On Rank Invariance of Schwarz-Pick-Potapov Block Matrices of Matricial Schur Functions

We derive statements on rank invariance of Schwarz-Pick-Potapov block matrices of matrix-valued Schur functions. The rank of these block matrices coincides with the rank of some block matrices built from the corresponding section matrices of Taylor coefficients. These results are applied to the discussion of a matrix version of the classical Schur-Nevanlinna algorithm.     

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     

矩阵值Caratheodory函数广义块Pick矩阵的秩不变性

Lasarow^[1]推导出矩阵值Carath\'{e}odory函数的第一、第二型广义块Pick矩阵及其变型的秩不变性. 这些矩阵由同一个Carath\'{e}odory函数的值与它的直到某阶的导数值确定. 利用文献[2]中提出的块Toeplitz向量方法, 该文断言,这些块矩阵的秩分别相关并重合于具有秩不变性的块Toeplitz矩阵的秩, 从而改进了这两类广义块Pick矩阵的秩不变性结论的证明.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.     

One-sided tangential interpolation for operator-valued Hardy functions on polydisks

All solutions of one-sided tangential interpolation problems with Hilbert norm constraints for operator-valued Hardy functions on the polydisk are described. The minimal norm solution is explicitly expressed in terms of the interpolation data.The research of this author is partially supported by NSF grant DMS 9800704, and by the Faculty Research Assignment grant from the College of William and Mary.     

Operator Matrices as Generators of Cosine Operator Functions

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in L^p(0, 1) and in L²(Ω) equipped with dynamical and acoustic-like boundary conditions, respectively.     

On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.     

Kergin Interpolants of Holomorphic Functions

Let D be a C-convex domain in C ⁿ . Let , and d = 0,1,2, ..., be an array of points in a compact set . Let f be holomorphic on and let K _d (f) denote the Kergin interpolating polynomial to f at A _d0 ,... , A _dd . We give conditions on the array and D such that . The conditions are, in an appropriate sense, optimal. This result generalizes classical one variable results on the convergence of Lagrange—Hermite interpolants of analytic functions. Date received: October 21, 1995. Date revised: May 1, 1996.     

Multivariate Jackson Inequality

By known multivariate versions of the classical Jackson theorem, every compact cube P in R^N admits Jackson’s inequality. The purpose of this note is to deliver other examples of Jackson sets in R^N. We shall show that a finite union of disjoint Jackson compact sets in R^N is also a Jackson set and that this in general fails to hold for an infinite union of Jackson sets. We also give a characterization of Jackson sets in the family of Markov compact sets in R^N which together with a Bierstone result permits one to show that Whitney regular compact subsets of R^N are Jackson.     

The Schur Algorithm for Generalized Schur Functions II: Jordan Chains and Transformations of Characteristic Functions

 In the first paper of this series (Daniel Alpay, Tomas Azizov, Aad Dijksma, and Heinz Langer: The Schur algorithm for generalized Schur functions I: coisometric realizations, Operator Theory: Advances and Applications 129 (2001), pp. 1–36) it was shown that for a generalized Schur function s(z), which is the characteristic function of a coisometric colligation V with state space being a Pontryagin space, the Schur transformation corresponds to a finite-dimensional reduction of the state space, and a finite-dimensional perturbation and compression of its main operator. In the present paper we show that these formulas can be explained using simple relations between V and the colligation of the reciprocal s(z)⁻¹ of the characteristic function s(z) and general factorization results for characteristic functions. Received October 31, 2001; in revised form August 21, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions. Received: August 5, 2007., Accepted: October 24, 2007.     

Generalized solutions to Cauchy problems

This paper continues previous attempts to find a convenient mathematical setting in which linear and nonlinear Cauchy problems have a unique global solution, that reduces to a classical solution when the latter exists.With 1 Figure     

On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

ON THE RATES OF CONVERGENCE FOR PROBABILITY TYPE OPERATORS SEQUENCE

Abstract. In this paper, the rates of convergence for some probability type operators sequence are obtained. The quantitative Poisson type limit theorem is established as an application.     

On integral bernstein operators in some classes of measurable bivariate functions

The two main theorems are concerned with the approximations of (complex-valued) functions on the real plane by sums of Bernstein pseudoentire functions. They are formulated and proved in Section 4, after prior determination of the suitable integral operators. Analogous results for pseudopolynomial approximations were obtained by Brudnyî, Gonska, and Jetter ([2],[3]).Research supported by KBN grant 2 1079 91 01.     

Functions Operating on Positive Semidefinite Quaternionic Matrices

 We study functions on the quaternionic unit ball which operate on positive semidefinite matrices in the sense that is positive semidefinite whenever is a positive semidefinite square matrix with entries . (Received 21 September 2000; in revised form 8 March 2001)