On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003     

On the Carathéodory superposition of multifunctions and an existence theorem

Let I ? ? be an interval and Y a reflexive Banach space. We introduce the (H) property of a multifunction F from I × Y to Y and prove that the Carathéodory superposition of F with each continuous function f from I to Y is a derivative provided that F has the (H) property. Some application of this theorem to the existence of solutions of differential inclusions f′(x) ∈ F(x, f(x)) is given.     

Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem.     

On 4-flattening theorems and the curves of Carathéodory, Barner and Segre

We discuss three classes of closed curves in the Euclidean space $\mathbb{R}^{3}$ which have non-vanishing curvature and at least 4 flattenings (points at which the torsion vanishes). Calling these classes (de.ned below) Barner, Segre and Carathéodory, we prove that Barner $\subset$ (Segre $\cap$ Carathéodory). We also prove that (Segre)\ (Segre $\cap$ Carathéodory) and (Carathéodory)\(Segre $\cap$ Carathéodory) are open sets in the space of closed smooth curves with the C-topology. Finally, we define a class of closed curves containing the class of Segre curves and -based on contact topology considerations, as the Huygens principle- we establish the conjecture that any curve of our class has at least 4 flattenings.     

A New Approach to Investigation of Carnot�CCarath��odory Geometry

We develop a new approach to studying the geometry of Carnot–Carathéodory spaces under minimal assumptions on the smoothness of basis vector fields. We obtain quantitative comparison estimates for the local geometries of two different local Carnot groups, as well as of a local Carnot group and the original space. As corollaries, we deduce some results that are well-known and basic for the “smooth” case: the generalized triangle inequality for d _∞, the local approximation theorem for the quasimetric d _∞, the Rashevskiǐ–Chow theorem, the ball-box theorem, and so on.     

On the Equivalences of Matrix Norms Related to the l_p Norms

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An operator-theoretic approach to theorems of the Pick-Nevanlinna and Carathéodory types

In this paper we deal with singularities of the linear systems of plane curves passing through S, where S is a zerodimensional closed subscheme of degree n of P ²=P _k ² ,k an algebraically closed field of any characteristic. We determine the least degree of a nonsingular curve passing through S, when S is in uniform position. This paper was written while the author was member of C.N.R., Sez. 3 of G.N.S.A.G.A. and was supported by M.P.I. funds     

Analytic Description of the Spaces Dual to Spaces of Holomorphic Functions of Given Growth on Carathéodory Domains

The spaces dual to spaces of holomorphic functions of given growth on Carathéodory domains are described by using the Cauchy transform of functionals. A pseudoanalytic extension of such transforms to the whole plane is constructed, which makes it possible to remove convexity constrains and consider spaces determined by weights of general form, rather than only by those whose dependence on the distance from a point of the domain to its boundary is one-dimensional.     

A Maximal Function Approach to Christoffel Functions and Nevai��s Operators

Let μ be a compactly supported positive measure on the real line, with associated Christoffel functions λ _n (d μ,⋅). Let g be a measurable function that is bounded above and below on supp[μ] by positive constants. We show that λ _n (g  d μ,⋅)/λ _n (d μ,⋅)→g in measure in {x:μ′(x)>0} and consequently in all L _p norms, p<∞. The novelty is that there are no local or global restrictions on μ. The main idea is a new maximal function estimate for the “tail” in Nevai’s operators.     

Continuity of Some Classes of Functions Related to Degenerate Parabolic Equations and Its Applications

We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.     

Boundedness of Area Functions Related to Schrödinger Operators and Their Commutators in Weighted Hardy Spaces

In this paper, we consider the area function $S_Q$ related to the Schrödinger operator $\mathcal{L}$ and its commutator $S_{Q,b}$, establish the boundedness of $S_Q$ from $H^p_\rho(w)$ to $L^p(w)$ or $WL^p(w),$ as well as the boundedness of $S_{Q,b}$ from $H^1_\rho(w)$ to $WL^1(w).$     

On the relation between the darlington realizations of matrix functions from the Carathéodory class and their J p,r -inner SI-dilations

The paper is devoted to establishing a two-sided relation between the Darlington realizations of matrix functions from the Carathéodory class and their J _p,r -inner SI-dilations.     

On Summation Formulas Due to Plana, Lindelöf and Abel, and Related Gauss-Christoffel Rules, III

This is the last part of a work that has been divided into three parts. It contains Sections 4 and 5. Part I and Part II were published in BIT, vol. 37. Some amendments to the earlier parts are given in an appendix of this part.The reader is referred to Part I for the abstract, the contents and the references of the whole work. A short list of references for this part is found at the end of this part. A few new references of interest for the earlier parts are included.     

On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal <Emphasis Type="Italic">d</Emphasis>-Sets

The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1?s,d/s} is obtained. In particular, when passing from d≥s to d<s there is a change of behaviour from d+1?s to d/s which implies that even highly nonsmooth functions defined on cubes in ? ⁿ have not so rough graphs when restricted to, say, rarefied fractals.     

Infinite-Dimensional Representations of the Lie Algebra \mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right) Related to Complex Analogs of the Gelfand–Tsetlin Patterns and General Hypergeometric Functions on the Lie Group GL\left( {n,\mathbb{C}} \right)

Complex analogs of the Gelfand–Tsetlin patterns are introduced. Infinite-dimensional representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ in the vector spaces spanned on these patterns are constructed. Exponentials of these representations are described. These exponentials are operators T(x), x∈GL(n,C), defined only in neighborhoods of the identity element of GL(n,C). A system of differential-difference equations for matrix elements of operators T(x) is constructed. Explicit formulas for matrix elements are obtained for the case x∈Z ^±, where Z ⁺ and Z ^? are the triangular unipotent subgroups. Representations of $\mathfrak{g}\mathfrak{l}\left( {n,\mathbb{C}} \right)$ are also constructed; bases of these representations consist of Gelfand–Tsetlin patterns having infinitely many rows.