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 also discussed.
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 `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.
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…
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 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…
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.
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.
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}.
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.
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…
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 $\G_3.$
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.
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.
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…
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…
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 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.
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…
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…
For each n,N>0 we construct a set of points x_1,...,x_M in D^n with the following property: if f is a rational inner function on D^n of degree strictly less than N and g is an analytic function mapping D^n to D that satisfies g(x_i)=f(x_i) for each i=1,...,M, then g=f on D^n. In terms of the Pick problem on D^n, our result implies that for any rational inner f of degree less than N, the Pick problem with data x_1,...,x_M and f(x_1),...,f(x_M) has a unique solution.
The purpose of this article is threefold. The first is to construct a Nevanlina theory for meromorphic mappings from a polydisc to a compact complex manifold. In particular, we give a simple proof of Lemma on logarithmic derivative for nonzero meromorphic functions on $\mathbb{C}^l.$ The second is to improve the definition of the non-integrated defect relation of H. Fujimoto \cite{F2} and to show two theorems on the new non-integrated defect relation of meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact complex manifold. The third is to give a unicity theorem for meromor…
We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^σ\left(\mathbb{B}_{d}\right)$, in the range $0 < σ\leq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $\mathrm{H}^2(\mathbb{D}^d)$ and its multiplier algebra $\mathrm{H}^\infty(\mathbb{D}^d)$: in th…
The purpose of this paper is to study the Schwarz-Pick type inequalities for harmonic or pluriharmonic functions. By analogy with the generalized Khavinson conjecture, we first give some sharp estimates of the norm of harmonic functions from the Euclidean unit ball in $\mathbb{R}^n$ into the unit ball of the real Minkowski space. Next, we give several sharp Schwarz-Pick type inequalities for pluriharmonic functions from the Euclidean unit ball in $\mathbb{C}^n$ or from the unit polydisc in $\mathbb{C}^n$ into the unit ball of the Minkowski space. Furthermore, we establish some sharp coefficien…
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", …
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.
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 $0<α<1$.
Recent work of Kosiński on the three point Pick interpolation problem on the polydisc proves the von Neumann inequality for 3x3 matrices. We give a detailed explanation of this using several standard reductions---credit for the main result should be attributed to Kosiński.
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 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 $κ\ge κ_{\rm min}$.
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 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…
