Vladimir Bolotnikov, Alexander Kheifets
Three boundary Nevanlinna-Pick interpolation problems at finitely many points are formulated for generalized Schur functions. For each problem, the set of all solutions is parametrized in terms of a linear fractional transformation with a Schur class parameter.
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We formulate three boundary Nevanlinna-Pick interpolation problems for generalized Nevanlinna functions. For each problem, we parameterize the set of all solutions in terms of a linear fractional transformation with an extended Nevanlinna class parameter.
The Nevanlinna-Pick problem and the simplest case of the Carathéodory-Fejér problem on the spectral ball $\Om_3$ are reduced to interpolation problems on the symmetrized three-disc $\G_3.$
We give an elementary proof of Sarason's solvability criterion for the Nevanlinna-Pick problem with boundary interpolation nodes and boundary target values. We also give a concrete parametrization of all solutions of such a problem. The proofs are based on a reduction method due to Julia and Nevanlinna. Reduction of functions corresponds to Schur complementation of the corresponding Pick matrices.
$q$-Rational functions and interpolation with complete Nevanlinna Pick Kernels
该研究提出矩阵值 $q$-有理函数，刻画其实现与代数性质，并将其用于 Schur 乘子及完全 Nevanlinna–Pick 核框架下的插值问题。
The uniqueness problem for the 3-Point Nevanlinna-Pick extremal interpolation problem in the unit Euclidean ball $\Bn{d}$
本文旨在解决任意维度的单位欧氏球中3点Nevanlinna-Pick极值插值问题的唯一性问题，推广了Agler和McCarthy在双圆盘情形下的研究。
We solve a three point Nevanlinna-Pick problem in the Euclidean ball. In particular, we determine a class of rational functions that interpolate this problem.
We give a characterization for the existence of a holomorphic interpolant on the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point Pick--Nevanlinna data. One of the key steps is a characterization for the existence of an interpolant that is a rational inner function on $\mathbb{D}^n.$ The latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a type of positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results, which are of independent interest, on the factoriza…
In this article, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example of such domains is the spectral unit ball. We present several results for these matricial domains. Our first result shows, generalizing a result of Ransford-White for the spectral unit ball, that the holomorphic automorphism group of these matricial domains does not act transitively. We also consider $2$-point and $3$-point Pick-Nevanlinna interpolation problem from the unit disc to these matricial domains. We present results providing necessary cond…
Inner functions and Pick-Nevanlinna interpolation in a multi-connected domain
该研究针对多连通域场景，刻画有限阶内函数集合，给出Pick-Nevanlinna插值问题的优化形式，进一步将其推广到Hermite型插值。
In \cite{ds_hfs}, a geometric procedure for constructing a Nevanlinna-Pick problem on $\D^n$ 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 $\DT$ to have a unique solution. We use the results of \cite{ds_hfs} and Bezout's theorem to establish three special cases of this conjecture.
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.
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
We provide a short argument to establish a Beurling-Lax-Halmos theorem for reproducing kernel Hilbert spaces whose kernel has a complete Nevanlinna-Pick factor. We also record factorization results for pairs of nested invariant subspaces.
In the paper `Distinguished Varieties,' Agler and McCarthy used Hilbert function spaces to study the uniqueness properties of the Nevanlinna-Pick problem on the bidisc. In this work we give a geometric procedure for constructing a Nevanlinna-Pick problems on D^n with a specified set of uniqueness. On the way to establishing this procedure, we prove a result about Hilbert function spaces and partially answer a question posed by Agler and McCarthy.
A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function $n$ with a suitable asymptotic expansion at $\infty$, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation $p$ times we find a Nevanlinna function $n_p$ which is a fractional linear transformation of the given function $n$. The main results concern the effect of this transformation to the realizations of $n$ an…
Isometric Gelfand transforms of complete Nevanlinna-Pick spaces
本文证明乘子代数具有等距Gelfand变换或交换C*-包络的完全Nevanlinna-Pick空间本质上必为开单位圆盘上的Hardy空间。
We establish some multivariate generalizations of the Beurling-Lax-Halmos theorem.
Complex structure that admits complete Nevanlinna-Pick spaces of Hardy type
本文刻画了不可约完备Nevanlinna-Pick空间乘子范数与上确界范数相等的集合，并证明Hardy型该空间的既约复空间必双全纯同构于去掉零解析容量集的单位圆盘，从而确立Hardy空间的唯一性。
The main goal of the paper is to study $m$-extremal mappings in the symmetrized bidisc showing that they are rational and $\GG_2$-inner which, in particular, answers a question posed in \cite{Agl-Lyk-You 2013}.
It is very elementary to observe that functions interpolating an extremal two-point Pick problem on the polydisc are just left inverses to complex geodesics. In the present article we show that the same property holds for a three-point Pick problem on polydiscs, i.e. it may be expressed it in terms of three-complex geodesics. Using this idea we are able to solve that problem obtaining formulas and a uniqueness theorem for solutions of extremal problems. In particular, we determine a class of rational inner functions interpolating that problem. Possible extensions and further investigations are…
This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.
A modification of the well-known step-by-step process for solving Nevanlinna-Pick problems in the class of $\bR_0$-functions gives rise to a linear pencil $H-λJ$, where $H$ and $J$ are Hermitian tridiagonal matrices. First, we show that $J$ is a positive operator. Then it is proved that the corresponding Nevanlinna-Pick problem has a unique solution iff the densely defined symmetric operator $J^{-1/2}HJ^{-1/2}$ is self-adjoint and some criteria for this operator to be self-adjoint are presented. Finally, by means of the operator technique, we obtain that multipoint diagonal Padé approximants t…
We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert funct…
Given a collection $K$ of positive integers, let $H^{\infty}_K(\mathbb{D})$ denote the set of all bounded analytic functions defined on the unit disk $\mathbb{D}$ in $\mathbb{C}$ whose $k^{\text{th}}$ derivative vanishes at zero, for all $k \in K$. In this paper, we establish a Nevanlinna-Pick interpolation result for the subalgebra $H^{\infty}_K(\mathbb{D})$, where $K = \{1,2,\dotsc,k\}$, which is a slight generalization of the interpolation theorem that Davidson, Paulsen, Raghupathi, and Singh proved for the algebra $H^{\infty}_{\{1\}}(\mathbb{D})$. Furthermore, we provide a sufficient condi…
We study reproducing kernel Hilbert spaces on the unit ball with the complete Nevanlinna-Pick property through the representation theory of their algebras of multipliers. We give a complete description of the representations in terms of the reproducing kernels. While representations always dilate to $*$-representations of the ambient $C^*$-algebra, we show that in our setting we automatically obtain coextensions. In fact, we show that in many cases, this phenomenon characterizes the complete Nevanlinna-Pick property. We also deduce operator theoretic dilation results which are in the spirit of…
Dynamic Nevanlinna-Pick Theory, Covariance Dilations, and Non-commutative Varieties
该研究推广了Nevanlinna-Pick插值理论，建立了相关算子理论框架，证明了H^∞(𝔻)成员关系的普遍检测方法，并揭示非交换内簇上Arveson型完全谱集表示失效的特性。
The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson's notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson--Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are i…
Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk
本文研究对称双圆盘上可解 Nevanlinna-Pick 插值问题的唯一性集合，证明其为由插值数据规范构造的唯一性簇，并给出其中特征簇的完整代数与几何刻画。
Nevanlinna-Pick interpolation in the right half-plane
本文研究右半平面上的 Nevanlinna-Pick 插值，证明 Szegö-Dirichlet 核的正整数次幂均无两点标量 Pick 性质，并给出 $\mathbb H_0$ 的网络实现公式。
Moment Problems and Spectral Functions
本文研究了Nevanlinna-Pick插值和动量问题，证明了因果数据空间的凸性，并回顾了相关结果。
We give necessary and sufficient conditions for solving the spectral Nevanlinna--Pick lifting problem. This reduces the spectral Nevanlinna--Pick problem to a jet interpolation problem into the symmetrized polydisc.
We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Loewner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Loewner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive…
The notion of a unitary realization is used to estimate derivatives of arbitrary order of functions in the Schur-Agler class on the polydisk and unit ball.
We prove the interpolation condition of S. Takahashi to the Nevanlinna-Pick-Caratheodory problem by using Sarason's theory.
The nondegenerate Nevanlinna-Pick-Carathéodory-Fejer interpolation problem with finitely many interpolation conditions always has infinitely many solutions in a generalized Schur class $\cS_κ$ for every $κ\ge κ_{\rm min}$ where the integer $κ_{\rm min}$ equals the number of negative eigenvalues of the Pick matrix associated to the problem and completely determined by interpolation data. A linear fractional description of all $\cS_{κ_{\rm min}}$ solutions of the (nondegenerate) problem is well known. In this paper, we present a similar result for an arbitrary $κ\ge κ_{\rm min}$.
We introduce a family of domains --- which we call the $μ_{1,n}$-quotients --- associated with an aspect of $μ$-synthesis. We show that the natural association that the symmetrized polydisc has with the corresponding spectral unit ball is also exhibited by the $μ_{1,n}$-quotient and its associated unit "$μ_E$-ball". Here, $μ_E$ is the structured singular value for the case $E = \{[w]\oplus(z I_{n-1})\in \mathbb{C}^{n\times n}: z,w\in \mathbb{C}\}$, n = 2, 3, 4,... Specifically: we show that, for such an $E$, the Nevanlinna-Pick interpolation problem with matricial data in a unit "$μ_E$-ball", …
We consider the multi-point Schwarz-Pick lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick lemma give Schur's continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk.
A certain kernel (sometimes called the Pick kernel) associated to Schur functions on the disk is always positive semi-definite. A generalization of this fact is well-known for Schur functions on the polydisk. In this article, we show that the Pick kernel on the polydisk has a great deal of structure beyond being positive semi-definite. It can always be split into two kernels possessing certain shift invariance properties.
Consider a scaled Nevanlinna-Pick interpolation problem and let $Π$ be the Blaschke product whose zeros are the nodes of the problem. It is proved that if $Π$ belongs to a certain class of inner functions, then the extremal solutions of the problem or most of them, are in the same class. Three different classical classes are considered: inner functions whose derivative is in a certain Hardy space, exponential Blaschke products and also the well known class of $α$-Blaschke products, for $0<α<1$.
First, an abstract scheme of constructing biorthogonal rational systems related to some interpolation problems is proposed. We also present a modification of the famous step-by-step process of solving the Nevanlinna-Pick problems for Nevanlinna functions. The process in question gives rise to three-term recurrence relations with coefficients depending on the spectral parameter. These relations can be rewritten in the matrix form by means of two Jacobi matrices. As a result, a convergence theorem for multipoint Padé approximants to Nevanlinna functions is proved.
Two different problems are considered here. First, a version of Schwarz-Pick Lemma for $n$ points leads to an interpolation problem for analytic functions from the disc into itself, which may be considered as a particular case of the classical Nevanlinna-Pick interpolation problem. Second, a characterization of sampling sequences for this class of functions is given.
Sufficient boundary asymptotic conditions are established for a generalized Schur function $f$ to be identically equal to a given rational function $g$ unimodular on the unit circle. Similar rigidity statements are presented for generalized Carathéodory and generalized Nevanlinna functions.
Sampling and interpolation for analytic selfmappings of the disc
本文研究了圆盘解析自映射的两个问题：给出了采样序列的刻画，并基于$n$点Schwarz-Pick引理导出了一个可视为经典Nevanlinna-Pick插值特例的插值问题。
We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the 2-torus. We prove the existence of a generalized model with certain properties corresponding to such a singularity and use this result to solve two function-theoretic problems. The first of these is to characterise the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carathéodory condition holds. The seco…
A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $κ_1, \ldots, κ_N$, quaternions $p_1, \ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u \neq 1$ for $u = 1,\ldots, N$, and quaternions $s_1, \ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad {\rm for} \quad u=1,\ldots, N,$$ and $$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\leκ_…
The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Hamburger moment problems in the most general case. In the odd case, these results are completely new for the matrix case, whereas the scalar version was recently treated by V. A. Derkach, S. Hassi and H. S. V. de Snoo. The even case was studied earlier by G.-N. Chen and Y.-J. Hu. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked …
We give an elementary proof of a solvability criterion for the {\em boundary Carathéodory-Fejér problem}: given a point $x \in \R$ and, a finite set of target values, to construct a function $f$ in the Pick class such that the first few derivatives of $f$ take on the prescribed target values at $x$. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem. The proofs are based on a reduction method due to Julia and Nevanlinna.
We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point $x \in \mathbb{R}$ and, a finite set of target values $a^0,a^1,...,a^n \in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \to x$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.
The Schur-Agler class in infinitely many variables
本文定义了无穷多变量下的Schur-Agler类，证明了Agler分解的推广及传递函数实现，给出了Pick插值型定理，并探讨了其与Dirichlet级数的联系。
I describe a verifiable criterion for the solvability of the 2 by 2 spectral Nevanlinna-Pick problem with two interpolation points, and likewise for three other special cases of the mu-synthesis problem. The problem is to construct an analytic 2 by 2 matrix function F on the unit disc subject to a finite number of interpolation constraints and a bound on the cost function sup {mu(F(z)): |z| < 1}, where mu is an instance of the structured singular value of a 2 by 2 matrix.
Boundary measures of holomorphic functions on the imaginary domain
本研究围绕Pick函数的Herglotz-Nevanlinna积分表示，提出虚域上全纯函数的边界测度概念，并阐释了该测度的基本性质。
We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foiaş. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X a…
Approximation, interpolation, and lifting on the unit ball
本文通过内函数和极值函数解决了单位球上的Nevanlinna-Pick插值问题，并推广了交换子提升定理，解决了多个著名问题。
We consider a Nevanlinna-Pick interpolation problem on finite sequences of the unit disc D constrained by Hardy and radial-weighted Bergman norms. We find sharp asymptotics on the corresponding interpolation constants. As another application of our techniques we prove embedding theorems for rational functions. We find that the embedding of H $\infty$ into Hardy or radial-weighted Bergman spaces in D is invertible on the subset of rational functions of a given degree n whose poles are separated from the unit circle and obtain asymptotically sharp estimates of the corresponding embedding constan…
We analyse the 3-extremal holomorphic maps from the unit disc $\mathbb{D}$ to the symmetrised bidisc $ \mathcal{G}$, defined to be the set $ \{(z+w,zw): z,w\in\mathbb{D}\}$, with a view to the complex geometry and function theory of $\mathcal{G}$. These are the maps whose restriction to any triple of distinct points in $\mathbb{D}$ yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most 4. It is shown that there are two qualitatively different classes of rational $\mathcal{G}$-inner functions of degree at most 4, to be call…
Commutant lifting, interpolation, and perturbations on the polydisc
针对Sarason换位提升基本定理不适用于多圆盘的问题，本文提出多变量换位提升的两种分类，解决了多圆盘上有界解析函数的Nevanlinna-Pick插值与扰动问题，推进了希尔伯特函数空间理论的两个著名问题。
There are three new things in this paper about the open symmetrized bidisk $\mathbb G = \{(z_1+z_2, z_1z_2) : |z_1|, |z_2| < 1\}$. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extensio…
Function theory on the annulus in the dp-norm
本文借助实现理论研究圆环上 dp-范数定义的 dp-Schur 类，引入 DP Szegő 核并建立 Pick 插值判据，还证明可解问题具有有理解。
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