Interpolation by holomorphic maps from the disc to the tetrablock

The tetrablock is the set $$ \mathcal{E}=\{x \in \mathbb{C}^3: \quad 1-x_1z-x_2w+x_3z w \neq 0 \quad whenever \quad |z|\leq 1, |w|\leq 1\}. $$ The closure of $\mathcal{E}$ is denoted by $\overline{\mathcal{E}}$. A tetra-inner function is an analytic map $x$ from the unit disc $ \mathbb{D} $ to $\overline{\mathcal{E}}$ such that, for almost all points $\lambda$ of the unit circle $ \mathbb{T}$, \[ \lim_{r\uparrow 1} x(r \lambda) \mbox{ exists and lies in } b \overline{\mathcal{E}}, \] where $b \overline{\mathcal{E}}$ denotes the distinguished boundary of $\overline{\mathcal{E}}$. There is a natural notion of degree of a rational tetra-inner function $ x$; it is simply the topological degree of the continuous map $ x|_\mathbb{T} $ from $ \mathbb{T} $ to $ b \overline{\mathcal{E}} $. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree $n$. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if $x= (x_1, x_2, x_3)$ is a rational tetra-inner function of degree $n$, then $x_1 x_2 - x_3$ either is identically $0$ or has precisely $n$ zeros in the closed unit disc $\overline{\mathbb{D}}$, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function $x= (x_1, x_2, x_3)$ consists of the points in $\overline{\mathbb{D}}$ for which $x_1 x_2 - x_3=0$ and the values of $x$ at these points.


Introduction
In this paper we present an algorithm for the construction of a general rational inner function from D to the tetrablock. The algorithm is based on a known solution of the Nevanlinna-Pick interpolation problem on D. Different versions of the Nevanlinna-Pick interpolation problem have been studied by many authors, beginning with G. Pick in 1916 [29] and continuing with R. Nevanlinna in 1922 [28], and they still attract interest, because they are natural questions in function theory and because of their applications to engineering, particular electric networks and control theory, see [7] for some references. We should mention particularly papers of J. A. Ball and J. W. Helton [8], D. Sarason [32], D. R. Georgijević [20], G.-N. Chen and Y.-J. Hu [18] and V. Bolotnikov and A. Kheifets [15], and the books of J. A. Ball, I. C. Gohberg and L. Rodman [9], and of V. Bolotnikov and H. Dym [14]. There are many further papers on this interesting topic and applications (see, for example, [19,33]).
The closed tetrablock E is the set in C 3 defined by The original motivation for the study of E was an attempt to solve a µ-synthesis problem [1], which is itself motivated by basic unsolved problems in H ∞ control theory [21,22]. The tetrablock has attracted considerable interest in recent years. It has interesting complex geometry [1,23,34,25,26], rich function theory [16,5,6] and associated operator theory [11]. The solvability of the µ-synthesis problem connected to E can be expressed in terms of the existence of rational inner functions from the open unit disc D in the complex plane C to the closure of E [16].
Observe that if x : D → E is analytic then, by Fatou's Theorem, for almost all λ ∈ T with respect to Lebesgue measure, the radial limit lim r↑1 x(rλ) exists. We say that an analytic map x : D → E is a tetra-inner function, or alternatively, an E-inner function if, for almost all λ ∈ T with respect to Lebesgue measure, lim r↑1 x(rλ) lies in the distinguished boundary bE of E. The distinguished boundary bE of E is homeomorphic to the solid torus D × T, which has a boundary [1]. The E-inner functions constitute a natural analogue (in the context of the tetrablock) of the inner functions introduced by A. Beurling [10], which play an important role in the function theory of the unit disc and in operator theory [17].
A basic question about rational inner functions ϕ from D to D was studied by W. Blaschke [12]. Specifically, he obtained (inter alia) a formula for the general rational inner function ϕ of degree n in terms of its zeros. Indeed, by the Argument Principle, any rational inner function ϕ of degree n has exactly n zeros in D, counted with multiplicity. From this fact one can see that ϕ is a "finite Blaschke product", having the form for some unimodular constant c and some α 1 , ..., α n ∈ D. The α j are the zeros of ϕ. It is evident from equation (1.1) that ϕ extends to a continuously differentiable function on D, given by the same formula. In this paper our aim is to write down a formula analogous to equation (1.1) for the general rational E-inner function of degree n. The first question that arises is: what data should replace the α j , the zeros of ϕ? We have found that an effective choice is the set of royal nodes of the tetra-inner function, which we shall now define. It was shown in [5] that if x = (x 1 , x 2 , x 3 ) is a rational E-inner function of degree n then x 3 − x 1 x 2 either is identically 0 or has exactly n zeros in the closed unit disc D, counted with an appropriate notion of multiplicity. Here, the degree of a rational E-inner function x is defined to be the topological degree of the restriction of x that maps T continuously to bE. Since bE is homeomorphic to the solid torus D × T, which is homotopic to T, the fundamental group π 1 (bE) is Z, and so the degree of x is an integer; it will be denoted by deg(x). The variety has an important role in the function theory of E; it is called the royal variety of E. For any rational tetra-inner function x = (x 1 , x 2 , x 3 ), the zeros of x 1 x 2 −x 3 in D are the points λ ∈ D such that x(λ) ∈ RĒ . We call these points the royal nodes of x. If σ ∈ D is a royal node of x, so that x(σ) = (η,η, ηη) for some η,η ∈ D, then we call η,η the royal values of x corresponding to the royal node σ of x. In this paper, in Theorem 1.7, we give a prescription for the construction of a general rational tetra-inner function of degree n in terms of its royal nodes and royal values. We shall make use of a known solution of an interpolation problem for finite Blaschke products. The E-inner functions x such that x(D) ⊆ R E are simply the functions of the form (ϕ 1 , ϕ 2 , ϕ 1 ϕ 2 ) where ϕ 1 , ϕ 2 are inner functions. These E-inner functions behave differently from the others, and we shall often specifically exclude them from consideration.
(1.4) Problem 1.3 has been analysed by several authors [31,32,8]. In the absence of the tangential conditions (1.4) the problem would be ill-posed, in that the solvability of the problem would depend only on the interpolation conditions at nodes in D, and the conditions at σ 1 , . . . , σ k would be irrelevant. With the conditions (1.4), however, the problem has a pleasing solution. The existence of a solution of the Blaschke interpolation problem can be characterized in terms of an associated "Pick matrix", and all solutions ϕ are parametrized by a linear fractional expression in terms of a parameter ζ ∈ T. There are polynomials a, b, c and d of degree at most n such that the general solution of Problem 1.3 is (ii) for all but at most k values of ζ ∈ T, the function is a solution of Problem 1.3; (iii) for some point τ ∈ T \{σ 1 , ..., σ k }, (iv) every solution ϕ of Problem 1.3 has the form (1.5) for some ζ ∈ T.
The connection between Problems 1.6 and 1.3 can be described with the aid of a certain 1-parameter family of rational functions on E that is parametrized by the unit circle T. These functions play a central role in the function theory of E (see [1,16]). They are defined, for ω ∈ T, by Ψ ω is holomorphic on E, except at points x ∈ E where x 2 ω − 1 = 0, and maps any point of E at which it is defined into D.
The main theorem of this paper is the following.
Theorem 1.7. For royal tetra-interpolation data (σ, η,η, ρ) of type (n, k) the following two statements are equivalent: where a, b, c and d are the polynomials in the normalized parametrization ϕ = aζ + b cζ + d of the solution of Problem 1.3.
The theorem follows from Theorems 5.1 and 6.4. The proofs of these theorems are given in Section 5 and Section 6 respectively. Theorem 6.4 provides us with a formula for a solution x of Problem 1.6 in terms of s 0 , p 0 , a, b, c and d. The formula is in terms of the polynomials a, b, c and d computed in [3,Theorem 3.9] (see Remark 4.11). In this way we derive an explicit solution of Problem 1.6. Theorem 6.4. Let (σ, η, ρ) be Blaschke interpolation data of type (n, k), and let (σ, η,η, ρ) be royal tetra-interpolation data, whereη j ∈ T, for j = 1, ..., k, andη j ∈ D, for j = k + 1, ..., n. Suppose that the Blaschke interpolation problem (Problem 1.3) with these data is solvable and the solutions ϕ of Problem 1.3 have normalized parametrization Suppose that there exist scalars =η j for j = 1, ..., n. (1.7) Then there exists a rational tetra-inner function x = (x 1 , x 2 , x 3 ) given by, , (1.10) for λ ∈ D, such that (i) x is a solution of the royal tetra-interpolation problem with the data (σ, η,η, ρ), that is, x(σ j ) = (η j ,η j , η jηj ) for j = 1, ..., n, and Ax 1 (σ j ) = ρ j for j = 1, ..., k, (ii) for all but finitely many ω ∈ T, the function Ψ ω • x is a solution of Problem 1.3. The solution sets of the royal tetra-interpolation problem and the corresponding Blaschke interpolation problem admit an explicit connection in terms of the functions Ψ ω . Corollary 6.5. Let (σ, η, ρ) be Blaschke interpolation data of type (n, k). Suppose that x is a solution of the Problem 1.6 with data (σ, η,η, ρ) for someη j ∈ D, j = 1, ..., n, and that x(D) ⊆ R E . For all ω ∈ T\{η 1 , ...,η k }, the function ϕ = Ψ ω • x is a solution of Problem 1.3 with Blaschke interpolation data (σ, η, ρ). Conversely, if ϕ is a solution of the Blaschke interpolation problem with data (σ, η, ρ), then there exists ω ∈ T such that ϕ = Ψ ω • x .
In [5] there is a construction of the general rational E-inner function x = (x 1 , x 2 , x 3 ) of degree n, in terms of different data, namely, the royal nodes of x and the zeros of x 1 and x 2 . A major step in the construction in [5] is to perform a Fejér-Riesz factorization of a non-negative trigonometric polynomial, which requires an iterative process, whereas, in contrast, the construction of x in this paper is purely algebraic and can be carried out entirely in rational arithmetic. The algorithm for the solution of the royal tetrainterpolation problem is presented in Section 7.
The authors are grateful to Nicholas Young for some helpful suggestions.
The following functions are important in the study of E.
For ω ∈ T, let Further we will need the following description of the tetrablock from [1].
and |x 2 | ≤ 1. Therefore, For a rational E-inner function x = (x 1 , x 2 , x 3 ) : D → E, we consider the rational functions ψ ω : D → D and υ ω : D → D given, for any ω ∈ T, by In [2] we introduced the terminology of the phasar derivative Af (z) for any differentiable function f : T → C \ {0} at z = e iθ ∈ T and wrote down some useful elementary properties of phasar derivatives, see Definition 1.2.
Let us recall that, by definition, σ ∈ T is a royal node of a tetra-inner function be a rational tetra-inner function and let σ ∈ T be a royal node of x. Then σ is a zero of the function x 3 − x 1 x 2 of multiplicity at least 2.
Therefore σ is a zero of (x 3 − x 1 x 2 ) of multiplicity at least 2.

Rational tetra-inner functions and royal polynomials
In this section we will show how to construct rationalĒ-inner functions with prescribed royal nodes and values. To describe this construction we need several theorems and definitions from [5]. Detailed proofs of these statements are given in [5,6].
For a polynomial p of degree less than or equal to n, where n ≥ 0, we define the polynomial p ∼n by  Proof. Since x is an E-inner function, by definition of tetra-inner functions, x(σ) ∈ bE for σ ∈ T. By Theorem 2.3, x 1 (σ) = x 2 (σ)x 3 (σ), |x 3 (σ)| = 1 and |x 2 (σ)| ≤ 1. By assumption σ is a royal node of x. Thus x 3 (σ) = x 1 (σ)x 2 (σ), and so |x 1 (σ)| = 1 and |x 2 (σ)| = 1 since |x 3 (σ)| = 1.  [5] Let x be a rational E-inner function of degree n and let R x be the royal polynomial of x. Then R x is 2n-symmetric and the zeros of R x on T have even order or infinite order.
be a rationalĒ-inner function such that x(D) R E and let R x be the royal polynomial of x. If σ is a zero of R x of order ℓ, we define the multiplicity #σ of σ (as a royal node of x) by We define the type of x to be the ordered pair (n, k), where n is the number of royal nodes of x that lie inD, counted with multplicity, and k is the number of royal nodes of x that lie in T, counted with multiplicity. R n,k denotes the collection of rationalĒ-inner functions of type (n, k).
is the homomorphism of fundamental groups induced by x when x is regarded as a continuous map from T to bE.
[5] If x ∈ R n,k is non-constant, then the degree of x is equal to n.
[5] Let x be a non-constant rational E-inner function of degree n. Then, either x(D) ⊆ R E or x(D) meets R E exactly n times.
Then the rational function has a cancellation at ζ ∈ D if and only if the following conditions are satisfied : ζ ∈ T, ζ is a royal node of x and ω = x 2 (ζ).

Criteria for the solvability of the Blaschke interpolation problem
In this section, for the convenience of the reader, we collect some known facts about finite Blaschke products that we need. They may be found in several places, but the most economical source for our purposes is [3], which assembles precisely the results which we require.
As mentioned in the Introduction, there is an extensive literature on boundary interpolation problems. A very valuable source of information about all manner of complex interpolation problems is the book of Ball, Gohberg and Rodman [9]. The authors of [8,32,15,9,14] make use of Krein spaces, moment theory, measure theory, reproducing kernel theory, realization theory and de Branges space theory. They obtain far-reaching results, including generalizations to matrix-valued functions and to functions allowed to have a limited number of poles in a disc or half-plane. See also papers of [13,27,30] for elementary treatments of interpolation problems. The monograph [14] by Bolotnikov and Dym is entirely devoted to boundary interpolation problems for the Schur class. They reformulate the problem within the framework of the Ukrainian school's Abstract Interpolation Problem and solve it by means of operator theory in de Branges-Rovnyak spaces.
The Blaschke interpolation Problem 1.3 as described in [3] is an algebraic variant of the classical Pick interpolation problem. One looks for a Blaschke product of degree n satisfying n interpolation conditions, rather than a Schur-class function as in the original Pick interpolation problem. We admit interpolation nodes both in the open unit disc and on the unit circle. There is a criterion for the solvability of the Blaschke interpolation problem in terms of the positivity of a "Pick matrix" formed from the interpolation data. To obtain a well-posed problem one imposes additional interpolation conditions, on phasar derivatives at the interpolation nodes on the unit circle. These bounds on the phasar derivatives appear as the diagonal entries of the Pick matrix.    The following is a refinement of the Sarason Interpolation Theorem [32]. if and only if M is minimally positive.
In [3] the authors described a strategy for the construction of the general solution of the Blaschke interpolation problem (Problem 1.3). It is to adjoin an additional boundary interpolation condition ϕ(τ ) = ζ where τ ∈ T \ {σ 1 , ..., σ k } and ζ ∈ T. This augmented problem has a unique solution. All the solutions of Problem 1.3 are then obtained in terms of a unimodular parameter. The Pick matrix B ζ,τ of the augmented problem is the (n + 1) × (n + 1) matrix, where ρ ζ,τ = M −1 u ζ,τ , u ζ,τ , M is the Pick matrix associated with Problem 1.3, and u ζ,τ is the n × 1 column matrix defined by The jth standard basis vector in C n will be denoted by e j . The exceptional set Z τ for Problem 1.3 is defined to be Z τ = {ζ ∈ T : for some j such that 1 ≤ j ≤ k, M −1 u ζ,τ , e j = 0} (4.7) Define n × 1 vectors x λ and y λ for λ ∈ D\{σ 1 , ..., σ k } by so that u ζ,τ = x τ − ζy τ (4.9)  (i) If ζ ∈ T \ Z τ , then there is a unique solution ϕ ζ of Problem 1.3 that satisfies ϕ ζ (τ ) = ζ. (ii) There exist unique polynomials a τ , b τ , c τ , and d τ of degree at most n such that and, for all ζ ∈ T, if ϕ is a solution of a Problem 1.3 such that ϕ(τ ) = ζ, then for all λ ∈ D. (iii) Ifã,b,c,d are rational functions satisfying the equation and such that for three distinct points ζ in T \ Z τ , the equation holds for all λ ∈ D, then there exists a rational function X such thatã = Xa τ ,b = Xb τ ,c = Xc τ andd = Xd τ .
(2) There exist polynomials a, b, c, d of degree at most n such that a normalized parametrization of the solutions of Problem 1.3 is (3) For any polynomials a, b, c, d as in (2), 4) and moreover, Proof.
Let Z τ be defined as in equation (5.12), let τ ∈ T \ {σ 1 , ..., σ k } be such that Z τ consists of at most k points, and let is a solution of Problem 1.3 which satisfies ϕ(τ ) = ζ.
Proof. By equation (5.8), for any ω ∈ T, Hence, as long as is a solution of Problem 1.3 which satisfies ϕ(τ ) = ζ. Condition (5.19) can also be written, for j = 1, 2, ..., k, For ζ ∈ T \ (Z τ ∪ Z ∼ τ ) where Z ∼ τ is defined in Lemma 5.2, we have two formulae for the unique solution of Problem 1.3 satisfying ϕ(τ ) = ζ, namely, the equations (5.14) and (5.18). Note that Because the set Z τ ∪ Z ∼ τ is finite, the linear fractional transformations in equations (5.14) and (5.18) are equal at infinitely many points and therefore coincide. It follows from the normalizing condition that Suppose that a, b, c and d are polynomials that parametrize the solutions of Problem 1.3, as in Theorem 5.1 (2). By the observation (5.15), there is a rational function X such that Let us find connections between x 1 , x 2 , x 3 and the polynomials a, b, c, d. Equations (5.23) and (5.24) for x 2 and X can be written as

(5.25)
Then, the solution of the system (5.25) is and Equations (5.21) and (5.22) give us the system (5.28) Then, the solution of the system (5.28) is Thus x 1 , x 2 , x 3 are given by equations (5.5), (5.6) and (5.7). The proof of Theorem 5.1 is complete.
Note that we can also prove a result similar to Theorem 5.1, using the function Υ ω instead of Ψ ω , where which is defined for every (x 1 , x 2 , x 3 ) in C 3 such that x 1 ω − 1 = 0. In this case we suppose that ρ j = Ax 2 (σ j ) for j = 1, 2, .., k.

From the Blaschke interpolation problem to the royal tetra-interpolation problem
In this section we will prove Theorem 6.4. This theorem shows that, if Blaschke interpolation data (σ, η, ρ) of type (n, k) are given and the corresponding Problem 1.3 is solvable, then we can construct a solution for the royal tetra-interpolation problem (σ, η,η, ρ), for someη = (η 1 , ...,η n ). We will start with some technical lemmas.
Proposition 6.2. Let a, b, c, d be polynomials in the variable λ and let Let rational functions x 1 , x 2 , x 3 be defined by .

(6.2)
and define a rational function ζ in the indeterminate ω by Then, as rational functions in (ω, λ), Proof. Let x 1 , x 2 , x 3 be defined by equations (6.2). Then Proposition 6.3. Let a, b, c, d be polynomials having no common zero in D, and satisfying |c| ≤ |d| on D . Suppose that Let rational functions x 1 , x 2 , x 3 be defined by , (6.4) and let (i) If, for all but finitely many values of λ ∈ D, for all but finitely many ζ ∈ T, then x • 1 c + d has no zero in D and x = (x 1 , x 2 , x 3 ) is an analytic map from D to E. (ii) If, for all but finitely many ζ ∈ T, the function ψ ζ is inner, then either x(D) ⊆ R E or x = (x 1 , x 2 , x 3 ) is a rational tetra-inner function.
Proof. (i) Suppose there is a finite subset E of D such that, for all λ ∈ D \ E, there is a finite subset F λ of T for which the inequality (6.6) holds for all ζ ∈ T \ F λ . Let us show that the denominator Pick a sequence (α j ) in D \ E such that α j → α. For each j, for ζ ∈ T \ F λ j , we have |ψ ζ (λ)| ≤ 1 on D \ E. Hence, for ζ ∈ T \ ∪ j F λ j , which is to say, for all but countably many ζ ∈ T, Because c(α j )ζ + d(α j ) → 0 uniformly almost everywhere for ζ ∈ T as j → ∞, the same holds for a(α j )ζ + b(α j ). Therefore a(α j ) → 0 and b(α j ) → 0. Hence a(α) = b(α) = 0. Thus a, b, c, d all vanish at α, contrary to our assumption. So x • 1 c + d has no zeros in D. Thus x 1 , x 2 , x 3 defined by equations (6.4) are rational functions having no poles in D.
Proof. By Corollary 4.12 (3), |c| ≤ |d| on D. Hence d(λ) c(λ) ≥ 1 for λ ∈ D. By assumption, which is a contradiction since d(λ) c(λ) ≥ 1 for all λ ∈ D, and |x • 1 | < 1 on D. Therefore, x • 1 c = −d on D. By Proposition 6.3, either x(D) ⊆ R E or x is a rational E-inner function. Because a, b, c, d are polynomials of degree at most n, the rational function x has degree at most n.
We have already observed that x is a rational E-inner function, deg(x) ≤ n and that x(D) is not in R E . Thus by Theorem 3.10, the number of royal nodes of x is equal to the degree of x. Consequently x has at most n royal nodes. Because the points σ j , j = 1, ..., n are royal nodes, they contain all n royal nodes of x, and so deg(x) = n. Observe that k of the σ j lie in T; thus x has exactly k royal nodes in T. Hence x ∈ R n,k .

The algorithm
In this section we present a concrete algorithm for the solution of the royal E-interpolation problem.
Let (σ, η,η, ρ) be royal interpolation data of type (n, k) for the tetrablock, as in Definition 1.5. One can consider the associated Blaschke interpolation data (σ, η, ρ) of type (n, k) . To construct a rational E-inner function x : D → E of degree n having royal nodes σ j for j = 1, ..., n, royal values η j ,η j , and phasar derivatives ρ j at σ j for j = 1, ..., k, we proceed as follows.
(1) Consider the Pick matrix M = [m i,j ] n i,j=1 for the data (σ, η, ρ). It has entries Assume that M is positive definite; otherwise the interpolation problem 1.3 is not solvable. Introduce the notation as in equations (4.8).
(3) Let and let polynomials a, b, c, d be defined by Observe that If there is no (x • 1 , x • 2 , x • 3 ) satisfying these conditions, then by Theorem 6.4, the royal Einterpolation problem is not solvable.