Paul Anthony Smith
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.
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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.
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.
It is known (implicit in [HMNT]) that when $Λ$ is an interpolating sequence for the Nevanlinna or the Smirnov class then there exist functions $f_λ$ in these spaces, with uniform control of their growth and attaining values 1 on $λ$ and 0 in all other $λ'\neqλ$. We provide an example showing that, contrary to what happens in other algebras of holomorphic functions, the existence of such functions does not imply that $Λ$ is an interpolating sequence.
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.
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。
We consider a free interpolation problem in Nevanlinna and Smirnov classes and find a characterization of the corresponding interpolating sequences in terms of the existence of harmonic majorants of certain functions. We also consider the related problem of characterizing positive functions in the disc having a harmonic majorant. An answer is given in terms of a dual relation which involves positive measures in the disc with bounded Poisson balayage. We deduce necessary and sufficient geometric conditions, both expressed in terms of certain maximal functions.
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。
Random interpolation in the Nevanlinna and Smirnov classes and related spaces
本文研究了Nevanlinna和Smirnov类及相关空间中随机插值序列的性质，揭示了其与Blaschke条件的关系，并探讨了大Hardy-Orlicz空间中插值条件的刻画问题。
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 show how Pick interpolation and interpolation on peak interpolation sets can be combined in an abstract uniform algebra setting. In particular as a special case, the Rudin-Carleson theorem can be combined with the classical Pick interpolation theorem on the disc.
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…
We prove the interpolation condition of S. Takahashi to the Nevanlinna-Pick-Caratheodory problem by using Sarason's theory.
On interpolation in Carathéodory hyperbolic domains
本文研究了Carathéodory超几何域中的插值问题，探讨了Pick体与有限维Hilbert空间收缩之间的关系，并提出了将Pick体映射到开放单位盘的条件。
Pick interpolation and invariant functions
本文建立Pick体与不变函数的联系，证明Pick体关联的不变函数可决定Pick插值问题可解性并推广Carathéodory伪距离，还在开单位圆盘描述该函数并导出广义Lempert函数，证明特定条件下二者相等，且双/三圆盘在特殊情况有类似Lempert定理的结果。
We provide a necessary condition for the existence of a 3-point holomorphic interpolant $F:\mathbb{D}\longrightarrowΩ_n$, $n\geq 2$. Our condition is inequivalent to the necessary conditions hitherto known for this problem. The condition generically involves a single inequality and is reminiscent of the Schwarz lemma. We combine some of the ideas and techniques used in our result on the $\mathcal{O}(\mathbb{D},Ω_n)$-interpolation problem to establish a Schwarz lemma -- which may be of independent interest -- for holomorphic correspondences from $\mathbb{D}$ to a general planar domain $Ω\Subset…
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 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 investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in $\mathbb{C}^d$.
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。
Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded as the most general hypergeometric orthogonal polynomials. Instead of a differential equation, they satisfy a second order Sturm-Liouville type difference equation in terms of the Wilson divided-difference operator. This suggests that in order to better understand the distinctive properties of Wilson polynomials and related topics, one should use a function theory that is more natural with respect to the Wilson operator. Inspired by the recent work of Halburd and Korhonen, we establish a full-fledged Nevanlinna theor…
We analyze the three point Pick interpolation problem on the bidisk
We give a solution to Pick's interpolation problem on the unit polydisc in $\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the $n$-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.
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 $0<α<1$.
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.
The main content of this paper is Lectures 5 and 6 that continue lecture notes [20]. Content of Lectures 1-4 of [20] is reviewed for the reader's convenience in sections 1-4, respectively. It is shown in Lecture 5 how residual parts of the minimal unitary extensions, that correspond to solutions of the problem, yield some boundary properties of the coefficient matrix-function. These results generalize the classical Nevanlinna - Adamjan - Arov~- Krein theorem. Lecture 6 discusses how further properties of the coefficient matrices follow from denseness of certain sets in the associated function …
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.
We generalize the notion of the Arov-Krein entropy functional for the case of generalized Nevanlinna functions and obtain a representation of these functionals on solutions of indefinite interpolation problems. The case of indefinite Caratheodory problem and application to Szegő limit formula for this nonclassical case are considered in greater detail.
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 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 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…
The multivariate Herglotz-Nevanlinna class: Rational approximation
本文研究多元Herglotz-Nevanlinna类中有界全纯函数的有理逼近问题，基于Takagi-Pfister斜特征向量构造Pade型逼近格式，证明Montessus de Ballore型收敛定理，并揭示Agler函数Taylor系数集的半代数性质。
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.
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.
Motivated by the recent progress of research on extending holomorphic functions defined on subvarieties of classical domains and its connections to the 3-point Pick interpolation, we study a special class of two-dimensional algebraic subvarieties $M_α$ of the unit tridisc, defined as the sets $$\lbrace (z_1,z_2,z_3)\in \mathbb{D}^3:α_1z_1+α_2z_2+α_3z_3=\overlineα_1z_2z_3+\overlineα_2z_1z_3+\overlineα_3z_1z_2\rbrace.$$ In this paper we show that given non-degenerated extremal maximal $3$-point Pick problem there exists an $α$ such that $M_α$ appears as its uniqueness variety. We also describe s…
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。
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…
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