Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

The Nevanlinna—Pick Interpolation Problem in Multiply Connected Domains

We simplify and strengthen Abrahamse's result on the Nevanlinna–Pick interpolation problem in a finitely connected planar domain, according to which the problem has a solution if and only if the Pick matrices associated with character-automorphic Hardy spaces are positive semidefinite for all characters in ^n–1/^n–1, where n is the connectivity of the domain. The main aim of the paper is to reduce the indicated procedure (verification of the positive semidefiniteness) for the entire real (n–1)-torus ^n–1/^n–1 to a part of it, whose dimension is, possibly, less than n–1. Bibliography: 14 titles.     

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     

Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc II

In [19], a geometric procedure for constructing a Nevanlinna–Pick problem on Dⁿ${\mathbb{D}}^n$ with a specified set of uniqueness was established. In this sequel we conjecture a necessary and a sufficient condition for a Nevanlinna–Pick problem on D²${\mathbb{D}}^2$ to have a unique solution. We use the results of [19] and Bezout?s theorem to establish three special cases of this conjecture.     

Nevanlinna–Pick interpolation on distinguished varieties in the bidisk

This article treats Nevanlinna–Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.     

Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi–Trudi formulas and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulas we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.     

Nevanlinna–Pick Problem and Uniqueness of Left Inverses in Convex Domains,Symmetrized Bidisc and Tetrablock

In the paper we discuss the problem of uniqueness of left inverses (solutions of two-point Nevanlinna–Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock.     

Linear Connectivity,Schwarz–Pick Lemma and Univalency Criteria for Planar Harmonic Mapping

In this paper, we first establish a Schwarz–Pick lemma for higher-order derivatives of planar harmonic mappings, and apply it to obtain univalency criteria. Then we discuss distortion theorems,Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.     

Crank–Nicolson Scheme for Abstract Linear Systems

This paper studies the Crank–Nicolson discretization scheme for abstract differential equations on a general Banach space. We show that a time-varying discretization of a bounded analytic C₀-semigroup leads to a bounded discrete-time system. On Hilbert spaces, this result can be extended to all bounded C₀-semigroups for which the inverse generator generates a bounded C₀-semigroup. The presentation is based on C₀-semigroup theory and uses a functional analysis approach.     

Input–Output Identifiability of Continuous-Time Linear Systems

In time-domain identification of linear systems the aim is to estimate the impulse response or transfer function of a linear system to within a given tolerance using a finite number of noisy observations of the output. Whether this is possible depends on the model set, that is, a given set to which the system is assumed to belong a priori. We give necessary and sufficient conditions on the model set to ensure that such identification is possible in the continuous-time case.     

Fractal Jacobi Systems and Convergence of Fourier–Jacobi Expansions of Fractal Interpolation Functions

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of \({\mathbb{R}}\) interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in \({\mathcal{L}^{2}(-1,1)}\) with respect to the weight function \({\rho^{(r,s)}(x)=(1-x)^{r} (1+x)^{s}}\), \({r > -1}\) and \({s > -1}\). In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of \({\alpha}\)-fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions \({\mathcal{L}_{\rho}^{2}(-1,1)}\) is found. The Fourier–Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier–Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier–Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.     

Solving Large Sparse Linear and Quadratic Generalized Eigenvalue Problems

We study the methods for solving the following large order eigenvelue problems occurring in the analysis of structural vibration: (1) (2) and (3) where M and C are both symmetric matrices, while (?) is skew symmetric. Moreover, M is positive definite, and the matrix K in (2) and (3) is also assumed to be symmetric positive definite. 1 The eigenvalue problem for normal matrices in generalized inner prduct Let B be an n×n Hermitian positive definite matrix. Then     

Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

On Polynomial Solutions of Generalized Moisil-Théodoresco Systems and Hodge-de Rham Systems

The aim of the paper is to study relations between polynomial solutions of generalized Moisil-Théodoresco (GMT) systems and polynomial solutions of Hodge-de Rham systems and, using these relations, to describe polynomial solutions of GMT systems. We decompose the space of homogeneous solutions of GMT system of a given homogeneity into irreducible pieces under the action of the group O(m) and we characterize individual pieces by their highest weights and we compute their dimensions.