk ?? up to order 2n + 1; 3) t₀‐isometry of the coefficients ofthe boundary asymptotics; 4) a certain structured matrix ? constructed from these coefficients being Hermitian. It is also shown that for an arbitrary analytic function, properties 2), 3), 4) are still equivalent to each other and imply 1) (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions

A general interpolation problem (which includes as particular cases the Nevanlinna–Pick and Carathéodory–Fejér interpolation problems) is considered in two classes of slice hyperholomorphic functions of the unit ball of the quaternions. In the Hardy space of the unit ball we present a Beurling–Lax type parametrization of all solutions, and the formula for the minimal norm solution. In the class of functions slice hyperholomorphic in the unit ball and bounded by one in modulus there (that is, in the class of Schur functions in the present framework) we present a necessary and sufficient condition for the problem to have a solution, and describe the set of all solutions in the indeterminate case.     

Higher‐order Sobolev‐type embeddings on Carnot–Carathéodory spaces

A sufficient condition for higher‐order Sobolev‐type embeddings on bounded domains of Carnot–Carathéodory spaces is established for the class of rearrangement‐invariant function spaces. The condition takes form of a one‐dimensional inequality for suitable integral operators depending on the isoperimetric function relative to the Carnot–Carathéodory structure of the relevant sets. General results are then applied to particular Sobolev spaces built upon Lebesgue, Lorentz and Orlicz spaces on John domains in the Heisenberg group. In the case of the Heisenberg group, the condition is shown to be necessary as well.     

On a Parametrization Formula for the Solution Set of a Completely Indeterminate Generalized Matricial Carathéodory–Fejér Problem

A parametrization formula for the solution set of a completely indeterminate generalized matricial Carathéodory–Fejér problem is given. The unique A–normalized γ–generating quadruple is constructed by a limit procedure. Moreover, its blocks are expressed as Gramians.     

Nonlinear Elliptic Equations with Singular Terms and Combined Nonlinearities

We consider nonlinear elliptic Dirichlet problems with a singular term, a concave (i.e., (p − 1)-sublinear) term and a Carathéodory perturbation. We study the cases where the Carathéodory perturbation is (p − 1)-linear and (p − 1)-superlinear near +∞. Using variational techniques based on the critical point theory together with truncation arguments and the method of upper and lower solutions, we show that if the L ^∞-coefficient of the concave term is small enough, the problem has at least two nontrivial smooth solutions.     

On Generalized Carathéodory's Conditions in Ordinary Differential Equations

In this paper we prove that generalized Carathéodory's conditions (so called (G) conditions) imply well - known general conditions which guarantee existence and some properties of solutions of the Cauchy problem, in the Carathéodory sense, as e.g. continuous dependence on initial conditions.     

Solution of a multiple Nevanlinna–Pick problem for Carathéodory functions via orthogonal rational functions in the matrix case

We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary

Abstract

In this work, we study the Carathéodory approximate solution for a class of one-dimensional perturbed stochastic differential equations with reflecting boundary (PSDERB). Based on the Carathéodory approximation procedure, we prove that PSDERB have a unique solution and show that the Carathéodory approximate solution converges to the solution of PSDERB whose both drift and diffusion coefficients are non-Lipschitz. After that, we establish an explicit rate of convergence in the case of PSDERB with Lipschitz coefficients.     

Boundary behaviour of invariant functions on planar domains

Precise behaviour of the Carathéodory, Kobayashi and Bergman metrics and distances near smooth boundary points of planar domains is found under various assumptions of regularity.     

A topological principle for retarded functional differential equations of Carathéodory type

The paper presents a Razumikhin-type version of Wa?ewski's principle for RFDEs of Carathéodory type, thus generalizing the author's previous results (J. Differential Equations36, 117–138 (1980)). A few facts from the theory of RFDEs are reviewed, and the concept of a regular polyfacial set with respect to an RFDE of Carathéodory type is introduced and the results are state and proved. Most of the examples given in the above work can easily be adapted to the more general setting of Carathéodory systems. An indication how to do it is given in only one case. Throughout the paper, standard notation is used.     

Ereditarietà delle misure di Caratheodory

Let μ be a diffuse Carathéodory measure. The purpose of this paper is to prove that, for every open setU inR, μ(·∩U) is a Carathéodory measure too.     

Pseudoconvexity and Gromov hyperbolicity

We give an estimate for the distance functions related to the Bergman, Carathéodory, and Kobayashi metrics on a bounded strictly pseudoconvex domain with C²-smooth boundary. Our formula relates the distance function on the domain with the Carnot- Carathéodory metric on the boundary. As a corollary we conclude that the domain equipped with the any of the standard invariant distances is hyperbolic in the sense of Gromov. When the boundary of the domain is C³-smooth, our estimate is exact up to a fixed additive term.     

On the Schröder-Bernstein problem for carathéodory vector lattices

In this note we prove that there exists a Carathéodory vector lattice V such that V ≌ V ³ and V ?V ². This yields that V is a solution of the Schröder-Bernstein problem for Carathéodory vector lattices. We also show that no Carathéodory Banach lattice is a solution of the Schröder-Bernstein problem.     

On a boundary analogue of the Carathéodory-Schur interpolation problem

Characterization of Schur functions in terms of their Taylor coefficients is due to C. Carathéodory and I. Schur. We discuss the boundary analogue of this problem.

     

On boundary value problems for an <Emphasis Type="Italic">n</Emphasis>th-order equation

For an nth-order differential equation with right-hand side satisfying the Carathéodory conditions, we consider a two-point boundary value problem with boundary conditions of which the first two have a general form and the remaining conditions are simplest. We show that the existence of a lower and an upper function and the compactness condition imply the solvability of this boundary value problem.     

Convex separation by regular pairs

It is known that the existence of a convex (resp., concave) separator between two given functions can be characterized via a simple inequality. The notion of convexity can be generalized applying regular pairs (in other words, two dimensional Chebyshev systems). The aim of the present note is to extend the above mentioned result to this setting. In the proof, a modified version of the classical Carathéodory’s theorem and the characterization of convex functions play the key role.     

The Carathéodory topology for multiply connected domains I

We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.