"Wrong" Side Interpolation by Low Degree Positive Real rational Functions

TL;DR

This paper demonstrates that positive real rational functions of degree at most m can interpolate m nodes in the left half-plane to any target points in the complex plane, providing a new parametrization method.

Contribution

It introduces a novel interpolation method using positive real rational functions and a simple parametrization for a large subset of such functions.

Findings

Any set of m nodes in the left half-plane can be mapped to any complex points by degree-m positive real functions.

A new parametrization in R^{2m+3} for a large subset of these interpolating functions.

The approach combines polynomial interpolation with structural properties of positive real functions.

Abstract

Using polynomial interpolation, along with structural properties of the family of positive real rational functions, we here show that a set of m nodes in the open left half of the complex plane, can always be mapped to anywhere in the complex plane by rational positive real functions whose degree is at most m. Moreover, we introduce an easy-to-find parametrization in $R^{2m+3}$ of a large subset of these interpolating functions.

Equations

$$y_j=f(x_j)j=1,\dots ,m.$$

$$\mathrm{Re}\left(f(s)\right)\ge 0\forall s\in {\mathbb{C}}_r.$$

$$\frac{y_j^{\ast }+y_k}{x_j^{\ast }+x_k}j,k=1,\dots ,m.$$

$$\frac{|x_j-x_k{|}^2}{\mathrm{Re}(x_j)\mathrm{Re}(x_k)}\ge \frac{|y_j-y_k{|}^2}{\mathrm{Re}(y_j)\mathrm{Re}(y_k)}m\ge j>k\ge 1.$$

$$\mathrm{Re}(x_j)\mathrm{Re}(y_j)>0j=1,\dots ,m,$$

$$f(s)=p(s)+r\Delta (s),$$

$$\eta (s):=\prod _{j=1}^m(s-x_j).$$

$${\varphi }_j(s)=\frac{\eta (s)}{s-x_j}=\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m(s-x_k)j=1,\dots ,m.$$

$$\sum _{j=1}^m{c}_j{\varphi }_j(s)c_j\in \mathbb{C}.$$

$$\tilde{d}(s)=b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s)b,c_j\in \mathbb{C}.$$

$$c_1\cdots c_m\ne 0,$$

$${\varphi }_j(s)=\sum _{k=0}^{m-1}{a}_{jk}{s}^k$$

$$a_j:=\left(\begin{array}{c}{a}_{jo}\\ a_{j1}\\ ⋮\\ a_{j,m-1}\end{array}\right).$$

$$\begin{array}{rl}\left(\begin{array}{ccccc}1& x_1& x_1^2& \dots & x_1^{m-1}\\ 1& x_2& x_2^2& \dots & x_2^{m-1}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& x_m& x_m^2& \dots & x_m^{m-1}\end{array}\right)\left(\begin{array}{cccc}{a}_{10}& a_{20}& \dots & a_{m0}\\ a_{11}& a_{21}& \dots & a_{m1}\\ a_{12}& a_{22}& \dots & a_{m2}\\ ⋮& ⋮& ⋮& ⋮\\ a_{1,m-1}& a_{2,m-1}& \dots & a_{m,m-1}\end{array}\right)& =\\ & =\left(\begin{array}{ccccc}\ast & 0& 0& \dots & 0\\ 0& \ast & 0& \dots & 0\\ 0& 0& \ast & \dots & 0\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ 0& 0& 0& \dots & \ast \end{array}\right)\end{array}$$

$$\tilde{d}(s{)}_{{|}_{s=x_j}}=c_j{\varphi }_j(s{)}_{{|}_{s=x_j}}={\left(c_j\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^m(x_j-x_k)\right)}_{{|}_{c_j\ne 0}}\ne 0.$$

$$\mathrm{Im}(x_j)>0⟹x_{j+1}=x_j^{\ast }.$$

$$\tilde{d}(s)=b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s)b\in \mathbb{R}\begin{array}{cccc}\mathrm{Im}(x_j)>0& & 0\ne & c_{j+1}=c_j^{\ast }\\ & & & \\ \mathrm{Im}(x_j)=0& & 0\ne & c_j\in \mathbb{R}.\end{array}$$

$$\begin{array}{ccc}{\tilde{d}}_o(s)& :={\tilde{d}(s)}_{{|}_{b=0}}& =\sum _{j=1}^m{\gamma }_j{\varphi }_j(s)\\ {\tilde{d}}_1(s)& :={\tilde{d}(s)}_{{|}_{b\ne 0}}& =b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s).\end{array}$$

$$\begin{array}{cccccccccc}{\tilde{d}}_o(s)& =& \sum _{j=1}^m{\gamma }_j{\varphi }_j(s)& & & & & \mathrm{deg}\left({\tilde{d}}_o(s)\right)& =& m-1\\ & & & & & & & & & \\ {\tilde{d}}_1(s)& =& b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s)& & & & & \mathrm{deg}\left({\tilde{d}}_1(s)\right)& =& m.\end{array}$$

$${\nu }_o(s)=\sum _{j=1}^m{y}_j{\gamma }_j{\varphi }_j(s){\nu }_1(s)=\sum _{j=1}^m{y}_j{c}_j{\varphi }_j(s).$$

$$\begin{array}{ccc}{\tilde{p}}_o(s):=\frac{{\nu }_o(s)}{{\tilde{d}}_o(s)}=\frac{\sum _{j=1}^m{y}_j{\gamma }_j{\varphi }_j(s)}{\sum _{j=1}^m{\gamma }_j{\varphi }_j(s)}& & {\tilde{p}}_1(s):=\frac{{\nu }_1(s)}{{\tilde{d}}_1(s)}=\frac{\sum _{j=1}^m{y}_j{c}_j{\varphi }_j(s)}{b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s)},\end{array}$$

$$\begin{array}{ccc}{\tilde{\Delta }}_o(s)& ={\left(\frac{{\tilde{d}}_o(s)}{\eta (s)}\right)}^{-1}& ={\left(\sum _{j=1}^m\frac{{\gamma }_j}{s-x_j}\right)}^{-1}\\ & & \\ {\tilde{\Delta }}_1(s)& ={\left(\frac{{\tilde{d}}_1(s)}{\eta (s)}\right)}^{-1}& ={\left(b+\sum _{j=1}^m\frac{c_j}{s-x_j}\right)}^{-1}\end{array}$$

$$\begin{array}{ccc}{\tilde{f}}_o(s)& :=& {\tilde{p}}_o(s)+r_o{\tilde{\Delta }}_o(s)=\frac{{\nu }_o(s)}{{\tilde{d}}_o(s)}+r_o\frac{\eta (s)}{{\tilde{d}}_o(s)}\\ & & \\ & =& \frac{\sum _{j=1}^m{y}_j{\gamma }_j{\varphi }_j(s)+r_o\eta (s)}{\sum _{j=1}^m{\gamma }_j{\varphi }_j(s)}\end{array}{r}_o\in \mathbb{R}\mathrm{parameter},$$

$$\begin{array}{ccc}{\tilde{f}}_1(s)& :=& {\tilde{p}}_1(s)+r_1{\tilde{\Delta }}_1(s)=\frac{{\nu }_1(s)}{{\tilde{d}}_1(s)}+r_1\frac{\eta (s)}{{\tilde{d}}_1(s)}\\ & & \\ & =& \frac{\sum _{j=1}^m{y}_j{c}_j{\varphi }_j(s)+r_1\eta (s)}{b\eta (s)+\sum _{j=1}^m{c}_j{\varphi }_j(s)}\end{array}{r}_1\in \mathbb{R}\mathrm{parameter}.$$

$$m=\mathrm{degree}\left(\eta \right)\ge \mathrm{degree}\left(\nu \right)=m-1.$$

$$\begin{array}{ccc}m& =& \mathrm{degree}\left(\tilde{\Delta }\right)\\ & =& \mathrm{degree}\left(\frac{\eta }{d}\right)\\ & =& \mathrm{degree}\left(d\right)+\mathrm{degree}\left(\eta \right)\\ & =& \mathrm{degree}\left(d\right)+\max \left(\mathrm{degree}\left(\eta \right),\mathrm{degree}\left(\nu \right)\right)\\ & \ge & \mathrm{degree}\left(d\right)+\mathrm{degree}\left(\nu +r\eta \right)\\ & \ge & \mathrm{degree}\left(\frac{\nu +r\eta }{d}\right)\\ & =& \mathrm{degree}\left(\tilde{f}\right).\end{array}$$

$$x_1,\dots ,x_m\in {\mathbb{C}}_l.$$

$$\mathrm{Re}\left(f(s)\right)>0\forall s\in {\overline{\mathbb{C}}}_r.$$

$$\frac{c_j}{s-x_j}+\frac{c_j^{\ast }}{s-x_j^{\ast }}=2\mathrm{Re}(c_j){\left(s+\frac{\left(\mathrm{Re}(c_j)\left(-\mathrm{Re}(x_j)\right)+\mathrm{Im}(c_j)\mathrm{Im}(x_j)\right)s+\mathrm{Re}(c_j)|x_j{|}^2}{\mathrm{Re}(c_j)s+\left(\mathrm{Re}(c_j)\left(-\mathrm{Re}(x_j)\right)-\mathrm{Im}(c_j)\mathrm{Im}(x_j)\right)}\right)}^{-1}.$$

$$\mathrm{Re}(c_j)>\frac{|\mathrm{Im}(c_j)\mathrm{Im}(x_j)|}{-\mathrm{Re}(x_j)}\ge 0,$$

Full text

1. Introduction

Problem Formulation

A framework for many classical interpolation problems is as follows. Given a set of distinct nodes $x_{1},~{}\ldots~{},~{}x_{m}$, image points $y_{1},~{}\ldots~{},~{}y_{m}$ (not necessarily distinct) and a family of functions $\mathcal{F}$, find whether there exist functions $f\in\mathcal{F}$ so that

[TABLE]

If yes, parameterize all of them, preferably within a degree bound. There is a vast literature on the subject see e.g. [2]-[7], [11]-[17]. To simplify the discussion, we here focus on scalar real rational functions. Thus degree simply means the maximum between the degree of the numerator and of the denominator polynomials. The polynomial (a.k.a. the Lagrange) interpolation [13] (in [14] it is attributed to [16]) is probably the best known problem in this framework. For the case where $\mathcal{F}$ is the set of rational functions see [3] and if in addition all functions in $\mathcal{F}$ are analytic in a disk of a prescribed radius in $\mathbb{C}$, the problem was addressed in [2].

We shall denote by $\mathbb{C}_{r}$ $(\overline{\mathbb{C}}_{r})$ the open (closed) right half plane (the subscript stands for “right”). The family of functions $\mathcal{F}$ we here focus on, is of  positive real,  i.e. analytically mapping the open right half plane to its closure. Namely, a real rational function $~{}f(s)~{}$ of a complex variable $~{}s~{}$ is said to be positive if

[TABLE]

Interpolation problem with rational positive real functions can be further classified by the domain the nodes $x_{1},~{}\ldots~{},~{}x_{m}$ belong to.

If the nodes $x_{j}$ are in $\mathbb{C}_{r}$, this amounts to the classical Nevanlinna-Pick interpolation problem, see e.g. [4, Theorem 18.1] and for real functions [17]. There, from the interpolation data one constructs the Pick matrix whose $j,k$ element is given by

[TABLE]

It is known that there exist interpolating functions if and only if the Pick matrix is positive semi-definite. Moreover, all interpolating functions may be parameterized through this Pick matrix. Recall that having the Pick matrix positive semi-definite implies that each 2-dimensional minor is non-negative, which in turn can be written as,

[TABLE]

This condition means that the map from the nodes $x_{1}~{},~{}\ldots~{},~{}x_{m}$ to the image points $y_{1}~{},~{}\ldots~{},~{}y_{m}$, is contractive in $\mathbb{C}_{r}$ in the sense of Eq. (1.3). This illustrates the fact that the interpolation problem cannot be solvable for arbitray data set.

If the nodes are confined to the imaginary axis, an interpolation scheme, elegant in its simplicity, appeared in [18].

If the nodes $x_{j}$ are in $\overline{\mathbb{C}}_{r}$ (with possibly some nodes on $i{\mathbb{R}}$) the problem is much harder, see e.g. [4, Chapter 21], [6], [7], [11] and [15].

If the interpolation data is in whole plane, provided that

[TABLE]

one can still resort to the classical Nevanlinna-Pick interpolation scheme: First, complete the data set so that if $x,y$ is an interpolation pair, then so is $-x,-y$. Then, from this extended data, take the $m$ nodes which are in $\mathbb{C}_{r}$, construct the the corresponding Pick matrix and proceed as usual. Finally, use the fact (see [17]) that whenever the Pick matrix is positive semi-definite, among the interpolation functions there exists some function with odd symmetry, i.e. $f(s)=-f(-s)~{}$ (a.k.a. Foster or lossless functions, see e.g. [5], [8], [17]). If instead of the left and right half planes, $\mathbb{C}$ is partitioned to the unit disk and its exterior, a similar idea is presented in [2, Section 5].

In this work we focus on the case where the nodes $x_{1},~{}\ldots~{},~{}x_{m}$ are all in $\mathbb{C}_{l}$ (the open left half plane). We parameterize a large subset of rational positive real interpolating functions whose degree is less or equal to $m$. In particular, it is shown that this set is never empty.

A key idea is the following: We construct two rational functions sharing the same denominator: (i) an interpolating function $p(s)$ (not necessarily positive real) and (ii) a strictly positive real rational function ${\scriptstyle\Delta}(s)$ vanishing at the nodes. Thus, for all $r\in{\mathbb{R}}$ the parametric rational function

[TABLE]

is interpolating. Moreover, for ${r}$ “sufficiently large”, $f(s)$ turns to be positive real.

Interestingly, we can mention two ideas conceptually similar to those in the current work, which have appeared within completely different interpolation frameworks: (i) The fact that for interpolation by low degree rational functions, one should separately treat numerators and denominators, appeared in the context of Schur functions in [9, Theorems 1, 2]. (ii) In [1] we have used Eq. (1.5) for interpolation by structured matrix-valued polynomials. For example, where $p(s)$ was an interpolating polynomial which on $i{\mathbb{R}}$, attained Hermitian values, while the polynomial ${\scriptstyle\Delta}(s)$, vanishing at the nodes, was positive definite on $i{\mathbb{R}}$, see [1, Eq. (1.6)].

In Section II we present a five steps interpolation procedure:

In Step 1 we parameterize all candidates for denominator polynomials of the sought interpolating functions. Namely, all real polynomials, of degree of at most $~{}m$, non-vanishing at the nodes.

In Step 2, to each of these denominator polynomials we match a numerator to obtain $p(s)$, a rational interpolating function (not necessarily positive real).

In Step 3, we construct ${\scriptstyle\Delta}(s)$ strictly positive real rational functions, vanishing at the nodes. We now restrict the denominators of $p(s)$, the rational functions from Step 2 to the subset of the resulting deniminators of ${\scriptstyle\Delta}(s)$.

To each of the resulting interpolating function $p(s)$, we add $r{\scriptstyle\Delta}(s)$, a weighted version of the strictly positive real rational functions, vanishing at the nodes (which shares the same denominar). Thus, $p(s)+r{\scriptstyle\Delta}(s)$ is an interpolating function, of degree of at most $m$. Furthermore, for $r$ “sufficiently large” it is positive real.

A closer scrutiny reveals that all interpolating, positive real rational functions obtained, are so that the degree of the denominator is larger or equal to the degree of the numerator.

In Step 5, we complete our the description of positive real interpolating functions as follows: We repeat the previous steps by constructing positive real interpolating functions from the original nodes $~{}x_{j}$ but to111Consider the case where $y_{j}=0$ for some $j$.  $\frac{1}{y_{j}}$. Finally, as the sought solution, we take the reciprocal of these functions.

In Section III we illustrate the above procedure by detailed examples. and add concluding remarks.

2. A Recipe

2.1. Step 1: Constructing all real monic polynomials,

of degree ${\mathbf{m}}$ and ${\mathbf{m}-1}$, non-vanishing at the nodes

We shall do it in stages.

1a Constructing all complex polynomials of degree of at most  $m$  with no roots at prescribed distinct points  $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$.

First, denote by  $\eta(s)$  the monic polynomial (of degree  $m$)  whose roots are the prescribed distinct points  $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$,

[TABLE]

Next, denote by  $\phi_{1}(s),~{}\ldots~{},~{}\phi_{m}(s)$  the monic divisors of $~{}\eta(s)~{}$ of degree $~{}m-1$, i.e.

[TABLE]

Lemma 2.1**.**

For distinct $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}~{}$ let $\eta(s)$ and $\phi_{1}(x)~{},~{}\ldots~{},~{}\phi_{m}(x)$ be as in (2.1) and (2.2), respectively.

1. The set of all polynomials of degree of at most $m-1$ can be parameterized by

[TABLE]

2. The set of all polynomials of degree of at most $m$ can be parameterized by

[TABLE]

3. If in addition,

[TABLE]

$\tilde{d}(s)$* forms the set of all polynomials of degree of at most $m$ which do not vanish at $x_{1}~{},~{}\ldots~{},~{}x_{m}$.*

Proof 1. For each $j$, $j\in[1,~{}m]$, let

[TABLE]

be identified with

[TABLE]

Now by construction,

[TABLE]

where $~{}*~{}$ denotes a non-zero element.

The leftmost is a Vandermonde matrix, which is non-singular, whenever $x_{1}~{},~{}\ldots~{},~{}x_{m}$ are distinct, see e.g. [10, Item 0.9.11]. Thus, the middle matrix whose entries are $a_{jk}$ must be non-singular, so in particular its columns form a basis to $\mathbb{C}^{m}$. In other words ${\phi}_{1}(s),~{}\ldots~{},~{}{\phi}_{m}(s)$ form a basis to all polynomials of degree of at most $m-1$.

2. In a similar way, the coefficients of $\eta(s)$ in (2.1) can be identified with a vector in $\mathbb{C}^{m+1}$. By adding a bottom zero to each the above vectors $a_{1}~{},~{}\ldots~{},~{}a_{m}$, they are embedded in $\mathbb{C}^{m+1}$. Thus, the problem is reduced to verifying that the last element in the vector associated with $\eta(s)$ is non-zero, but by construction $\eta(s)$ is of degree $m$, so indeed its bottom element is non-zero.

3. Having no roots at $x_{1}~{},~{}\ldots~{},~{}x_{m}$. Note that,

[TABLE]

In other words, having $~{}c_{1}{\cdots}c_{m}\not=0~{}$ is necessary and sufficient for these polynomials not to vanish at the original points $x_{1}~{},~{}\ldots~{},~{}x_{m}$. Thus the claim is established. ∎

In the sequel we focus on real polynomials and real rational functions. Hence, we have the next stage.

1b Assuming the prescribed distinct points $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$, are closed under complex conjugation, constructing all real polynomials of degree of at most  $m$,  with no roots at these points.

Assume hereafter that, if necessary, the original set of distinct points $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$  is complemented so it is closed under complex conjugation. Namely,

[TABLE]

Note that then in Eq. (2.1) the resulting  $\eta(s)$  is real.

We now construct the sought polynomials.

Lemma 2.2**.**

For distinct  $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$,  closed under complex conjugation, let  $\eta(s)$  and  $\phi_{1}(s),~{}\ldots~{},~{}\phi_{m}(s)$  be as in Eqs. (2.1) and (2.2), respectively. The set of all real polynomials $~{}\tilde{d}(s)~{}$ of degree, of at most $~{}m$, with no roots at the original points $x_{1},~{}\ldots~{},~{}x_{m}$, can be parametrized by

[TABLE]

The claim follows from Lemma 2.1 along with Eq. (2.4). Note that in particular,  $\tilde{d}(s)$  may have multiple roots.

In the sequel, the prescribed points  $x_{1},~{}\ldots~{},~{}x_{m}$  will be referred to as  nodes.

Without loss of generality, we shall find it convenient to distinguish in Eq. (2.4) between the cases $b=0$, $b\not=0$. Furthermore to ease the distinction, we differently denote the coefficients222Although as before,   $\begin{smallmatrix}{\rm Im}(x_{j})>0&~{}&0\not=&{\gamma}_{j+1}={\gamma}_{j}^{*}\\ ~{}\\ {\rm Im}(x_{j})=0&~{}&0\not=&{\gamma}_{j}\in{\mathbb{R}}.\end{smallmatrix}$

of the polynomials for $b=0$ and $b\not=0$, i.e.

[TABLE]

We now can state the following.

Theorem 2.3**.**

For distinct nodes  $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}$,  closed under complex conjugation, let  $\eta(s)$  and  $\phi_{1}(s),~{}\ldots~{},~{}\phi_{m}(s)$  be as in Eqs. (2.1) and (2.2), respectively. Let also  ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}~{}$ and $~{}c_{1},~{}\ldots~{},~{}c_{m}~{}$ along with $~{}b\in{\mathbb{R}}$, be all non-zero parameters as in Eq. (2.4).

The set of all real polynomials  $d(s)$  of degree, of at most $~{}m$, with no roots at these nodes, can be parametrized by two families,

[TABLE]

2.2. Step 2:

(Not necessarily positive real) interpolating functions with prescribed denominator.

With each of the denominators in Eq. (2.5), $d_{o}(s)$ and $d_{1}(s)$, we here match a numerator, denoted by $\nu_{o}(s)$ and $\nu_{1}(s)$, respectively, to obtain a rational (not necessarily positive real) interpolating function (from $x_{j}$ to $y_{j}$).

Theorem 2.4**.**

Let the interpolation data in Eq. (1.1) be closed under complex conjugation333Namely, if ${\rm Im}(x)_{j}>0$ then $x_{j+1}=x_{j}^{*}$ and $y_{j+1}=y_{j}^{*}$., where the nodes $x_{1},~{}\ldots~{},~{}x_{m}$ are distinct and in $\mathbb{C}_{l}$. Let also the (non-zero) coefficients ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}$ and $c_{1},~{}\ldots~{},~{}c_{m}$ be as in Eq. (2.5).

Construct the polynomials (where $\phi_{j}(s)$ as in Eq. (2.2))

[TABLE]

Then, the rational functions,

[TABLE]

(with $0\not=b\in{\mathbb{R}}$) interpolate between $x_{j}$ and $y_{j}$.

This result follows directly from the definition of $\phi_{j}(s)$ in Eq. (2.2).

We next construct additional interpolating rational functions of degree of at most $~{}m$.

Lemma 2.5**.**

*Let $\eta(s)$, $d_{k}(s)$, $\nu_{k}(s)$, and $p_{k}(s)$, (with $k=0,1$) from Eqs. (2.1), (2.5), (2.6), and (2.7) respectively.

The set of all real rational functions of degree $~{}m$, vanishing at the nodes is given by,*

[TABLE]

with $~{}{\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}$, $~{}b$, $c_{1},~{}\ldots~{},~{}c_{m}$ all non-zero.

Using $\tilde{\scriptstyle\Delta}_{k}(s)$, define the rational functions,

[TABLE]

and

[TABLE]

Then the following is true.

• (i)    

For arbitrary $~{}r_{o},r_{1}\in{\mathbb{R}}~{}$ the functions $~{}\tilde{f}_{o}(s)~{}$ and $~{}\tilde{f}_{1}(s)~{}$ in Eqs. (2.9) (2.10) are of degree of at most $~{}m$.

• (ii)  

For arbitrary $~{}r_{o},r_{1}\in{\mathbb{R}}~{}$ the functions $~{}\tilde{f}_{o}(s)~{}$ and $~{}\tilde{f}_{1}(s)~{}$ in Eqs. (2.9) (2.10) interpolate from $x_{1},~{}\ldots~{},~{}x_{m}$ to $y_{1},~{}\ldots~{},~{}y_{m}$.

Proof : For convenience, throughout the proof, we omit the dependence on $k=0,1$ and simply write $\tilde{\scriptstyle\Delta}(s)$, $\nu(s)$, $d(s)$ and $f(s)$.

$(i)$ Recall that by construction (see Eqs. (2.1), (2.5) and Theorem 2.3) the polynomials $\eta(s)$ and $d(s)$ are relatively prime. Recall also (see Theorem 2.4) that

[TABLE]

We thus have the following,

[TABLE]

$(ii)$ This is immediate from Theorem 2.4 along with the definitions of $\tilde{\scriptstyle\Delta}(s)$ and of $\tilde{f}(s)$, see Eqs. (2.8), (2.9) and (2.10), respectively. ∎

Following Eq. (1.1), we assume hereafter that the data set is closed under complex conjugation and that the distinct nodes are in the open left half plane, i.e.

[TABLE]

Thus, some of the resulting interpolating functions in Eqs. (2.9) and (2.10) may be positive real while other are not. Either way, the rest of this section is depvoted to extracting positive real functions out of them. Specifically, we shall devise a scheme of easily constructing large subsets of positive real interpolating functions. This simplicity, based on the structure of interpolating functions (see Corollary 2.7 below), comes on the expense of guranteeing finding  all  positive real interpolating functions.

2.3. Step 3:

All positive real rational functions of degree $~{}{\mathbf{m}}$, with prescribed denominator, and vanishing at the nodes.

We first recall well-known facts, which are fundamental to our construction.

Theorem 2.6**.**

The following is true.

• (i)  

For prescribed data, the family of interpolating rational functions is convex.

• (ii) 

The set of rational positive real functions forms a convex cone.

As a non-empty intersection of convex sets, is convex, one can conclude the following.

Corollary 2.7**.**

For prescribed data set, whenever not empty, the family of rational positive real interpolating functions, is convex.

We now resort to the following, see e.g. [8, Section 4.3], [12, Definition 6.4].

Definition 2.8**.**

A rational function $f(s)$ will be called  Strictly Positive Real if $~{}f(s-\epsilon)~{}$ is positive real, for some  $\epsilon>0$. **

The following well known properties will be useful in the sequel.

Theorem 2.9**.**

(i) If a rational function $~{}f(s)~{}$ is strictly positive real, then

[TABLE]

(ii) The set of positive real functions forms a Convex Invertible Cone.

Item (i) follows from Definition 2.8 and for item (ii) see [8, Proposition 4.1.1].

Whenever the functions in Eq. (2.8) are strictly positive real, the tilde will be omitted and they will be denoted by  $~{}{\scriptstyle\Delta}_{o}(s)$ and $~{}{\scriptstyle\Delta}_{1}(s)$. This is addressed next.

Example 2.10**.**

We next illustrate the fact that for any set of nodes in $\mathbb{C}_{l}$, the open left-half plane, one can choose the coefficients $~{}{\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}~{}$ and $~{}c_{1},~{}\ldots~{},~{}c_{m}$, so that in Eq. (2.8) one obtains  strictly positive real functions, ${\scriptstyle\Delta}_{o}(s)$ and ${\scriptstyle\Delta}_{1}(s)$.

As the reasoning is identical, we show it only for $c_{1},~{}\ldots~{},~{}c_{m}$.

Specifically, if $x_{j}\in{\mathbb{R}}_{-}$ then $\frac{c_{j}}{s-x_{j}}$ is strictly positive real for all $c_{j}\in{\mathbb{R}}_{+}~{}$.

If $~{}x_{j}\in\{\mathbb{C}_{l}\smallsetminus{\mathbb{R}}_{-}\}~{}$ (and from Eq. (2.3) $x_{j+1}=x_{j}^{*}$) then taking $c_{j+1}=c_{j}^{*}$ yields,

[TABLE]

Thus, choosing $c_{j}$ so that,

[TABLE]

is sufficient to guarantee that the function in Eq. (2.11) is strictly positive real. * ∎*

Motivated by the above example we can state the following.

Theorem 2.11**.**

Out of the polynomials $~{}\eta(s)$, $~{}\tilde{d}_{o}(s)~{}$ and $~{}\tilde{d}_{1}(s)~{}$ in Eqs. (2.1), (2.5) respectively, one can choose the coefficients $~{}c_{1}~{},~{}\ldots~{},~{}c_{m}$, $b~{}$ and $~{}{\gamma}_{1}~{},~{}\ldots~{},~{}{\gamma}_{m}~{}$ to construct all monic444As in Eqs. (2.9) (2.10) $~{}\tilde{\scriptstyle\Delta}(s)$ is scaled by $r$, without loss of generality one can take $d(s)$ to be moinc. polynomials,

[TABLE]

so that the rational functions from Eq. (2.8) are strictly positive real, i.e.

[TABLE]

*Moreover:

The coefficients of $~{}{d}_{o}(s)$: ${\gamma}_{1}~{},~{}\ldots~{},~{}{\gamma}_{m}$ form a convex set (excluding ${\gamma}_{j}=0$) within a hyperplane in $\mathbb{C}^{m}$.

The coefficients of $~{}{d}_{1}(s)$: $c_{1},~{}\ldots~{},~{}c_{m}$ form a convex subset of $\mathbb{C}^{m}$ (excluding $~{}m~{}$ hyperplanes $c_{j}=0$). This set is positively unbounded in the sense that if in Eqs. (2.5), (2.13)*

[TABLE]

is an admissible set of parameters, then so is555 provided that, to preserve complex conjugation, $c_{j+1}=c_{j}^{*}$ implies $\delta_{j+1}=\delta_{j}~{}$.

[TABLE]

Furthermore, the sets ${\gamma}_{1}~{},~{}\ldots~{},~{}{\gamma}_{m}$ and $c_{1},~{}\ldots~{},~{}c_{m}$ can be parameterized by a convex subset of ${\mathbb{R}}^{m}$ (excluding the axes).

Proof From Example 2.10 it follows that for arbitrary set of nodes, this family of ${\scriptstyle\Delta}(s)$ functions is not empty.

To simplify establishing structural properties, we begin by ignoring the condition that neither ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}$ nor $c_{1},~{}\ldots~{},~{}c_{m}$ vanish.

By Corollary 2.7 the set of interpolating positive real rational functions is convex.

Indeed, if $c_{1}~{},~{}\ldots~{},~{}c_{m}$ and $\hat{c}_{1}~{},~{}\ldots~{},~{}\hat{c}_{m}$ are two admissible sets in Eqs. (2.5), (2.13) then so is ${\scriptstyle\theta}{c_{1}}+(1-{\scriptstyle\theta})\hat{c}_{1}~{},~{}\ldots~{},~{}{\scriptstyle\theta}{c_{m}}+(1-{\scriptstyle\theta})\hat{c}_{m}$, for all ${\scriptstyle\theta}\in[0,~{}1]$.

The fact that $\sum\limits_{j=1}^{m}{\gamma}_{j}=1$ forms a hyper-plane in $\mathbb{C}^{m}$ is straightforward. Next, to show that the set $c_{1},~{}\ldots~{},~{}c_{m}$ is positively unbounded, one can resort again to the construction in Example 2.10.

Recall however that the restriction that $\prod\limits_{j=1}^{m}c_{j}\not=0$, implies that the set of coefficients $~{}c_{1},~{}\ldots~{},~{}c_{m}~{}$ forms an almost convex cone, as it excludes $~{}m~{}$ hyper-planes, $~{}c_{j}=0$.

Real coefficients: Recall that in (2.5) the coefficients $c_{j}$ are real or come in complex conjugate pairs. Specifically if there are $q$ coefficients in the upper half plane and $m-2q$ are real, they are described by a point in ${\mathbb{R}}^{m}$. ∎

The above analysis suggests that in the coefficient space, it is enough to find the boundary of the (almost convex) sets of admissible ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}~{}$ and $~{}c_{1},~{}\ldots~{},~{}c_{m}$.

In the next step, we combine Theorems 2.4 and 2.11 to construct positive real interpolating functions of degree of at most $m$.

2.4. Step 4: Positive real

interpolating functions.

To extract positive real functions, out of the set of interpolating functions $\tilde{f}(s)$ in Eqs. (2.9) and (2.10), we focus on those whose deniminator is given by Theorem 2.11. This is formalized next.

Lemma 2.12**.**

*Let $\eta(s)$, $d_{k}(s)$, $\nu_{k}(s)$, $p_{k}(s)$ and $~{}{\scriptstyle\Delta}_{k}(s)$, (with $k=0,1$) from Eqs. (2.1), (2.5), (2.6), (2.7) and (2.8) respectively.

Define the rational functions,*

[TABLE]

and

[TABLE]

Then, $~{}f_{k}(s)$ (and $p_{k}(s)$) are interpolating function with $~{}{\scriptstyle\Delta}_{k}(s)$ strictly positive real, vanishing at the nodes (all sharing the same denominator).

Furthermore, the quantities,

[TABLE]

are well defined.

Proof : For convenience, throughout the proof, we omit the dependence on $k=0,1$ and simply write $\nu(s)$, $d(s)$ and $f(s)$.

The construction in Theorem 2.11 guarantees that in Eq. (2.13)

[TABLE]

is  strictly positive real and thus by item (i) of Theorem 2.9

[TABLE]

Recalling that (i) the numerator $~{}\eta(s)$ vanishes only at $~{}m~{}$ points in ${\mathbb{C}}_{l}$, see Eq. (2.1) and (ii) in addition ${\rm degree}(\eta)\geq{\rm degree}(d)$ see Eq. (2.5), in fact

[TABLE]

Next, exploiting again the fact that $~{}{\scriptstyle\Delta}(s)~{}$ is strictly positive real, see Theorem 2.9, implies that $~{}d(s)~{}$ does not vanish in $\overline{\mathbb{C}}_{r}$. We can thus conclude that

[TABLE]

is well defined, so the claim is established. ∎

So far we have described interpolating rational functions $~{}f(s)~{}$ of degree of at most $~{}m$ whose denominator is so that $~{}{\scriptstyle\Delta}(s)~{}$ is strictly positive real. To proceed with the construction, the idea is very simple, see Eq. (1.5) :

With the same $~{}\eta(s)$, $~{}d(s)~{}$ and $~{}\nu(s)$ construct the rational functions

[TABLE]

where $r_{a}$ and $r_{b}$ are real parameters. On the one hand, from Lemma 2.12 it follows that $f_{a}(s)$ and $f_{b}(s)$ interpolate between with same data. On the other hand, Theorems 2.9 and 2.11 imply that,

[TABLE]

Next, recall that by Eq. (1.2) $~{}f(s)~{}$ is positive real whenever,

[TABLE]

Thus, one can formally define $\forall s\in\mathbb{C}_{r}$,

[TABLE]

We next combine the above definition of $\underline{r}$ along with Lemma 2.12.

Proposition 2.13**.**

Let the rational function $~{}f_{k}(s)~{}$ and the scalars $~{}\underline{r}_{k}$ (with $k=0,1$) be as in Eqs. (2.14), (2.15) and (2.16), respectively. Then,

[TABLE]

and $~{}f_{k}(s)~{}$ is positive real if and only if $~{}r_{k}\geq\underline{r}_{k}~{}$.

Proof : For simplicity, we omit both the dependence on $~{}s~{}$ and the subscript $~{}k$. Using Eqs. (2.14) (2.15) note that

[TABLE]

Now, $~{}f~{}$ is positive real if and only if

[TABLE]

Namely,

[TABLE]

in turn, using the fact that ${\frac{\eta}{d}}$ is strictly positive real, see Theorem 2.11, this means that

[TABLE]

Hence, one can conclude that $~{}f~{}$ in Eqs. (2.14), (2.15) is positive real, if and only if,

[TABLE]

and by Eq. (2.16), the proof is complete. ∎

Noting that ${\rm deg}(\eta_{k})=m-1$, for $k=0,1$, while ${\rm deg}(\psi)=m$ together with the fact that ${\scriptstyle\Delta}_{k}$ is strictly positive real, guarantees the following.

Observation 2.14**.**

In Proposition 2.13,

[TABLE]

Note that from Eqs. (2.14) and (2.15) it follows that for $~{}r_{k}>0$, with $k=0,1$ whenever there is no pole-zero cancelation, the degree of the numerator of $~{}f_{o}(s)~{}$ or of $~{}f_{1}(s)~{}$ is $~{}m$. Thus, all positive real interpolating functions $~{}f(s)~{}$ we have constructed are of degree at most $~{}m$, but under the restriction that the degree of the numerator is greater or equal to the degree of the denominator. In the next section we address the complementary case where the degree of the denominator is greater or equal to the degree of the numerator.

2.5. Step 5: Additional positive real

interpolating functions.

Taking the original data, if one considers a function, say $g(s)$, interpolating from $x_{1},~{}\ldots~{},~{}x_{m}$ to666Assuming $y_{j}\not=0$ $\frac{1}{y_{1}}~{},~{}\ldots~{},~{}\frac{1}{y_{m}}$, then $~{}\frac{1}{g(s)}~{}$ solves the original problem, where we have relied on the fact that the inverse of a positive real function, is positive real, see item (ii) of Theorem 2.9. Here are the details.

We follow the previous steps (while adding hat to the respective functions) and first mimic Theorem 2.4.

Theorem 2.15**.**

Let the interpolation data777Assuming $~{}y_{j}\not=0$. be as in Eq. (1.1), the (non-zero) denominator coefficients ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}~{}$ and $~{}c_{1},~{}\ldots~{},~{}c_{m}~{}$ be as in Eq. (2.5). Construct the polynomials (where $\phi_{j}(s)$ as in Eq. (2.2))

[TABLE]

Then, the rational functions,

[TABLE]

interpolate between $~{}x_{j}~{}$ and $~{}\frac{1}{y_{j}}~{}$ with $~{}j=1,~{}\ldots~{},~{}m$.

Note that indeed all the parameters are as before.

We next mimic Lemma 2.12 and construct the rational functions,

[TABLE]

and

[TABLE]

where $\eta(s)$, $d_{k}(s)$, ${\scriptstyle\Delta}_{k}(s)$ and $p_{k}(s)$, with $k=0,1$ are  as before, see Eqs. (2.1), (2.5) (2.13) and (2.7), respectively.

As before, the rational function $~{}\hat{f}_{k}(s)~{}$ (with $k=0,1$) see Eqs. (2.17), and (2.18) interpolate from $x_{1},~{}\ldots~{},~{}x_{m}$ to $y_{1},~{}\ldots~{},~{}y_{m}$ for all $~{}\hat{r}_{k}\in{\mathbb{R}}$. Out of this family, we next extract the positive real subset. To this end, we introduce the following notation,

[TABLE]

By using item (ii) of Theorem 2.9, we can next adapt Proposition 2.13 to guarantee that the sought interpolating functions are indeed positive real.

Proposition 2.16**.**

Let the rational function $~{}\hat{f}_{k}(s)~{}$ and the scalars $~{}\underline{\hat{r}}_{k}(s)$ (with $k=0,1$) be as in Eqs. (2.17), (2.18) and (2.19), respectively. Then,

[TABLE]

and $~{}\hat{f}_{k}(s)$ is positive real if and only if $~{}\hat{r}_{k}\geq\underline{\hat{r}}_{k}~{}$.

We have shown that $~{}\hat{f}_{o}(s)~{}$ and $~{}\hat{f}_{1}(s)~{}$ are positive real interpolating functions of degree at most $~{}m$, where the degree of the numerator is larger or equal to the degree of the denominator.

Simlar to the reasoning at end of Step 4, one can conclude the following.

Observation 2.17**.**

In Proposition 2.16,

[TABLE]

3. Examples and Concluding remarks

The above recipe is illustrated through simple examples.

A. We start by illustrating the role of $~{}f_{o}(s)~{}$ vs. $~{}\hat{f}_{o}(s)$ in Eqs. (2.14) and (2.17), respectively to obtain interpolating functions having at $s=\infty$ either pole or zero.

(i) Find a minimal degree positive real function $~{}f(s)~{}$ mapping $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}_{l}$ to $y_{1}=x_{1},~{}\ldots~{},~{}y_{m}=x_{m}$. Clearly the sought solution is

[TABLE]

We now follow the above recipe and substitute in Eq. (2.14)

[TABLE]

(ii) Find a minimal degree positive real function $~{}f(s)~{}$ mapping $x_{1},~{}\ldots~{},~{}x_{m}\in\mathbb{C}_{l}$ to $y_{1}=\frac{1}{x_{1}},~{}\ldots~{},~{}y_{m}=\frac{1}{x_{m}}$. Clearly the sought solution is

[TABLE]

We now follow the above recipe and substitute in Eq. (2.17)

[TABLE]

B. Parametrize all positive real rational functions, of degree of at most two, so that

[TABLE]

where $y_{1},y_{2}\in{\mathbb{R}}$ are arbitrary.

First for reference, a direct computation reveals that all rational functions, of degree of at most one, are given by

[TABLE]

These functions are positive real whenever,

[TABLE]

The conditions in Eq. (3.2) may be satisfied for all $~{}y_{1},~{}y_{2}\in{\mathbb{R}}$  umless, $~{}0>y_{1}=y_{2}$.

This implies that for $y_{1}=y_{2}\geq 0$ there is a zero degree positive real interpolating function, see item (iii) below. For $~{}0>y_{1}=y_{2}$, the positive real interpolating functions are of degree of at least two, see item (vi) below. In all other cases, there exist positive real interpolating functions of degree one and above.

We now follow the recipe from the previous section.

From Step 1

[TABLE]

and

[TABLE]

From Step 2, and using Eq. (2.5) yields

[TABLE]

and

[TABLE]

where $c_{1}$ and $c_{2}$ are such that $1+{\frac{c_{1}}{s+1}}+{\frac{c_{2}}{s+3}}$ is strictly positive real. For $d_{1}(s)$ the set of admissible parameters is convex and positively unbounded888From Theorem 2.11 it follows that in particular it contains the whole first quadrant of the $\{c_{2},c_{1}\}$ plane (excluding the axes $c_{1}=0$ and $c_{2}=0$), it is given by

[TABLE]

From Step 3

[TABLE]

Now from Step 4

[TABLE]

with $\gamma\in[0,~{}4]\smallsetminus\{1,3\}$, and with ${c_{1},~{}c_{2}}$ from Eq. (3.3),

[TABLE]

One can verify that taking $~{}r_{o}$, $~{}r_{1}~{}$ “sufficiently large” renders $f_{o}(s)$, $~{}f_{1}(s)~{}$ positive real.

Next, from Step 5, assuming $~{}{y_{1}y_{2}}\not=0$ and $\gamma\in[0,~{}4]\smallsetminus\{1,3\}$,

[TABLE]

and with ${c_{1},~{}c_{2}}$ from Eq. (3.3),

[TABLE]

Again, taking $~{}\hat{r}_{o}$, $~{}\hat{r}_{1}$ “sufficiently large” renders $~{}\hat{f}_{o}(s)$, $~{}\hat{f}_{1}(s)~{}$ positive real.

Here are five particular cases.

(i) Recall that in the Introduction we pointed out that if $y_{1},y_{2}\in{\mathbb{R}}_{-}$, see Eq. (1.4), one can still try to resort to the classical Nevanlinna-Pick interpolation, seeking positive real odd fuctions so that

[TABLE]

Now, the solvability condition in Eq. (1.3) reads,

[TABLE]

and the resulting positive real odd interpolating functions (of degree of at most two) are

[TABLE]

We now show, that these positive real odd functions, are special cases of the above recipe:

Indeed, assuming the condition in Eq. (3.8) is staisfied, from Eq. (3.5)

[TABLE]

and from Eq. (3.7)

[TABLE]

To further emphasize that our approach is different, in the four following special cases (ii), (iii) and (v), the condition in Eq. (3.8) is not satisfied, so the classical Nevanlinna-Pick interpolation is not applicable.

(ii) Take the special case where $y_{1}=1$ and $y_{2}=3$.

Clearly, $f(s)=-s$ is a real, anti-positive, minimal degree, interpolating function. We next seek minimal degree positive real interpolating functions.

Substituting these image points in $~{}f_{o}(s)~{}$ in Eq. (3.4) yields the following positive real interpolating functions,

[TABLE]

To guarantee minimal degree, further substitute $~{}{\underline{r}_{o}}=0$, to obtain interpolating functions with zero at infinity,

[TABLE]

Comparing with Eqs. (3.1) and (3.2) reveals that in this case our recipe yields all minimal degree (equals one) positive real interpolating functions.

Similarly for $~{}\hat{f}_{o}(s)~{}$ in Eq. (3.6)

[TABLE]

with $\gamma\in[0,~{}4]\smallsetminus\{1,3\}$. As before, to single out interpolating functions of degree one, we focus on cases where $~{}\underline{\hat{r}}_{o}=0$. However, then to guarantee positive realness, the range of the parameter $~{}{\gamma}~{}$ needs to be further restricted, i.e.

[TABLE]

Here, at infinity, the interpolating function has neither pole nor zero.

Finally note that comparison with Eqs. (3.1) and (3.2) reveals that in this case, the recipe produced  all  interpolating functions of degree one.

(iii) Take the special case where $y_{1}=y_{2}\geq 0$.

One can substitute in Eq. (3.4) $~{}\underline{r}_{o}=0~{}$ to obtain the minimal (=zero) degree interpolating function $f_{o}(s)\equiv y_{1}$.

Similarly, one can substitute in Eq. (3.6) $~{}\underline{\hat{r}}_{o}=0~{}$ to obtainn the minimal (=zero) degree interpolating function $\hat{f}_{o}(s)\equiv y_{1}$.

(iv) Take the special case where $0>y_{1}=y_{2}~{}$.

Recall that from Eqs. (3.1) and (3.2) we know that there are no positive real interpolating function of degree less than two.

To obtain interpolating functions use the recipe and substitute in Eq. (3.4) to obtain,

[TABLE]

Note that $\underline{r}_{o}$ turns to be unbounded, as $\gamma$ approaches 4.

(v) Take the special case where $y_{1}=2$, $y_{2}=0$.

As before, substituting these image points in Eqs. (3.1) and (3.2) (with $\frac{b}{a}={\scriptstyle\gamma}$) reveals that all  minimal degree (equals one) positive real interpolating functions are of the form999Substituting in Eq. (3.4) $r_{o}=0$, yields the subset of the interpolating functions in Eq. (3.9), where $4\geq{\scriptstyle\gamma}$

[TABLE]

Next, address the case where the interpolating function is so that the degree of the denominator is strictly larger then the the degree of the numerator. Now, recall that since the set of image points contains zero, Step 5 of the recipe cannot be used. Nevertheless, all required interpolating functions are obtained.

We start with a straightforward considerations: Since at $x=-1$, the numerator is non-zero, but it vanishes at $x=-3$, it must be (at least) of degree one. Thus, the denominator is (at least) of degree two. Indeed, to obtain all minimal degree interpolating functions of the required nature, substitute in Eq. (3.5)

[TABLE]

where adapting Eq.(3.3),

[TABLE]

C.  In the previous item the interpolation nodes were real. We here illustrate the fact that the recipe is identical for the non-real case, assuming the interpolation nodes are closed under complex conjugation.

Assume that the interpolation nodes are $x_{1}=-\gamma+i\delta$ and $x_{2}=-\gamma-i\delta$ where $\gamma>0$ and $0\not=\delta\in{\mathbb{R}}$. Hence,

[TABLE]

We now construct the denominator polynomials.

Following Theorem 2.11 a degree one numerator polynomial $~{}d_{o}(s)~{}$ is given by the condition that the following rational function is strictly positive real,

[TABLE]

namely,

[TABLE]

Hence one arrives at the following parametrization,

[TABLE]

∎

Concluding remarks

1. As already pointed out in Corollary 2.7, for arbitrary prescribed data set in $\mathbb{C}$, the family of all positive real interpolating functions is convex (whenever not empty).

In contrast, the set of rational functions of a degree of at most $m$ is a cone, but highly non-convex. In fact, the degree of a sum of two rational functions is higher than the degree of each of the summands, unless one of the denominators divides the other.

When the interpolation nodes are in $\mathbb{C}_{l}$, the open left half plane, we here introduce an easy-to-compute parametrization of positive real interpolating functions as a subset of ${\mathbb{R}}^{2m+3}$, see item 2 for details.

2. For arbitrary interpolating data set in Eq. (1.1), closed under complex conjugation, with nodes in $\mathbb{C}_{l}$, a large subset of positive real interpolating functions of degree of at most $m$ may be conveniently parametrized a union of convex subsets within ${\mathbb{R}}^{2m+3}$.

Indeed the coefficients in Eq. (2.12) are so that $c_{1},~{}\ldots~{},~{}c_{m}$ form a positively unbounded convex subset of ${\mathbb{R}}^{m}$, which in particular contains ${\mathbb{R}}_{+}^{m}$, excluding the axes (see e.g. Figure 1). Next, ${\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}$ form a hyper-plane in ${\mathbb{R}}^{m-1}$. Finally, each of the four parameters $\underline{r}_{o}$, $\underline{r}_{1}$, $\underline{\hat{r}}_{o}$, $\underline{\hat{r}}_{1}$, lies in $\overline{{\mathbb{R}}}_{+}$.

3. Step 4 of the recipe relies on the fact that positive real rational functions form a convex cone and that the set of interpolating functions is convex. Steps 3 and 5 rely on the fact that the set of positive real rational functions is closed under inversion.

4. The parametrization through $f_{o}(s)$, $f_{1}(s)$, $\hat{f}_{o}(s)$, $\hat{f}_{1}(s)$ is motivated by  simplicity. It is neither minimal, as the same interpolation function may be obtained in more than one way, see e.g. Example B(iii), nor is it comprehensive, as some of the minimal degree interpolating functions may be missing, see e.g. Example B(v).

5. While the parametrization through $f_{o}(s)$, $f_{1}(s)$, $\hat{f}_{o}(s)$, $\hat{f}_{1}(s)$ is convenient, focusing on  minimal degree interpolating functions involves “fine tuning” of the parameters $~{}{\gamma}_{1},~{}\ldots~{},~{}{\gamma}_{m}$, $~{}c_{1},~{}\ldots~{},~{}c_{m}$, $~{}r_{o}$, $~{}r_{1}$, $~{}\hat{r}_{o}~{}$ and $~{}\hat{r}_{1}$, see Examples A, B.

Acknowledgement

The authors thank Prof. V. Bolotnikov form the Math. Dept. at the College of William and Mary, Williamsburg, Virginia, USA for providing useful, constructive remarks at early stage of this work.