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# Strong asymptotics of multi-level Hermite-Padé polynomials

L. G. González Ricardo  Department of Mathematics, Universidad Carlos iii de
Madrid, Avenida de la Universidad, 30 CP-28911, Leganés, Madrid, Spain.
<EMAIL_ADDRESS>and  G. López Lagomasino  Department of Mathematics,
Universidad Carlos iii de Madrid, Avenida de la Universidad, 30 CP-28911,
Leganés, Madrid, Spain<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We obtain the strong asymptotics of multiple orthogonal polynomials which
arise in a mixed type Hermite-Padé approximation problem defined on a Nikishin
system of functions. The results obtained allow to give exact estimates of the
rate of convergence of the approximating rational functions and the strong
asymptotics of Cauchy biorthogonal polynomials.

The first author was supported in part by a research fellowship from
Department of Mathematics of Universidad Carlos III de Madrid, Spain.

The second author was supported in part by the research grant
PGC2018-096504-B-C33 of Ministerio de Ciencia, Innovación y Universidades,
Spain.

Keywords: multiple orthogonal polynomials, biorthogonal polynomials, Hermite-
Padé approximation, strong asymptotics, fixed point theorems

AMS classification: Primary: 42C05; 30E10, Secondary: 41A21, 47H10

##  1\. Introduction

###  1.1. Background

Szegő’s theory of orthogonal polynomials [27, Chap. X-XII] is a gem of real
and complex analysis not only for its intrinsic beauty but also because of its
many consequences, extensions, and applications in different fields (see [24,
25]). The natural framework in which the theory is developed is for
polynomials orthogonal with respect to measures supported on the unit circle.
Then, it may be transposed to polynomials orthogonal with respect to measures
supported on bounded intervals of the real line. To illustrate, we present
what is perhaps the most important result of Szegő’s theory in the context of
orthogonal polynomials on segments of the real line. First, we need some
notation.

Let $\mu$ be a finite positive Borel measure supported on an interval
$\Delta=[a,b]\subset\mathbb{R}$. We say that $\mu$ satisfies Szegő’s condition
and write $\mu\in\mathcal{S}(\Delta)$ if

(1.1)
$\int_{\Delta}{\ln\mu^{\prime}(x)}\operatorname{d}\eta_{\Delta}(x)>-\infty,$

where

$\operatorname{d}\eta_{\Delta}(x)=\frac{\operatorname{d}x}{\sqrt{(b-x)(x-a)}},\qquad
x\in\Delta,$

and $\mu^{\prime}$ denotes the Radon-Nikodym derivative of $\mu$ with respect
to the Lebesgue measure. When $\mu\in\mathcal{S}(\Delta)$

$\mathsf{G}(\mu,z):=\exp\left[\frac{\sqrt{(z-b)(z-a)}}{2\pi}\int_{\Delta}\frac{\ln(\sqrt{(b-x)(x-a)}\,\mu^{\prime}(x))}{x-z}\operatorname{d}\eta_{\Delta}(x)\right],$

is called the Szegő function of $\mu$. The square root outside the integral is
taken to be positive for $z>b$ and those inside the integral are positive when
$x\in(a,b)$. The Szegő function verifies

(1.2) $\begin{cases}\mathsf{G}(\mu,z)\neq 0\textrm{ for
}z\in\overline{\mathbb{C}}\setminus\Delta,\\\ \mathsf{G}(\mu,\infty)>0,\\\
\lim_{y\to
0}|\mathsf{G}(\mu,x+iy)|^{2}=(\sqrt{(b-x)(x-a)}\,\mu^{\prime}(x))^{-1},\quad\textrm{for
a.e.}\,\,x\in\Delta.\end{cases}$

Let $\Psi$ be the conformal map from $\overline{\mathbb{C}}\setminus\Delta$
onto the complement (in $\overline{\mathbb{C}}$) of the closed unit disk such
that $\Psi(\infty)=\infty$, $\Psi^{\prime}(\infty)>0$. Let
$l_{n}(z)=\kappa_{n}z^{n}+\cdots$ be the $n$-th orthonormal polynomial with
respect to $\mu$ with positive leading coefficient; that is, $\kappa_{n}>0$
and

$\int l_{n}(x)l_{k}(x)\operatorname{d}\mu(x)=\delta_{n,k},\qquad n,k\geq 0$

where $\delta_{n,k}$ denotes the Kronecker $\delta$. We have

######  Theorem 1.1 (G. Szegő).

Assume that $\mu\in\mathcal{S}(\Delta)$, then

$\lim_{n\to\infty}\frac{l_{n}(z)}{\Psi^{n}(z)}=\frac{1}{\sqrt{2\pi}}\mathsf{G}(\mu,z),$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta$, and

$\lim_{n\to\infty}\frac{\kappa_{n}}{C^{n}}=\frac{\mathsf{G}(\mu,\infty)}{\sqrt{2\pi}},$

where $C^{-1}=(b-a)/4$ is the logarithmic capacity of $\Delta$.

The second formula follows directly from the first one applied at $z=\infty$.
The first formula expresses that asymptotically, the $n$-th orthonormal
polynomial behaves like the $n$-th power of the fixed function $\Psi$.

Our goal is to obtain a similar result for the multiple orthogonal polynomials
(which share orthogonality relations with a system of measures) arising in a
certain mixed type approximation problem. Previously, this type of extension
of Szegő’s theory was considered by A.I. Aptekarev [1, 2] for type ii Hermite-
Padé polynomials of so called Angelesco and Nikishin systems of measures. Our
work is related with Nikishin systems of measures but the orthogonality
relations differ from the type ii case. However, as we explain later, our main
result Theorem 1.4 generalizes in several directions that of A.I. Aptekarev in
[2, Theorem 2].

In the study of multi-peakon solutions of the Degasperis-Procesi equation, a
mixed type Hermite-Padé approximation problem was introduced in [17]. The same
construction showed to be useful in the analysis of the two matrix model [6]
and Cauchy biorthogonal polynomials [5]. The approximated objects are systems
of two functions, defined in terms of Cauchy transforms of two measures
supported on disjoint intervals of the real line. This application motivated
the introduction in [15] of an approximation problem for an arbitrary Nikishin
system generated by $m$ measures (which reduces to the previous case when
$m=2$) and study the convergence of the method.

Recently, the approximation problem was generalized further by V.G. Lysov in
[18] where he studied the asymptotic zero distribution and the logarithmic
asymptotics of the associated multiple orthogonal polynomials. The ratio
asymptotics of these multiple orthogonal polynomials was given in [12]. Here,
we wish to find their strong asymptotic behavior. Before stating our main
result, let us briefly present the so called Nikishin systems of measures and
Lysov’s definition of multi-level Hermite-Padé approximants.

###  1.2. Nikishin systems and their multi-level Hermite-Padé polynomials

Nikishin systems of measures were first introduced by E.M. Nikishin in [20].
Throughout the years, it has been shown that polynomials which share
orthogonality relations with respect to such systems of measures behave
asymptotically similar to standard orthogonal polynomials (with respect to a
single measure). A brief overview of the subject may be found in [16].

Let $\mathcal{M}(\Delta)$ denote the class of all finite positive Borel
measures $s$ with compact support $\operatorname{supp}s\subset\Delta=[a,b]$
containing an infinite set of points. If $s$ satisfies Szegő’s condition (1.1)
we write $s\in\mathcal{S}(\Delta)$. By

$\widehat{s}(z):=\int\frac{\operatorname{d}s(x)}{z-x}$

we denote the Cauchy transform of the measure $s$. Obviously,
$\widehat{s}\in\mathcal{H}(\overline{\mathbb{C}}\setminus\Delta)$ (holomorphic
on $\overline{\mathbb{C}}\setminus\Delta$).

Let $\Delta_{\alpha}$, $\Delta_{\beta}$ be two intervals contained in the real
line such that $\Delta_{\alpha}\cap\Delta_{\beta}=\varnothing$. Consider the
measures $\sigma_{\alpha}\in\mathcal{M}(\Delta_{\alpha})$ and
$\sigma_{\beta}\in\mathcal{M}(\Delta_{\beta})$. With these two measures we
construct a third one as follows (using differential notation)

$\operatorname{d}\langle\sigma_{\alpha},\sigma_{\beta}\rangle(x):=\widehat{\sigma}_{\beta}(x)\operatorname{d}\sigma_{\alpha}(x).$

Notice that
$\langle\sigma_{\alpha},\sigma_{\beta}\rangle\neq\langle\sigma_{\beta},\sigma_{\alpha}\rangle$.
We can extend this operation inductively to more than two measures. For
example, if $\Delta_{\gamma}\cap\Delta_{\alpha}=\varnothing$ and
$\sigma_{\gamma}\in\mathcal{M}(\Delta_{\gamma})$, we define

$\langle\sigma_{\gamma},\sigma_{\alpha},\sigma_{\beta}\rangle:=\langle\sigma_{\gamma},\langle\sigma_{\alpha},\sigma_{\beta}\rangle\rangle.$

######  Definition 1.2.

Take a collection $\Delta_{j}$, $j=1,\ldots,m,m\geq 2,$ of intervals such that

(1.3) $\Delta_{j}\cap\Delta_{j+1}=\varnothing,\qquad j=1,\ldots,m-1.$

Let $(\sigma_{1},\ldots,\sigma_{m})$ be a system of measures such that
$\sigma_{j}\in\mathcal{M}(\Delta_{j})$, $j=1,\ldots,m$. We say that
$(s_{1,1},\ldots,s_{1,m})=\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$, where

$s_{1,1}=\sigma_{1},\;s_{1,2}=\langle\sigma_{1},\sigma_{2}\rangle,\;s_{1,3}=\langle\sigma_{1},\langle\sigma_{2},\sigma_{3}\rangle\rangle,\,\ldots,\;s_{1,m}=\langle\sigma_{1},\langle\sigma_{2},\ldots,\sigma_{m}\rangle\rangle$

is the Nikishin system of measures generated by
$(\sigma_{1},\ldots,\sigma_{m})$. The vector
$(\widehat{s}_{1,1},\ldots,\widehat{s}_{1,m})$ is called the Nikishin system
of functions.

Notice that any sub-system of $(\sigma_{1},\ldots,\sigma_{m})$ of consecutive
measures is also a generator of some Nikishin system. In the sequel, for
$1\leq j\leq k\leq m$, we will write

$s_{j,k}:=\langle\sigma_{j},\sigma_{j+1},\ldots,\sigma_{k}\rangle,\qquad
s_{k,j}:=\langle\sigma_{k},\sigma_{k-1},\ldots,\sigma_{j}\rangle.$

Without further notice, throughout the paper we consider that $m\geq 2$.

Let $(\mathbb{Z}_{+}^{m})^{*}$ be the set of all $m$-dimensional vectors with
non-negative integer components, not all equal to zero. For
$\vec{n}=(n_{1},\ldots,n_{m})\in(\mathbb{Z}_{+}^{m})^{*}$ we define
$|\vec{n}|=n_{1}+\cdots+n_{m}$ and $\eta_{\vec{n},j}=n_{1}+\cdots+n_{j}$.

######  Definition 1.3.

Consider the Nikishin system $\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$ and
$\vec{n}=(n_{1},\ldots,n_{m})\in(\mathbb{Z}_{+}^{m})^{*}$. There exist
polynomials $a_{\vec{n},0},a_{\vec{n},1},\ldots,a_{\vec{n},m}$, where $\deg
a_{\vec{n},j}\leq|\vec{n}|-1$, $j=0,1,\ldots,m-1,$ and $\deg
a_{\vec{n},m}\leq|\vec{n}|$, not all identically equal to zero, called multi-
level (ML) Hermite-Padé polynomials of
$\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$ with respect to $\vec{n}$, that
verify

(1.4)
$\mathcal{A}_{\vec{n},j}(z):=\left((-1)^{j}a_{\vec{n},j}+\sum_{k=j+1}^{m}(-1)^{k}a_{\vec{n},k}\widehat{s}_{j+1,k}\right)(z)=\mathcal{O}\left(\frac{1}{z^{n_{j+1}+1}}\right),\qquad
z\to\infty,$

where $j=0,\ldots,m-1$ (the asymptotic expansion of $\mathcal{A}_{\vec{n},j}$
at $\infty$ begins with $z^{-n_{j+1}-1}$, or higher). For completeness, set
$\mathcal{A}_{\vec{n},m}:=(-1)^{m}a_{\vec{n},m}$.

We warn the reader that with our terminology in [18, Problem A] the ML
Hermite-Padé polynomials were defined with respect to the system
$\mathcal{N}(\sigma_{m},\ldots,\sigma_{1})$. Initially, in [15] ML Hermite-
Padé approximants were defined only for multi-indices of the form
$(n,0,\ldots,0)$.

Following Mahler’s denomination [19], a multi-index
$\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$ is said to be normal if $\deg
a_{{\vec{n}},j}=|\vec{n}|-1$, $j=0,\ldots,m-1,$ and $\deg
a_{{\vec{n}},m}=|\vec{n}|$. The system of functions is said to be perfect when
all the multi-indices are normal. In [18, Theorem 1.1], it was proved that
Nikishin systems of functions are perfect for this approximation problem.
Normality implies that the vector $(a_{\vec{n},0},\ldots,a_{\vec{n},m})$ is
uniquely determined up to a multiplicative factor. In the sequel, we normalize
this vector so that $a_{\vec{n},m}$ has leading coefficient equal to one.

The zeros of the forms $\mathcal{A}_{\vec{n},j}$ have some interesting
properties (see [12, Lemma 2.2] or Subsection 2.1 below). The form
$\mathcal{A}_{\vec{n},0}$ has no zero in $\mathbb{C}\setminus\Delta_{1}$. For
$j=1,\ldots,m,$ $\mathcal{A}_{\vec{n},j}$ has exactly $\eta_{\vec{n},j}$ zeros
in $\mathbb{C}\setminus\Delta_{j+1}$, $(\Delta_{m+1}=\varnothing)$, they are
all simple and lie in $\mathring{\Delta}_{j}$ (the interior of $\Delta_{j}$
with the Euclidean topology of $\mathbb{R}$). If $Q_{\vec{n},j}$,
$j=1,\ldots,m,$ denotes the monic polynomial whose roots are the simple zeros
which $\mathcal{A}_{\vec{n},j}$ has in $\mathring{\Delta}_{j}$, then

$\frac{\mathcal{A}_{\vec{n},j}(z)}{Q_{\vec{n},j}(z)}=\mathcal{O}\left(\frac{1}{z^{\eta_{\vec{n},j+1}+1}}\right)\in\mathcal{H}(\mathbb{C}\setminus\Delta_{j+1}),\qquad
z\to\infty,\qquad j=1,\ldots,m-1.$

For each $j=0,\ldots,m-1$ the order of interpolation at infinity prescribed in
(1.4) is exact.

A sequence of distinct multi-indices $\Lambda\subset(\mathbb{Z}_{+}^{m})^{*}$
is said to be a ray sequence if

$\lim_{\vec{n}\in\Lambda}n_{j}/|\vec{n}|=p_{j},\qquad j=1,\ldots,m.$

(Consequetly, $\sum_{j=1}^{m}p_{j}=1$.) The main results in [18, Theorem 1.2,
Corollary 1.1] give the asymptotic zero distribution and the logarithmic
asymptotics of ray sequences of the polynomials $Q_{\vec{n},j}$,
$j=1,\ldots.m,$ when $\sigma_{j}^{\prime}\neq 0$, $j=1,\ldots,m$ (this
assumption can be relaxed to requiring that the measures $\sigma_{j}$ be
regular, see [26, Chap. 3] for the definition of regular measure). With the
assumptions in Lysov’s paper it is possible to prove ratio asymptotics (see
[12, Theorem 3.4, Corollary 3.5]).

$\Lambda\subset(\mathbb{Z}_{+}^{m})^{*}$ is said to be a straight ray sequence
when

(1.5) $\frac{n_{j}}{|\vec{n}|}=p_{j},\qquad
j=1,\ldots,m,\qquad\vec{n}\in\Lambda.$

Our goal is to give the strong asymptotics of sequences of polynomials
$Q_{\vec{n},j}$, $a_{\vec{n},j-1}$, $j=1,\ldots,m,$ and forms
$\mathcal{A}_{\vec{n},j}$, $j=0,\ldots,m$, for $\vec{n}$ belonging to a
straight ray sequence of multi-indices assuming that
$\sigma_{j}\in\mathcal{S}(\Delta_{j})$, $j=1,\ldots,m$.

###  1.3. Statement of the main results

Let $(p_{1},\ldots,p_{m})\in\mathbb{R}_{\geq 0}^{m}$, $p_{1}>0.$ Set

(1.6) $P_{j}=\sum_{k=1}^{j}p_{k},\qquad j=1,\ldots,m.$

Notice that $P_{j}>0$, $j=1,\ldots,m$. Let
$\vec{\Delta}=(\Delta_{1},\ldots,\Delta_{m})$ be a collection of bounded
intervals $\Delta_{j}=[a_{j},b_{j}]$ of the real line which satisfies (1.3).
By $\mathcal{M}_{1}(\vec{\Delta})$ we denote the cone of all vector
probability measures $\vec{\mu}=(\mu_{1},\ldots,\mu_{m})$ such that
$\operatorname{supp}(\mu_{j})\subset\Delta_{j}$, $j=1,\ldots,m$.

Given a finite Borel measure $\mu$ with compact support its logarithmic
potential is defined as

$V^{\mu}(z):=\int\ln\frac{1}{|z-x|}\operatorname{d}\mu(x).$

In the present paper, we need the solution
$\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{m})\in\mathcal{M}_{1}(\vec{\Delta})$
of the vector equilibrium problem

(1.7)
$V^{\lambda_{j}}(x)-\frac{1}{2}\left[\frac{P_{j-1}}{P_{j}}\,V^{\lambda_{j-1}}(x)+\frac{P_{j+1}}{P_{j}}\,V^{\lambda_{j+1}}(x)\right]\begin{cases}=\omega_{j},&x\in\operatorname{supp}\lambda_{j},\\\
\geq\omega_{j},&x\in\Delta_{j},\end{cases}$

for $j=1,\ldots,m$. Here, $V^{\lambda_{m+1}}\equiv V^{\lambda_{0}}\equiv 0$,
$P_{0}=P_{m+1}=0$.

The problem (1.7) is equivalent to

(1.8)
$P_{j}^{2}V^{\lambda_{j}}(x)-\frac{1}{2}\left[P_{j}P_{j-1}\,V^{\lambda_{j-1}}(x)+P_{j}P_{j+1}\,V^{\lambda_{j+1}}(x)\right]\begin{cases}=P_{j}^{2}\omega_{j},&x\in\operatorname{supp}\lambda_{j},\\\
\geq P_{j}^{2}\omega_{j},&x\in\Delta_{j},\end{cases}$

for $j=1,\ldots,m$. It is well known that there exists a unique
$\vec{\lambda}\in\mathcal{M}_{1}(\vec{\Delta})$ such that (1.8) holds for each
$j=1,\ldots,m$. $\vec{\lambda}$ is called the unitary vector equilibrium
measure and $\vec{\omega}:=(\omega_{1},\ldots,\omega_{m})$ the vector
equilibrium constant corresponding to the vector equilibrium problem with
interaction matrix

(1.9)
$\mathcal{C}:=\left(\begin{array}[]{ccccc}P_{1}^{2}&-\frac{P_{1}P_{2}}{2}&0&\cdots&0\\\
-\frac{P_{2}P_{1}}{2}&P_{2}^{2}&-\frac{P_{2}P_{3}}{2}&\ddots&0\\\
0&-\frac{P_{3}P_{2}}{2}&P_{3}^{2}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&0&\cdots&P_{m}^{2}\end{array}\right)$

on the system of intervals $\vec{\Delta}$.

The proof of the existence of unique $\vec{\lambda}$ and $\vec{\omega}$ in the
present situation follows from results contained in [21, Chap. 5, §4] (see
also [3, Section 4]) due to the fact that $\mathcal{C}$ is a symmetric
positive definite matrix whose entries $c_{j,k}$ are non-negative whenever
$\Delta_{j}\cap\Delta_{k}\neq\varnothing$. We will need that
$\operatorname{supp}(\lambda_{j})=\Delta_{j}$, $j=1,\ldots,m$, in order to
have equality in (1.7) throughout all $\Delta_{j}$. It will be shown in Lemma
2.4 that this is true whenever $p_{1}>0$ and $p_{1}\geq\cdots\geq p_{m}$.

In the sequel,

(1.10) $\Phi_{j}:=e^{-v_{j}},\qquad
v_{j}:=V^{\lambda_{j}}+i\widetilde{V}^{\lambda_{j}},\qquad
C_{j}=e^{\omega_{j}},$

where $\widetilde{V}^{\lambda_{j}}$ denotes the harmonic conjugate of
$V^{\lambda_{j}}$ on $\mathbb{C}\setminus\Delta_{j}$, $j=1,\ldots,m,$ which
equals zero on $(b_{j},+\infty)$. The function $\Phi_{j}$ is a single-valued
analytic function in $\mathbb{C}\setminus\Delta_{j}$ (notice that the values
of $-v_{j}$ vary by a multiple of $2\pi i$ as we circulate around $\Delta_{j}$
along a closed Jordan curve) with a simple pole at $\infty$. These functions
will play the same role as $\Psi$ in Szegő’s Theorem 1.1. The functions
$\Phi_{j}^{\eta_{\vec{n},j}}$ will be used to compare the polynomials
$Q_{\vec{n},j}$. We still need to introduce the limiting functions.

Assume that
$\vec{\sigma}=(\sigma_{1},\ldots,\sigma_{m})\in\mathcal{S}(\vec{\Delta})$;
that is, $\sigma_{j}\in\mathcal{S}(\Delta_{j})$, $j=1,\ldots,m$. There exists
a unique system of Szegő functions
$\vec{\mathsf{G}}=(\mathsf{G}_{1},\ldots,\mathsf{G}_{m})$ (see Theorem 1.6
below), where
$\mathsf{G}_{k}\in\mathcal{H}(\overline{\mathbb{C}}\setminus\Delta_{k})$,
which verifies the system of boundary value equations

(1.11)
$|\mathsf{G}_{j}(x)|^{2}=\frac{\sqrt{|x-b_{j+1}||x-a_{j+1}|}\,\mathsf{G}_{j-1}(x)\mathsf{G}_{j+1}(x)}{\sqrt{(b_{j}-x)(x-a_{j})}\sigma_{j}^{\prime}(x)},\quad\mbox{a.e.
on}\,\,\Delta_{j},\quad j=1,\ldots,m.$

Here, $\mathsf{G}_{0}\equiv\mathsf{G}_{m+1}\equiv 1$ and when $j=m$,
$\sqrt{|x-b_{m+1}||x-a_{m+1}|}$ is substituted by $1$. Notice that
$\mathsf{G}_{j}(z)$ is the Szegő function with respect to a measure on
$\Delta_{j}$ whose Radon-Nikodym devivative is (compare (1.11) and (1.2))

$\frac{\sigma_{j}^{\prime}(x)}{\sqrt{|x-b_{j+1}||x-a_{j+1}|}\,\mathsf{G}_{j-1}(x)\mathsf{G}_{j+1}(x)}.$

######  Theorem 1.4.

Let $\Lambda=\Lambda(p_{1},\ldots,p_{m})$ be a straight ray sequence of multi-
indices such that (1.5) takes place, and $p_{1}\geq\cdots\geq p_{m}$. Assume
that $\vec{\sigma}\in\mathcal{S}(\vec{\Delta})$. Then

(1.12)
$\lim_{\vec{n}\in\Lambda}\frac{Q_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}_{j}(z)}{\mathsf{G}_{j}(\infty)},\qquad
j=1,\ldots,m,$

uniformly on each compact subset of
$\overline{\mathbb{C}}\setminus\Delta_{j}$, where
$\vec{\mathsf{G}}=(\mathsf{G}_{1},\ldots,\mathsf{G}_{m})$ is the vector of
Szegő functions whose components are determined by the boundary conditions
(1.11) and the $\Phi_{j}$ are as defined in (1.10) constructed from the vector
equilibrium problem associated with the given vector $(p_{1},\ldots,p_{m})$.
The normalizing constants $\kappa_{\vec{n},j}$, $j=1,\ldots,m,$ defined in
(2.7) verify

(1.13)
$\lim_{n\in\Lambda}\frac{\kappa_{\vec{n},j}}{C_{j}^{\eta_{\vec{n},j}}}=\frac{\mathsf{G}_{j}(\infty)}{\sqrt{2\pi}},$

where the $C_{j}$ were given in (1.10). Additionally, the ML Hermite-Padé
polynomials satisfy

(1.14)
$\lim_{\vec{n}\in\Lambda}\frac{a_{\vec{n},j}(z)}{\Phi_{m}^{|\vec{n}|}(z)}=\frac{\mathsf{G}_{m}(z)}{\mathsf{G}_{m}(\infty)}\widehat{s}_{m,j+1}(z),\qquad
j=0,\ldots,m-1,$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{m}$.

The asymptotic formulas satisfied by the forms $\mathcal{A}_{\vec{n},j}$,
$j=0,\ldots,m-1,$ are more entangled and we give them in Corollary 3.1.

The proof of Theorem 1.4 makes use of fixed point theorems. This method was
employed by A.I. Aptekarev in [1] for the study of the strong asymptotics of
type ii Hermite-Padé polynomials of Angelesco systems of two measures and used
again in [2] to obtain the strong asymptotics of type ii Hermite-Padé
polynomials of Nikishin systems with an arbitrary number of measures. Here, we
adapt the scheme proposed in [2] to the case of ML Hermite-Padé approximation,
simplify some proofs, and correct some statements.

Type II Hermite-Padé polynomials and ML Hermite-Padé polynomials in general do
not coincide, but for the multi-indices considered in Theorem 1.4 for which
$n_{1}\geq n_{2}\geq\cdots\geq n_{m}$ their construction is equivalent as
follows from V.G. Lysov’s result [18, Proposition 1.1]. In [2] the measures
are assumed to be absolutely continuous with respect to Lebesgue measure and
the multi-indices are of the form $\vec{n}=(n,n,\ldots,n),n\geq 1$. Therefore,
our result covers that of A.I. Aptekarev with weaker assumptions. Theorem 1.4
extends Szegő’s classical result on the strong asymptotic of orthogonal
polynomials to the case when $m>1$.

Let $\mathcal{C}({\vec{\Delta}})$ be the linear space of all vector functions
$\vec{f}=(f_{1},\ldots,f_{m})$ such that $f_{j}$ is continuous on
$\Delta_{j-1}\cup\Delta_{j+1}$, $(\Delta_{0}=\Delta_{m+1}=\varnothing)$. Set

$\|\vec{f}\|_{\vec{\Delta}}:=\max_{1\leq j\leq
m}\left\\{\|f_{j}\|_{\Delta_{j-1}\cup\Delta_{j+1}}\right\\}$

where $\|\cdot\|_{\Delta_{j-1}\cup\Delta_{j+1}}$ denotes the sup norm on the
indicated set. It is well known that
$(\mathcal{C}({\vec{\Delta}}),\|\cdot\|_{\vec{\Delta}})$ is a Banach space.

Let $\mathcal{C}^{+}({\vec{\Delta}})$ denote the set of all vector functions
$\vec{f}=(f_{1},\ldots,f_{m})$ such that $f_{j}$ is positive and continuous on
$\Delta_{j-1}\cup\Delta_{j+1}$. On $\mathcal{C}^{+}({\vec{\Delta}})$ we define
the distance

$d(\vec{f},\vec{g})=\max_{1\leq j\leq
m}\left\\{\|\ln(f_{j}/g_{j})\|_{\Delta_{j-1}\cup\Delta_{j+1}}\right\\}.$

Since the logarithm establishes a homeomorphism between
$\mathcal{C}^{+}({\vec{\Delta}})$ and $\mathcal{C}({\vec{\Delta}})$, it is
easy to verify that $(\mathcal{C}^{+}({\vec{\Delta}}),d)$ is a complete metric
space. Furthermore, we have an important relation between the distance
$d(\cdot,\cdot)$ and the norm $\|\cdot\|_{\vec{\Delta}}$ both acting on
$\mathcal{C}^{+}({\vec{\Delta}})$; namely, for any sequence of vector
functions $(\vec{g}_{n})_{n\geq 0}\subset\mathcal{C}^{+}({\vec{\Delta}})$ and
$\vec{g}\in\mathcal{C}^{+}({\vec{\Delta}})$, we have

(1.15)
$\lim_{n\to\infty}\|\vec{g}_{n}-\vec{g}\|_{\vec{\Delta}}=0\quad\Leftrightarrow\quad\lim_{n\to\infty}d(\vec{g}_{n},\vec{g})=0.$

######  Definition 1.5.

Let $\vec{w}=(w_{1},\ldots,w_{m})$ be a vector of non-negative functions such
that $\ln w_{j}\in L_{1}(\eta_{\Delta_{j}})$, $j=1,\ldots,m$. Define

$\vec{T}_{\vec{w}}=(T_{1},\ldots,T_{m}):\mathcal{C}^{+}({\vec{\Delta}})\longrightarrow\mathcal{C}^{+}({\vec{\Delta}}),$

where for each $\vec{f}=(f_{1},\ldots,f_{m})$, $T_{j}\vec{f}$ is the Szegő
function on $\overline{\mathbb{C}}\setminus\Delta_{j}$ whose boundary values
on $\Delta_{j}$ verify

(1.16)
$|T_{j}\vec{f}(x)|^{2}=\frac{f_{j-1}(x)f_{j+1}(x)}{w_{j}(x)},\qquad\mbox{a.e.
on}\,\,\Delta_{j},\qquad j=1,\ldots,m$

($f_{0}\equiv f_{m+1}\equiv 1$).

Recall that Szegő functions are positive on the real line outside of the
support of the defining measure. Let ${\overline{m}}:=\lceil m/2\rceil$ be the
smallest integer greater or equal to $m/2$. As usual, $\vec{T}_{\vec{w}}^{j}$
denotes the composition of $\vec{T}_{\vec{w}}$ with itself $j$ times.

######  Theorem 1.6.

The map $\vec{T}_{\vec{w}}^{\overline{m}}$ is a contraction on the complete
metric space $({C}^{+}({\vec{\Delta}}),d)$; more precisely,

(1.17)
$d(\vec{T}_{\vec{w}}^{\overline{m}}\vec{f},\vec{T}_{\vec{w}}^{\overline{m}}\vec{g})\leq\gamma_{m}\,d(\vec{f},\vec{g})$

where

$\gamma_{m}:=\left\\{\begin{array}[]{cc}(2^{\overline{m}}-1)/2^{\overline{m}},&m\,\,\mathrm{even},\\\
(2^{\overline{m}}-2)/2^{\overline{m}},&m\,\,\mathrm{odd}.\end{array}\right.$

The map $\vec{T}_{\vec{w}}$ has a unique fixed point.

In [2, Proposition 1.1] the author proves that
$d(\vec{T}_{\vec{w}}\vec{f},\vec{T}_{\vec{w}}\vec{g})<d(\vec{f},\vec{g})$
showing that $\vec{T}_{\vec{w}}$ is non expansive and draws the erroneous
conclusion that $\vec{T}_{\vec{w}}$ is contractive. Theorem 1.6 corrects this
last statement.

###  1.4. Outline of the paper

In Subsections 2.1 and 2.2, we present some properties of the forms
$\mathcal{A}_{\vec{n},j}$ and of the polynomials $Q_{\vec{n},j}$ which carry
their zeros. In particular, it is shown that for each
$\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$ the polynomials $Q_{\vec{n},j}$ are
orthogonal with respect to certain varying measures (see (2.6)). Notice that
in the orthogonality relations the polynomials $Q_{\vec{n},j}$ appear at the
same time as orthogonal and in the denominator of the varying part of the
measure (with the subindices shifted). This inconvenience is settled
introducing an operator $\tilde{T}_{\vec{n}}$ on a set of vector polynomials,
denoted $\mathcal{P}_{\vec{n}}$, for which
$(Q_{\vec{n},1},\ldots,Q_{\vec{n},m})$ is a unique fixed point. In Subsection
2.3 we normalize the orthogonal polynomials and the varying measures allowing
us to use some known results of the theory of orthogonal polynomials with
respect to varying measures developed in [7, 8, 11] to obtain in Subsection
2.5 certain asymptotic formulas for multiple orthogonal polynomials which are
capital in the proof of Theorem 1.4 for the following reasons. Lemma 2.5
allows to reduce the proof of the strong asymptotics on compact subsets of
$\mathbb{C}\setminus\Delta_{j}$ to the strong asymptotics on
$\Delta_{j-1}\cup\Delta_{j+1}$ (since (2.24) implies (2.25)) and hints the
introduction of the operator $\vec{T}_{\vec{w}}$ (with $\vec{w}$ as in (3.4))
whose fixed point determines the limiting functions of the strong limits of
Theorem 1.4. In order to show that the fixed point of $\vec{T}_{\vec{w}}$
attracts fixed points of the operators $\tilde{T}_{\vec{n}}$ a local fixed
point theorem due to Brouwer is used and Lemma 2.6 is needed to guarantee that
the neighborhoods on which Brouwer’s Theorem is applied are non empty. In
Subsection 2.4 the solution of an equilibrium problem, which allows to
construct the comparison functions, is studied. Theorem 1.6 is proved in
Subsection 3.1 with assumptions more general than needed in this paper having
in mind future applications. Theorem 1.4 is obtained in Subsection 3.2.
Subsections 3.3 and 3.4 contain applications of the main result to give
precise estimates of the rate of convergence of ML Hermite-Padé approximations
and the strong asymptotics of Cauchy biorthogonal polynomials, respectively.

##  2\. Auxiliary results

###  2.1. Some useful relations

The polynomials $Q_{\vec{n},j}$ play an important role throughout the paper so
for the benefit of the reader we summarize their construction and some of its
properties (for details, see [12, Lemmas 2.1, 2.2, and 3.1]).

Let $(s_{1,1},\ldots,s_{1,m})=\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$ be
given. Assume that there exist polynomials with real coefficients
$a_{0},\ldots,a_{m}$ and a monic polynomial $Q$ with real coefficients whose
roots lie in $\mathbb{C}\setminus\Delta_{1}$ such that

$\frac{\mathcal{A}_{0}(z)}{Q(z)}\in\mathcal{H}(\mathbb{C}\setminus\Delta_{1})\quad\textrm{
and
}\quad\frac{\mathcal{A}_{0}(z)}{Q(z)}=\mathcal{O}\left(\frac{1}{z^{N}}\right),\quad
z\rightarrow\infty,$

where
$\displaystyle\mathcal{A}_{0}:=a_{0}+\sum_{k=1}^{m}a_{k}\widehat{s}_{1,k}$ and
$N\geq 1$. Let
$\displaystyle\mathcal{A}_{1}:=a_{1}+\sum_{k=2}^{m}a_{k}\widehat{s}_{2,k}$.
Using Cauchy’s integral formula on a contour surrounding $\Delta_{1}$ and
Fubini’s Theorem it readily follows (see [12, Lemma 2.1]) that

(2.1)
$\frac{\mathcal{A}_{0}(z)}{Q(z)}=\int\frac{\mathcal{A}_{1}(x)}{z-x}\frac{\operatorname{d}\sigma_{1}(x)}{Q(x)},$

and if $N\geq 2$, from Cauchy’s theorem, Fubini’s theorem, and Cauchy’s
integral formula, we have

(2.2) $\int
x^{\nu}\mathcal{A}_{1}(x)\frac{\operatorname{d}\sigma_{1}(x)}{Q(x)}=0,\qquad\nu=0,1,\ldots,N-2.$

In particular, $\mathcal{A}_{1}$ has at least $N-1$ sign changes in
$\mathring{\Delta}_{1}$ or it is identically equal to zero.

Fix $\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$. The forms $\mathcal{A}_{\vec{n},j}$
in Definition 1.3 are symmetric with respect to the real line (that is,
$\mathcal{A}_{\vec{n},j}(\overline{z})=\overline{\mathcal{A}_{\vec{n},j}({z})}$);
therefore, their roots come in conjugate pairs. Fix
$\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$. If $\mathcal{A}_{\vec{n},0}$ has a zero
in $\mathbb{C}\setminus\Delta_{1}$ take $Q_{\vec{n},0}$ as a monic polynomial
of degree $\geq 1$ with real coefficients whose roots are zeros of
$\mathcal{A}_{\vec{n},0}$ in $\mathbb{C}\setminus\Delta_{1}$; otherwise, take
$Q_{\vec{n},0}\equiv 1$. Using (2.1)-(2.2) and (1.4) for $j=0$, it follows
that

$\frac{\mathcal{A}_{\vec{n},0}(z)}{Q_{\vec{n},0}(z)}=\int\frac{\mathcal{A}_{\vec{n},1}(x)}{z-x}\frac{\operatorname{d}\sigma_{1}(x)}{Q_{\vec{n},0}(x)},$

and

$\int
x^{\nu}\mathcal{A}_{\vec{n},1}(x)\frac{\operatorname{d}\sigma_{1}(x)}{Q_{\vec{n},0}(x)}=0,\qquad\nu=0,1,\ldots,\eta_{\vec{n},1}-1.$

In the second relation there is at least one more orthogonality relation if
either the order in (1.4) for $j=0$ is higher than $\eta_{\vec{n},1}$ or $\deg
Q_{\vec{n},0}\geq 1$.

Therefore, there exists a monic polynomial with real coefficients
$Q_{\vec{n},1}$ of degree $\geq\eta_{\vec{n},1}$ (or strictly greater if
either the order in (1.4) for $j=0$ is higher than $\eta_{\vec{n},1}$ or $\deg
Q_{\vec{n},0}\geq 1$) whose roots are zeros of $\mathcal{A}_{\vec{n},1}$ in
$\mathbb{C}\setminus\Delta_{2}$ such that
$\mathcal{A}_{\vec{n},1}/Q_{\vec{n},1}$ is holomorphic in
$\mathbb{C}\setminus\Delta_{2}$. Using (2.1)-(2.2) and (1.4) for $j=1$, it
follows that

$\frac{\mathcal{A}_{\vec{n},1}(z)}{Q_{\vec{n},1}(z)}=\int\frac{\mathcal{A}_{\vec{n},2}(x)}{z-x}\frac{\operatorname{d}\sigma_{2}(x)}{Q_{\vec{n},1}(x)}$

and

$\int
x^{\nu}\mathcal{A}_{\vec{n},2}(x)\frac{\operatorname{d}\sigma_{2}(x)}{Q_{\vec{n},1}(x)}=0,\qquad\nu=0,1,\ldots,\eta_{\vec{n},2}-1.$

There is at least one more orthogonality relation if either the order in (1.4)
for $j=0,1$ is higher than prescribed, $\deg Q_{\vec{n},0}\geq 1$, or $\deg
Q_{\vec{n},1}>\eta_{\vec{n},1}$.

One can repeat the arguments proving that there exist monic polynomials
$Q_{\vec{n},j}$, $\deg Q_{\vec{n},j}\geq\eta_{\vec{n},j}$, $j=0,\ldots,m-1$,
$(\eta_{\vec{n},0}=0)$ with real coefficients whose roots lie in
$\mathbb{C}\setminus\Delta_{j+1}$ and are zeros of $\mathcal{A}_{\vec{n},j}$,
such that

(2.3)
$\frac{\mathcal{A}_{\vec{n},j}(z)}{Q_{\vec{n},j}(z)}=\int\frac{\mathcal{A}_{\vec{n},j+1}(x)}{z-x}\frac{\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)},$

and

(2.4) $\int
x^{\nu}\mathcal{A}_{\vec{n},j+1}(x)\frac{\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)}=0,\qquad\nu=0,1,\ldots,\eta_{\vec{n},j+1}-1,$

with an additional orthogonality relation if for some $j$ either the order in
(1.4) is greater than that prescribed or $\deg
Q_{\vec{n},j}>\eta_{\vec{n},j}$. Since
$\mathcal{A}_{\vec{n},m}=(-1)^{m}a_{\vec{n},m}$ and $\deg
a_{\vec{n},m}\leq|\vec{n}|=\eta_{\vec{n},m}$ the orthogonality relations for
$j=m-1$ imply that there can be no additional orthogonality relation unless
$a_{\vec{n},m}\equiv 0$. But this is not possible because (1.4) would imply
that $a_{\vec{n},j}\equiv 0$, $j=0,\ldots,m$ and the solution of that system
of equations would be the trivial one against our assumption. Consequently,
all the orders in (1.4) are exact, $\deg Q_{\vec{n},j}=\eta_{\vec{n},j}$,
$j=0,\ldots,m-1$. and $\deg a_{\vec{n},m}=|\vec{n}|$. In the sequel,
$Q_{\vec{n},m}:=a_{\vec{n},m}$.

Set

$\mathcal{H}_{\vec{n},j}(z):=\frac{Q_{\vec{n},j+1}(z)\mathcal{A}_{\vec{n},j}(z)}{Q_{\vec{n},j}(z)},\quad
j=0,\ldots,m-1,\qquad\mathcal{H}_{\vec{n},m}(z)\equiv(-1)^{m}.$

From (2.3)-(2.4), for $j=0,\ldots,m-1$, we get

(2.5)
$\mathcal{H}_{\vec{n},j}(z)=\int\frac{Q_{\vec{n},j+1}^{2}(x)}{z-x}\frac{\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)},$

where $Q_{\vec{n},0}\equiv Q_{\vec{n},m+1}\equiv 1$, and

(2.6) $\int
x^{\nu}Q_{\vec{n},j+1}(x)\frac{\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)}=0,\qquad\nu=0,1,\ldots,\eta_{\vec{n},j+1}-1.$

Define $K_{\vec{n},m}:=1$,

$K_{\vec{n},j}:=\left(\int(Q_{\vec{n},j+1}(x))^{2}\frac{|\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)|}{|{Q}_{\vec{n},j}(x){Q}_{\vec{n},j+2}(x)|}\right)^{-1/2},\quad
j=0,\ldots,m-1$

and

(2.7) $\kappa_{\vec{n},j+1}:=\frac{K_{\vec{n},j}}{K_{\vec{n},j+1}},\qquad
j=0,\ldots,m-1.$

Set

$q_{\vec{n},j}:=\kappa_{\vec{n},j}Q_{\vec{n},j},\quad
j=1,\ldots,m,\qquad{h}_{\vec{n},j}:=(K_{n,j})^{2}\mathcal{H}_{\vec{n},j},\quad
j=0,\ldots,m.$

With the new notation, (2.5)-(2.6) become

(2.8)
$h_{\vec{n},j}(z)=\varepsilon_{\vec{n},j+1}\int\frac{q_{\vec{n},j+1}^{2}(x)}{z-x}\frac{|h_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)|}{|Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)|},\qquad
j=0,\ldots,m-1$

where $\varepsilon_{\vec{n},j+1}$ denotes the constant sign which the varying
measure
$\frac{\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)}$
adopts on $\Delta_{j+1}$, and

$\int
x^{\nu}q_{\vec{n},j+1}(x)\frac{|h_{n,j+1}(x)\operatorname{d}\sigma_{j+1}(x)|}{|Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)|}=0,\qquad\nu=0,1,\ldots,\eta_{\vec{n},j+1}-1.$

Notice that $q_{\vec{n},j+1}$ is orthonormal with respect to
$\frac{|h_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)|}{|Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)|}$.

###  2.2. The operator $\tilde{T}_{\vec{n}}$

Fix $\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$. Let $\mathcal{P}_{\vec{n}}$ denote
the set of all vector polynomials $(Q_{1},\ldots,Q_{m})$ where
$Q_{j},j=1,\ldots,m,$ is a monic polynomial with real coefficients of degree
$\eta_{\vec{n},j}$ whose zeros lie in
$\mathbb{C}\setminus(\Delta_{j-1}\cup\Delta_{j+1})$. Recall that
$\Delta_{0}=\Delta_{m+1}\equiv\varnothing$.

######  Definition 2.1.

Define
$\tilde{T}_{\vec{n}}:\mathcal{P}_{\vec{n}}\longrightarrow\mathcal{P}_{\vec{n}}$
such that for each
${\vec{Q}}=({Q}_{1},\ldots,{Q}_{m})\in\mathcal{P}_{\vec{n}}$,
$\tilde{T}_{\vec{n}}(\vec{Q})=\vec{Q}^{*}=({Q}^{*}_{1},\ldots,{Q}^{*}_{m})\in\mathcal{P}_{\vec{n}}$
satisfies

(2.9) $\int
x^{\nu}Q_{j}^{*}(x)\frac{\mathcal{H}_{j}({\vec{Q}};x)\operatorname{d}\sigma_{j}(x)}{{Q}_{j-1}(x){Q}_{j+1}(x)}=0,\qquad\nu=0,\ldots,\eta_{\vec{n},j}-1,\qquad
j=1,\ldots m,$

where $\mathcal{H}_{m}({\vec{Q}};x)\equiv(-1)^{m}$, $Q_{0}=Q_{m+1}\equiv 1$,
and

(2.10)
$\mathcal{H}_{j}({\vec{Q}};z)=\int\frac{(Q^{*}_{j+1}(x))^{2}}{z-x}\frac{\mathcal{H}_{j+1}({\vec{Q}};x)\operatorname{d}\sigma_{j+1}(x)}{{Q}_{j}(x){Q}_{j+2}(x)},\qquad
j=0,\ldots,m-1.$

The components of $\vec{Q}^{*}$ as well as the weights
$\mathcal{H}_{j}({\vec{Q}};z)$ must be calculated inductively starting from
the value $j=m$ down to $j=0$. Since $\mathcal{H}_{m}({\vec{Q}};x)$ is given,
with (2.9)-(2.10) you can determine $Q_{m}^{*}$ and
$\mathcal{H}_{m-1}({\vec{Q}};x)$ univocally. With this information,
(2.9)-(2.10) determine $Q_{m-1}^{*}$ and $\mathcal{H}_{m-2}({\vec{Q}};x)$ and
so forth. Therefore, $\tilde{T}_{\vec{n}}$ is correctly defined.

We can define a distance
$d_{\vec{n}}:\mathcal{P}_{\vec{n}}\times\mathcal{P}_{\vec{n}}\longrightarrow\mathbb{R}_{\geq
0}$ as follows.

$d_{\vec{n}}(\vec{Q}_{1},\vec{Q}_{2})=\max_{1\leq j\leq
m}\\{\|Q_{1,j}-Q_{2,j}\|_{\Delta_{j-1}\cup\Delta_{j+1}}\\},$

where $\vec{Q}_{1}=(Q_{1,1},\ldots,Q_{1,m})$ and
$\vec{Q}_{2}=(Q_{2,1},\ldots,Q_{2,m})$ are arbitrary vector polynomials in
$\mathcal{P}_{\vec{n}}$ and the norms are uniform on the indicated sets
(recall that by convention $\Delta_{0}=\Delta_{m+1}=\varnothing$). It is not
difficult to prove that $\tilde{T}_{\vec{n}}$ is continuous on
$\mathcal{P}_{\vec{n}}$ with this distance (for details see the corresponding
result in [11, Subsection 3.6]). This property is used when we apply Brouwer’s
Theorem in Subsection 3.2 below.

######  Lemma 2.2.

Fix $\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$. The operator $\tilde{T}_{\vec{n}}$
has a unique fixed point which coincides with
$\vec{Q}_{\vec{n}}:=(Q_{\vec{n},1},\ldots,Q_{\vec{n},m})$.

###### Proof.

Indeed, $\vec{Q}_{\vec{n}}\in\mathcal{P}_{\vec{n}}$ and (2.5)-(2.6) show that
it is a fixed point of $\tilde{T}_{\vec{n}}$ (compare with (2.9)-(2.10)). Let
$\vec{\tilde{Q}}=(\tilde{Q}_{1},\ldots,\tilde{Q}_{m})$ be an arbitrary fixed
point of $\tilde{T}_{\vec{n}}$. We must show that
$\vec{\tilde{Q}}=\vec{Q}_{\vec{n}}$.

From the definition of the operator $\tilde{T}_{\vec{n}}$ and of a fixed
point, it follows that

(2.11) $\int
x^{\nu}\tilde{Q}_{j}(x)\frac{\mathcal{H}_{j}({\vec{\tilde{Q}}};x)\operatorname{d}\sigma_{j}(x)}{\tilde{Q}_{j-1}(x)\tilde{Q}_{j+1}(x)}=0,\qquad\nu=0,\ldots,\eta_{\vec{n},j}-1,\qquad
j=1,\ldots m,$

where $\mathcal{H}_{m}({\vec{\tilde{Q}}};x)\equiv(-1)^{m}$,
$\tilde{Q}_{0}=\tilde{{Q}}_{m+1}\equiv 1$, and

(2.12)
$\mathcal{H}_{j}({\vec{\tilde{Q}}};z)=\int\frac{(\tilde{Q}_{j+1}(x))^{2}}{z-x}\frac{\mathcal{H}_{j+1}({\vec{\tilde{Q}}};x)\operatorname{d}\sigma_{j+1}(x)}{\tilde{Q}_{j}(x)\tilde{Q}_{j+2}(x)},\qquad
j=0,\ldots,m-1.$

Notice that $\mathcal{H}_{j}({\vec{\tilde{Q}}};z)$, $j=0,\ldots,m$ is
analytic, symmetric with respect to the real line, and never equals zero in
$\mathbb{C}\setminus\Delta_{j+1}\,(\Delta_{m+1}=\varnothing)$. For $j=m$ this
is trivial. Then, from (2.12) for $j=m-1$ the assertion readily follows. If
for some $j+1$ the statement holds, from (2.12) the same is true for $j$.

Define

(2.13)
$\tilde{\mathcal{A}}_{j}(z):=\frac{\tilde{Q}_{j}(z)\mathcal{H}_{j}(\vec{\tilde{Q}};z)}{\tilde{Q}_{j+1}(z)},\qquad
j=0,\ldots,m.$

From this definition and the properties of
$\mathcal{H}_{j}(\vec{\tilde{Q}};z)$, it follows that
$\tilde{\mathcal{A}}_{j}$, $j=0,\ldots,m$ is analytic and symmetric in
$\mathbb{C}\setminus\Delta_{j+1}$, and its zeros in this region coincide with
the roots of $\tilde{Q}_{j}$ all of which are simple and lie on $\Delta_{j}$.
Let us show that there exist polynomials $a_{0},a_{1},\ldots,a_{m}$, where
$\deg a_{j}\leq|\vec{n}|-1$, $j=0,1,\ldots,m-1,$ $a_{m}$ monic, and $\deg
a_{m}\leq|\vec{n}|$, not all identically equal to zero, such that for each
$j=0,\ldots,m-1$

$\tilde{\mathcal{A}}_{j}(z)=\left((-1)^{j}a_{j}+\sum_{k=j+1}^{m}(-1)^{k}a_{k}\widehat{s}_{j+1,k}\right)(z)=\mathcal{O}\left(\frac{1}{z^{n_{j+1}+1}}\right),\qquad
z\to\infty.$

If this is so, from the uniqueness of the multi-level Hermite-Padé
approximants it follows that

$(a_{0},a_{1},\ldots,a_{m})=(a_{\vec{n},0},a_{\vec{n},1},\ldots,a_{\vec{n},m}),\qquad(\tilde{\mathcal{A}}_{0},\tilde{\mathcal{A}}_{1},\ldots,\tilde{\mathcal{A}}_{m-1})=({\mathcal{A}}_{\vec{n},0},{\mathcal{A}}_{\vec{n},1},\ldots,{\mathcal{A}}_{m-1})$

and

$(\tilde{Q}_{1},\ldots,\tilde{Q}_{m})=(Q_{\vec{n},1},\ldots,Q_{\vec{n},m}).$

Take $a_{m}:=\tilde{Q}_{m}=(-1)^{m}\tilde{\mathcal{A}}_{m}$. Obviously,
$a_{m}$ is a monic polynomial of degree $|\vec{n}|$. With the notation
introduced in (2.13), (2.11)-(2.12) adopt the form

(2.14) $\int
x^{\nu}\tilde{\mathcal{A}}_{j}(x)\frac{\operatorname{d}\sigma_{j}(x)}{\tilde{Q}_{j-1}(x)}=0,\qquad\nu=0,\ldots,\eta_{\vec{n},j}-1,\qquad
j=1,\ldots m,$

and

(2.15)
$\tilde{\mathcal{A}}_{j}(z)=\frac{\tilde{Q}_{j}(z)}{\tilde{Q}_{j+1}(z)}\int\frac{\tilde{Q}_{j+1}(x)}{z-x}\frac{\tilde{\mathcal{A}}_{j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{\tilde{Q}_{j}(x)},\qquad
j=0,\ldots,m-1.$

Since $\deg\tilde{Q}_{j}=\eta_{\vec{n},j}$ and
$\deg\tilde{Q}_{j+1}=\eta_{\vec{n},j+1}$ this relation entails that

$\tilde{\mathcal{A}}_{j}(z)=\mathcal{O}(1/z^{n_{j+1}+1}),\qquad z\to\infty,$

as needed. Using (2.14)-(2.15), we get

(2.16)
$\tilde{\mathcal{A}}_{j}(z)=\int\frac{\tilde{\mathcal{A}}_{j+1}(x)}{z-x}\operatorname{d}\sigma_{j+1}(x).$

For $j=m-1$, (2.16) gives

$\tilde{\mathcal{A}}_{m-1}(z)=\int\frac{\tilde{\mathcal{A}}_{m}(x)-\tilde{\mathcal{A}}_{m}(z)}{z-x}\operatorname{d}\sigma_{m}(x)+\int\frac{\tilde{\mathcal{A}}_{m}(z)}{z-x}\operatorname{d}\sigma_{m}(x)$
$=(-1)^{m-1}a_{m-1}(z)+(-1)^{m}a_{m}(z)\widehat{s}_{m,m},$

where

$a_{m-1}(z):=\int\frac{{a}_{m}(z)-{a}_{m}(x)}{z-x}\operatorname{d}\sigma_{m}(x),\qquad\deg
a_{m-1}=|\vec{n}|-1.$

Assume that there exist polynomials $a_{j+1},\ldots,a_{m-1}$, $\deg
a_{k}=|\vec{n}|-1$, $k=j+1,\ldots,m-1,$ such that

$\tilde{\mathcal{A}}_{j+1}=(-1)^{j+1}a_{j+1}+\sum_{k=j+2}^{m}(-1)^{k}a_{k}\widehat{s}_{j+2,k}.$

Substituting this expression inside the integral in (2.16), we get

$\tilde{\mathcal{A}}_{j}(z)=\int\frac{\tilde{\mathcal{A}}_{j+1}(x)\mp\left((-1)^{j+1}a_{j+1}(z)+\sum_{k=j+2}^{m}(-1)^{k}a_{k}(z)\widehat{s}_{j+2,k}(x)\right)}{z-x}\operatorname{d}\sigma_{j+1}(x)=$
$(-1)^{j+1}\int\frac{a_{j+1}(x)-a_{j+1}(z)}{z-x}\operatorname{d}\sigma_{j+1}(x)+\sum_{k=j+2}^{m}(-1)^{k}\int\frac{a_{k}(x)-a_{k}(z)}{z-x}\operatorname{d}s_{j+1,k}(x)+$
$\sum_{k=j+1}^{m}(-1)^{k}a_{k}(z)\widehat{s}_{j+1,k}(z).$

obtaining for $\tilde{\mathcal{A}}_{j}$ an expression similar to the one we
had for $\tilde{\mathcal{A}}_{j+1}$ with the same $a_{k}$, $k=j+1,\ldots,m$
plus

$a_{j}(z):=\int\frac{a_{j+1}(z)-a_{j+1}(x)}{z-x}\operatorname{d}{s}_{j+1,j+1}(x)+\sum_{k=j+2}^{m}(-1)^{k-j+1}\int\frac{a_{k}(z)-a_{k}(x)}{z-x}\operatorname{d}s_{j+1,k}(x),$

which obviously is a polynomial of degree $|\vec{n}|-1$. We have concluded the
proof. ∎

In the definition of the operator $\tilde{T}_{\vec{n}}$ one could be tempted
to replace the weights $\mathcal{H}_{j}(\vec{Q};x)$ with
$\mathcal{H}_{\vec{n},j}(x)$ (independent of $\vec{Q}$). The resulting
operator also has $\vec{Q}_{\vec{n}}$ as a fixed point but it is not clear
whether it has other fixed points or not.

###  2.3. Normalization

Fix $\vec{n}\in(\mathbb{Z}_{+}^{m})^{*}$. Let
$\vec{Q}=(Q_{1},\ldots,Q_{m})\in\mathcal{P}_{\vec{n}}$ and
$\vec{Q}^{*}=(Q_{1}^{*},\ldots,Q_{m}^{*})=\tilde{T}_{\vec{n}}(\vec{Q})$.
Define

$K_{j}(\vec{Q}^{*}):=\left(\int(Q^{*}_{j+1}(x))^{2}\frac{|\mathcal{H}_{j+1}(\vec{Q};x)\operatorname{d}\sigma_{j+1}(x)|}{|{Q}_{j}(x){Q}_{j+2}(x)|}\right)^{-1/2},\quad
j=0,\ldots,m-1,$
$K_{m}(\vec{Q}^{*}):=1,\qquad\kappa_{j+1}(\vec{Q}^{*}):=\frac{K_{j}(\vec{Q}^{*})}{K_{j+1}(\vec{Q}^{*})},\qquad
j=0,\ldots,m-1.$

Set

$q_{j}^{*}:=\kappa_{j}(\vec{Q}^{*})Q_{j}^{*},\quad
j=1,\ldots,m,\qquad{h}_{j}(\vec{Q};x):=(K_{j}(\vec{Q}^{*}))^{2}\mathcal{H}_{j}(\vec{Q};x),\quad
j=0,\ldots,m.$

With this normalization, (2.9)–(2.10) imply

$\int
x^{\nu}q_{j}^{*}(x)\frac{|{h}_{j}({\vec{Q}};x)\operatorname{d}\sigma_{j}(x)|}{|{Q}_{j-1}(x){Q}_{j+1}(x)|}=0,\quad\nu=0,\ldots,\eta_{\vec{n},j}-1,\quad
j=1,\ldots m,$

where ${h}_{m}({\vec{Q}};x)\equiv(-1)^{m}$, $Q_{0}=Q_{m+1}\equiv 1$, and

(2.17)
${h}_{j}({\vec{Q}};z)=\varepsilon_{j+1}(\vec{Q})\int\frac{(q^{*}_{j+1}(x))^{2}}{z-x}\frac{|{h}_{j+1}({\vec{Q}};x)||\operatorname{d}\sigma_{j+1}(x)|}{|{Q}_{j}(x){Q}_{j+2}(x)|},\qquad
j=0,\ldots,m-1,$

where $\varepsilon_{j+1}(\vec{Q})$ denotes the constant sign which the varying
measure
$\frac{{h}_{j+1}({\vec{Q}};x)\operatorname{d}\sigma_{j+1}(x)}{{Q}_{j}(x){Q}_{j+2}(x)}$
adopts on $\Delta_{j+1}$. Notice that $q^{*}_{j+1}$ is orthonormal with
respect to
$\frac{|{h}_{j+1}({\vec{Q}};x)\operatorname{d}\sigma_{j+1}(x)|}{|{Q}_{j}(x){Q}_{j+2}(x)|}$.

######  Lemma 2.3.

Suppose that $\sigma_{j}^{\prime}\neq 0$ a.e. on $\Delta_{j}=[a_{j},b_{j}]$,
$j=1,\ldots,m$. Let $\Lambda\subset(\mathbb{Z}_{+}^{m})^{*}$ be a sequence of
distinct multi-indices such that for some fixed $N\in\mathbb{Z}_{+}$

(2.18) $n_{j+1}\leq n_{j}+N,\qquad
j=1,\ldots,m-1,\qquad\vec{n}=(n_{1},\ldots,n_{m})\in\Lambda,$

and let
$(\vec{\tilde{Q}}_{\vec{n}})_{\vec{n}\in\Lambda},\vec{\tilde{Q}}_{\vec{n}}\in\mathcal{P}_{\vec{n}},$
be an arbitrary sequence of vector polynomials such that for each
$j=1,\ldots,m$ the zeros of the $j-th$ component $\tilde{Q}_{\vec{n},j}$ of
$\vec{\tilde{Q}}_{\vec{n}}$ remain uniformly bounded away from
$\Delta_{j-1}\cup\Delta_{j+1}$ for $\vec{n}\in\Lambda$. Then

(2.19)
$\lim_{\vec{n}\in\Lambda}|{h}_{j}({\vec{\tilde{Q}}}_{\vec{n}};z)|=\frac{1}{\sqrt{|z-b_{j+1}||z-a_{j+1}|}}=:\tilde{h}_{j}(z),\qquad
j=0,\ldots,m-1,$

uniformly on $\Delta_{j}\cup\Delta_{j+2}$,
$(\Delta_{0}=\Delta_{m+1}=\varnothing)$.

###### Proof.

In the sequel,
$\vec{{Q}}_{\vec{n}}^{*}=\tilde{T}_{\vec{n}}(\vec{\tilde{Q}}_{\vec{n}})$,
where
$\vec{{Q}}_{\vec{n}}^{*}=({Q}_{\vec{n},1}^{*},\ldots,{Q}_{\vec{n},m}^{*})$,
$\vec{\tilde{Q}}_{\vec{n}}=(\tilde{Q}_{\vec{n},1},\ldots,\tilde{Q}_{\vec{n},m})$,
and $({q}_{\vec{n},1}^{*},\ldots,{q}_{\vec{n},m}^{*})$ is the vector of
orthonormal polynomials associated with
$({Q}_{\vec{n},1}^{*},\ldots,{Q}_{\vec{n},m}^{*})$ using the normalization
introduced previously. According to (2.17)

(2.20)
${h}_{j}({\vec{\tilde{Q}}}_{\vec{n}};z)=\varepsilon_{j+1}(\vec{\tilde{Q}}_{\vec{n}})\int\frac{(q^{*}_{\vec{n},j+1}(x))^{2}}{z-x}\frac{|{h}_{j+1}({\vec{\tilde{Q}}}_{\vec{n}};x)\operatorname{d}\sigma_{j+1}(x)|}{|\tilde{Q}_{\vec{n},j}(x)\tilde{Q}_{\vec{n},j+2}(x)|},\qquad
j=0,\ldots,m-1.$

As usual, $\tilde{Q}_{\vec{n},0}=\tilde{Q}_{\vec{n},m+1}\equiv 1$.

By assumption, the zeros of the polynomials
$\tilde{Q}_{\vec{n},j}\tilde{Q}_{\vec{n},j+2}$ are uniformly bounded away from
$\Delta_{j+1}$. Due to (2.18), we have

$\deg\tilde{Q}_{\vec{n},j}(x)+\deg\tilde{Q}_{\vec{n},j+2}(x)-2\deg
q^{*}_{\vec{n},j+1}=n_{j+2}-n_{j+1}\leq N$

for all $\vec{n}\in\Lambda$, $q^{*}_{\vec{n},j+1}(x)$ is orthonormal with
respect to the varying measure
$\frac{|{h}_{j+1}({\vec{\tilde{Q}}}_{n};x)\operatorname{d}\sigma_{j+1}(x)|}{|\tilde{Q}_{\vec{n},j}(x)\tilde{Q}_{\vec{n},j+2}(x)|}$,
and $\sigma_{j+1}^{\prime}>0$ a.e. on $\Delta_{j+1}$. We have all the
ingredients to use [8, Theorem 8] (see also [7]) and we obtain

$\lim_{\vec{n}\in\Lambda}\int\frac{(q^{*}_{\vec{n},j+1}(x))^{2}}{z-x}\frac{|{h}_{j+1}({\vec{\tilde{Q}}}_{\vec{n}};x)\operatorname{d}\sigma_{j+1}(x)|}{|\tilde{Q}_{\vec{n},j}(x)\tilde{Q}_{\vec{n},j+2}(x)|}=\\\
\frac{1}{\pi}\int_{a_{j+1}}^{b_{j+1}}\frac{1}{z-x}\frac{\operatorname{d}x}{\sqrt{(b_{j+1}-x)(x-a_{j+1})}}=\frac{1}{\sqrt{(z-b_{j+1})(z-a_{j+1})}},$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{j+1}$,
provided $|{h}_{j+1}({\vec{\tilde{Q}}}_{\vec{n}};x)|$ converges uniformly to a
positive function on $\Delta_{j+1}$. This can be proved by induction.

When $j=m-1$, $|{h}_{m}({\vec{\tilde{Q}}}_{\vec{n}};x)|\equiv 1$ and the
property holds; therefore,

$\lim_{\vec{n}\in\Lambda}|{h}_{m-1}({\vec{\tilde{Q}}}_{\vec{n}};z)|=\lim_{\vec{n}\in\Lambda}\left|\int\frac{(q^{*}_{\vec{n},m}(x))^{2}}{z-x}\frac{|{h}_{m}({\vec{\tilde{Q}}}_{\vec{n}};x)\operatorname{d}\sigma_{m}(x)|}{|\tilde{Q}_{\vec{n},m-1}(x)|\leavevmode\nobreak\
}\right|=\frac{1}{|\sqrt{(z-a_{m})(z-b_{m})}|},$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{m}$; in
particular, on $\Delta_{m-1}$. Now, if we assume that the property holds for
some $j+1$, repeating the arguments and using (2.20), we obtain that it also
holds for $j$. ∎

###  2.4. Some properties of the vector equilibrium measure

The equilibrium measure which is relevant in our study satisfies some
additional properties.

######  Lemma 2.4.

Let $(p_{1},\ldots,p_{m})\in(\mathbb{Z}_{+}^{m})^{*}$ where $p_{1}>0$ and
$p_{1}\geq\cdots\geq p_{m}.$ Let
$\vec{\lambda}\in\mathcal{M}_{1}(\vec{\Delta})$ be the unitary vector
equilibrium measure which solves (1.7) with interaction matrix (1.9) over the
system of intervals $\vec{\Delta}$ and $P_{j},j=1,\ldots,m,$ is defined by
(1.6). Then, for each $j=1,\ldots,m$,

(2.21) $\operatorname{supp}(\lambda_{j})=\Delta_{j}=[a_{j},b_{j}],$

where $\vec{\lambda}=(\lambda_{1},\ldots,\lambda_{m})$. Moreover,
$\operatorname{d}\lambda_{j}(x)=\lambda_{j}^{\prime}(x)\operatorname{d}x$,
$j=1,\ldots,m,$ is absolutely continuous with respect to Lebesque measure on
$\Delta_{j}$ and there exist constants $A>0,\beta=-1/2,$ and $\beta_{0}$ such
that

(2.22) $A^{-1}((b_{j}-x)(x-a_{j}))^{\beta_{0}}\leq\lambda_{j}^{\prime}(x)\leq
A((b_{j}-x)(x-a_{j}))^{\beta},\qquad x\in(a_{j},b_{j}).$

###### Proof.

Fix $j$, $1\leq j\leq m$. We can rewrite (1.7) in the form

(2.23)
$V^{\lambda_{j}}(x)-\frac{1}{2}V^{\tau_{j}}(x)\begin{cases}=\omega_{j},&x\in\operatorname{supp}\lambda_{j},\\\
\geq\omega_{j},&x\in\Delta_{j},\end{cases}$

where

$\tau_{j}=\frac{P_{j-1}}{P_{j}}\lambda_{j-1}+\frac{P_{j+1}}{P_{j}}\lambda_{j+1}$

is a measure whose support is contained in $\Delta_{j-1}\cup\Delta_{j+1}$
which is a set disjoint from $\Delta_{j}$. The total mass of $\tau_{j}$ is

$\frac{P_{j-1}}{P_{j}}+\frac{P_{j+1}}{P_{j}}=:c_{j}.$

Obviously $c_{j}\leq 2$ since by hypothesis
$P_{j-1}+P_{j+1}-2P_{j}=p_{j+1}-p_{j}\leq 0$. This fact together with (2.23)
imply that $2\lambda_{j}$ is the balayage of $\tau_{j}$ onto $\Delta_{j}$ plus
$(2-c_{j})$ times the equilibrium measure of the interval $\Delta_{j}$ (see
[23, Ex. 4.8, pg.118]). The support of such a measure verifies (2.21) (see
[23, Cor. 4.12, pg.122]).

On the other hand, the external field $-V^{\tau_{j}}/2$ acting on $\Delta_{j}$
(see (2.23)) is infinitely differentiable on that interval. Using [9, Theorem
1.34] and [14, Lemma 4] the analytic properties of $\lambda_{j}$ readily
follow.∎

###  2.5. Auxiliary asymptotic relations

The operator $\vec{T}_{\vec{n}}$ has the remarkable property that it amplifies
the region where strong asymptotics takes place in the sense expressed by the
following lemma.

Everywhere below, if $w\operatorname{d}\mu$ is in the Szegő class, where $w$
is a weight and $\mu$ a positive measure, then $\mathsf{G}(w\mu,z)$ denotes
the Szegő function with respect to the measure $w\operatorname{d}\mu$.

######  Lemma 2.5.

Let $\Lambda=\Lambda(p_{1},\ldots,p_{m})\subset(\mathbb{Z}_{+}^{m})^{*}$ be a
ray sequence with $p_{1}\geq\cdots\geq p_{m},$ and (1.5) takes place. Let
$\vec{\lambda}\in\mathcal{M}_{1}(\vec{\Delta})$ be the unitary vector
equilibrium measure which solves (1.7) with interaction matrix (1.9) over the
system of intervals $\vec{\Delta}$ and $P_{j},j=1,\ldots,m,$ is defined by
(1.6). Take $\Phi_{j}$ and $C_{j},j=1,\ldots,m$ as in (1.10). Let
$(\vec{\tilde{Q}}_{\vec{n}})_{\vec{n}\in\Lambda},\vec{\tilde{Q}}_{\vec{n}}\in\mathcal{P}_{\vec{n}},$
be a sequence of vector polynomials such that for each $j=1,\ldots,m$ the
zeros of the $j-th$ component $\tilde{Q}_{\vec{n},j}$ of
$\vec{\tilde{Q}}_{\vec{n}}$ remain uniformly bounded away from
$\Delta_{j-1}\cup\Delta_{j+1}$ for $\vec{n}\in\Lambda$. Set

$\vec{f}_{\vec{n}}=\left(\frac{\tilde{Q}_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\tilde{Q}_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right).$

Assume that there exists
$\vec{f}=(f_{1},\ldots,f_{m})\in\mathcal{C}^{+}(\vec{\Delta})$ such that

(2.24)
$\lim_{\vec{n}\in\Lambda}\|\vec{f}_{\vec{n}}-\vec{f}\|_{\vec{\Delta}}=0.$

Let
$\vec{Q}_{\vec{n}}^{*}=\tilde{T}_{\vec{n}}(\vec{\tilde{Q}}_{\vec{n}})=(Q_{\vec{n},1}^{*},\ldots,Q_{\vec{n},j}^{*})$
and
${q}_{\vec{n},j}^{*}=\kappa_{j}(\vec{Q}_{\vec{n}}^{*}){Q}_{\vec{n},j}^{*}$,
$j=1,\ldots,m$. Suppose that $\vec{\sigma}\in\mathcal{S}(\vec{\Delta})$. Then

$\lim_{\vec{n}\in\Lambda}\frac{q^{*}_{\vec{n},j}(z)}{[C_{j}\Phi_{j}(z)]^{\eta_{\vec{n},j}}}=\frac{1}{\sqrt{2\pi}}\mathsf{G}\left((f_{j-1}f_{j+1})^{-1}\tilde{h}_{j}\sigma_{j},z\right),\qquad
j=1,\ldots,m,$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{j}$
($f_{0}\equiv f_{m+1}\equiv 1$) where $\tilde{h}_{j}$, $j=1,\ldots,m-1$ is
given in (2.19) and $\tilde{h}_{m}\equiv 1$. In addition,

$\lim_{\vec{n}\in\Lambda}\frac{\kappa_{j}(\vec{Q}_{\vec{n}}^{*})}{C_{j}^{\eta_{\vec{n},j}}}=\frac{1}{\sqrt{2\pi}}\mathsf{G}\left((f_{j-1}f_{j+1})^{-1}\tilde{h}_{j}\sigma_{j},\infty\right),\qquad
j=1,\ldots,m.$

Consequently,

(2.25)
$\lim_{\vec{n}\in\Lambda}\frac{Q^{*}_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}\left((f_{j-1}f_{j+1})^{-1}\tilde{h}_{j}\sigma_{j},z\right)}{\mathsf{G}\left((f_{j-1}f_{j+1})^{-1}\tilde{h}_{j}\sigma_{j},\infty\right)},\qquad
j=1,\ldots,m,$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{j}$.

###### Proof.

Due to the assumptions, Lemma 2.4 guarantees that
$\operatorname{supp}\lambda_{j}=\Delta_{j}$, $j=1,\ldots,m$ and we have
equality in (1.7) on all $\Delta_{j}$ for each $j=1,\ldots,m$.

To obtain the limit relations we apply [11, Th. 1.2] which is a theorem on the
strong asymptotics of orthogonal polynomials with respect to varying measures.
In that theorem, $n$ is a parameter which runs through all the natural
numbers. That result remains valid if $n$ runs through any infinite sequence
of distinct natural numbers. For our purpose what is important is that $n$
controls the degree of the orthogonal polynomial under consideration.

Fix $j\in\\{1,\ldots,m\\}$ and consider the sequence of varying measures

$(|{{h}_{j}({\vec{\tilde{Q}}_{\vec{n}}};x)|\operatorname{d}\sigma_{j}(x)/|\tilde{Q}_{\vec{n},j-1}(x)\tilde{Q}_{\vec{n},j+1}(x)}|),\qquad\vec{n}\in\Lambda,$

where $\tilde{Q}_{\vec{n},0}=\tilde{Q}_{\vec{n},m+1}\equiv 1$ and
$|{h}_{m}({\vec{\tilde{Q}}_{\vec{n}}};x)|\equiv 1$. We have to show that this
sequence of varying measures satisfies all the assumptions i)-iv) of [11, Th.
1.2] taking

$\operatorname{d}\mu_{n}(x)=|{h}_{j}({\vec{\tilde{Q}}_{\vec{n}}};x)|\operatorname{d}\sigma_{j}(x),\qquad
w_{2n}(x)=\tilde{Q}_{\vec{n},j-1}(x)\tilde{Q}_{\vec{n},j+1}(x),\qquad n=\deg
Q_{\vec{n},j}^{*}=\eta_{\vec{n},j}.$

where $\mu_{n}$ and $w_{2n}$ follow the notation in [11].

Because of Lemma 2.3,
$\lim_{\vec{n}\in\Lambda}|{h}_{j}({\vec{\tilde{Q}}_{\vec{n}}};x)|=\tilde{h}_{j}(x)$
uniformly on $\Delta_{j}$ and $\sigma_{j}^{\prime}\neq 0$ a.e. on
$\Delta_{j}$. Consequently, i) readily follows with
$\operatorname{d}\mu=\tilde{h}_{j}\operatorname{d}\sigma_{j}$. Since
$\sigma_{j}\in\mathcal{S}(\Delta_{j})$ and
$\tilde{h}_{j}(x)>0,x\in\Delta_{j}$, ii) is also immediate. The components of
$\vec{n}$ are decreasing, so

$\deg
w_{2n}=\deg(\tilde{Q}_{\vec{n},j-1}\tilde{Q}_{\vec{n},j+1})=\eta_{\vec{n},j-1}+\eta_{\vec{n},j+1}=2\deg
Q_{\vec{n},j}+n_{j+1}-n_{j}\leq 2\deg Q_{\vec{n},j}=2n$

and the zeros of $w_{2n}$ are bounded away from $\Delta_{j}$ for
$\vec{n}\in\Lambda$; therefore, iii) holds. Finally, taking into account
(1.10) and (1.5), we get

$\left|\Phi_{j-1}^{\eta_{\vec{n},j-1}}(x)\Phi_{j+1}^{\eta_{\vec{n},j+1}}(x)\right|^{-1}=\left|\Phi_{j-1}^{P_{j-1}/P_{j}}(x)\Phi_{j+1}^{P_{j+1}/P_{j}}(x)\right|^{-\eta_{\vec{n},j}}=$
$\exp\left[\eta_{\vec{n},j}\left(\frac{P_{j-1}}{P_{j}}V^{\lambda_{j-1}}(x)+\frac{P_{j+1}}{P_{j}}V^{\lambda_{j+1}}(x)\right)\right],\qquad
x\in\Delta_{j}.$

Set

$\varphi_{j}(x):=\exp\left(\frac{P_{j-1}}{P_{j}}V^{\lambda_{j-1}}(x)+\frac{P_{j+1}}{P_{j}}V^{\lambda_{j+1}}(x)\right).$

Using these identities and (2.24), we obtain

$\lim_{\vec{n}\in\Lambda}\varphi_{j}^{\eta_{\vec{n},j}}(x)|w_{2n}(x)|=$
$\lim_{\vec{n}\in\Lambda}\frac{|\tilde{Q}_{\vec{n},j-1}(x)|}{|\Phi_{j-1}^{\eta_{\vec{n},j-1}}(x)|}\frac{|\tilde{Q}_{\vec{n},j+1}(x)|}{|\Phi_{j+1}^{\eta_{\vec{n},j+1}}(x)|}=f_{j-1}(x)f_{j+1}(x)=:1/\psi_{j}(x),$

uniformly on $\Delta_{j}$, and iv) takes place with $\varphi=\varphi_{j}$ and
$\psi=\psi_{j}$. Thus, the external field of the equilibrium problem in [11,
(1.5)] which allows to describe the limit relations in [11, Th. 1.2] is

$-\frac{1}{2}\ln\varphi_{j}(x)=-\frac{1}{2}\left(\frac{P_{j-1}}{P_{j}}V^{\lambda_{j-1}(x)}+\frac{P_{j+1}}{P_{j}}V^{\lambda_{j+1}(x)}\right)$

which coincides with the one in (2.23). The equilibrium measure and
equilibrium constant are $\lambda_{j}$ and $w_{j}$, respectively. Now, the
conclusions of Lemma 2.5 follow directly from [11, Th. 1.2]. ∎

A clear advantage of this Lemma is that in order to find the strong
asymptotics for sequences of fixed points of the operators
$\tilde{T}_{\vec{n}}$ it is sufficient to restrict the analysis to the
intervals $\Delta_{j}$ to obtain (2.24). This result also indicates that if
the sequence of fixed points of the operators $\tilde{T}_{\vec{n}}$ satisfy
(2.24), then $\vec{f}$ has to be a fixed point of the operator
$\vec{T}_{\vec{w}}$ where $\vec{w}$ is given by (3.4).

The analogue of Lemma 2.5 in [2] is Proposition 1.2. For its proof, Aptekarev
uses a different technique close to Widom’s approach in [28] for the study of
the strong asymptotics of orthogonal polynomials on arcs and curves of the
complex plane. It has the advantage that one obtains $L_{2}$ estimates on the
intervals $\Delta_{j}$ of the strong asymptotic formula, but it has the
drawback that one must restrict the analysis to Nikishin systems generated by
weights (instead of measures) with Szegő’s condition. We could have followed
that approach but prefered to maintain the generality and the use of the
theory of orthogonal polynomials with respect to varying measures.

A natural question is if there are sequences of vector polynomials for which
(2.24) holds. As the next lemma shows, such sequences of vector polynomials
may be given for a large class of predetermined limiting vector functions
$\vec{f}$. This is relevant in the proof of Theorem 1.4.

######  Lemma 2.6.

Assume that $\vec{\mu}=(\mu_{1},\ldots,\mu_{m})\in\mathcal{S}(\vec{\Delta})$,
and for each $\vec{n}\in(\mathbb{Z}^{m})^{*}$,
$(\tilde{q}_{\vec{n},1},\ldots,\tilde{q}_{\vec{n},m})$ denotes the vector
polynomial whose components have positive leading coefficients, verify
$\deg\tilde{q}_{\vec{n},j}=\eta_{\vec{n},j}$, $j=1,\ldots,m,$ and

(2.26) $\int
x^{\nu}\tilde{q}_{\vec{n},j}(x)\frac{\operatorname{d}\mu_{j}(x)}{|\Phi_{j}^{2\eta_{\vec{n},j}}(x)|}=0,\qquad
0\leq\nu<\eta_{\vec{n},j},\qquad\int[\tilde{q}_{\vec{n},j}(x)]^{2}\frac{\operatorname{d}\mu_{j}(x)}{|\Phi_{j}^{2\eta_{\vec{n},j}}(x)|}=1.$

Then, for each ray sequence $\Lambda=\Lambda(p_{1},\ldots,p_{m})$ such that
(2.18) takes place,

(2.27)
$\lim_{\vec{n}\in\Lambda}\frac{\tilde{q}_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}(\mu_{j},z)}{\sqrt{2\pi}},$

and

(2.28)
$\lim_{\vec{n}\in\Lambda}\frac{\tilde{Q}_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}(\mu_{j},z)}{\mathsf{G}(\mu_{j},\infty)},$

uniformly on each compact subset of
$\overline{\mathbb{C}}\setminus\Delta_{j},j=1,\ldots,m,$ where the $\Phi_{j}$
were introduced in (1.10) from the equilibrium problem corresponding to
$(p_{1},\ldots,p_{m})$ and $\tilde{Q}_{\vec{n},j}$, $j=1,\ldots,m$ is
$\tilde{q}_{\vec{n},j}$ renormalized to be monic.

###### Proof.

Let $\Lambda$ be a ray sequence with the indicated restrictions. In
particular, (2.18) implies that $p_{1}\geq\cdots\geq p_{m}$ with $p_{1}>0$.
Then, the assumptions of Lemma 2.4 are satisfied and, consequently, (2.21) and
(2.22) take place. This means that for each $j=1,\ldots,m$ the hypotheses of
[11, Theorem 1.3] hold true and that result guarantees that the sequences of
polynomials $(\tilde{q}_{\vec{n},j})_{\vec{n}\in\Lambda}$ defined by (2.26)
fulfill (2.27) and (2.28). ∎

In [2], Theorem 2 is the analogue of Lemma 2.6. The proof of [2, Theorem 2] is
based on an intricate theoretical construction using meromorphic functions on
a Riemann surface which in our opinion has its weak points. Here Aptekarev
also uses that the generating measures of the Nikishin system are absolutely
continuous with respect to Lebesgue measure. Our proof gives an explicit
solution in terms of orthogonal polynomials, the arguments are quite simple,
and do not require absolute continuity of the generating measures.

##  3\. Proof of main results

###  3.1. Proof of Theorem 1.6

For simplicity, in the proof we write $\vec{T}_{\vec{w}}=\vec{T}$. Let us
assume that (1.17) takes place. It is known that if such an $\overline{m}$
exists then $\vec{T}$ has a unique fixed point (see, for example, [13, Ex.
(A.1), pg. 17]). The proof is simple and for completeness we include it.

Since $({C}^{+}(\vec{\Delta}),d)$ is a complete metric space and
$\gamma_{m}<1$, by Banach’s fixed point theorem $\vec{T}^{\overline{m}}$ has a
unique fixed point, say $\vec{f}^{*}$. In particular,
$\vec{T}^{\overline{m}}\vec{f}^{*}=\vec{f}^{*}$. Applying $\vec{T}$ one more
time, we get

$\vec{T}^{\overline{m}+1}\vec{f}^{*}=\vec{T}\vec{f}^{*}=\vec{T}^{\overline{m}}\vec{T}\vec{f}^{*}.$

Consequently, $\vec{T}\vec{f}^{*}$ is also a fixed point of
$\vec{T}^{\overline{m}}$. But the fixed point of $\vec{T}^{\overline{m}}$ is
unique, so $\vec{T}\vec{f}^{*}=\vec{f}^{*}$; therefore, $\vec{f}^{*}$ is a
fixed point of $\vec{T}$. On the other hand, if $\vec{f}$ is any fixed point
of $\vec{T}$ we have

$\vec{T}^{\overline{m}}\vec{f}=\vec{T}^{\overline{m}-1}\vec{f}=\cdots=\vec{T}\vec{f}=\vec{f}.$

So, $\vec{f}$ is also a fixed point of $\vec{T}^{\overline{m}}$. Consequently,
$\vec{T}$ cannot have two distinct fixed points since they would be distinct
fixed points of $\vec{T}^{\overline{m}}$.

Let us prove (1.17). For general $m$ the notation becomes a bit cumbersome so
to fix ideas we begin with the simple case $m=2$. Thus $\overline{m}=1$ and we
must show that $\vec{T}$ is contractive with $\gamma_{2}=1/2$. Take arbitrary
functions
$\vec{f}=(f_{1},f_{2}),\vec{g}=(g_{1},g_{2})\in{C}^{+}({\vec{\Delta}})$, and
$\vec{\Delta}=(\Delta_{1},\Delta_{2})$. We have

$d(\vec{T}\vec{f},\vec{T}\vec{g}):=\max\left\\{\|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)\|_{\Delta_{2}},\|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)\|_{\Delta_{1}}\right\\}.$

Now, $\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)$ and
$\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)$ are harmonic functions on
$\overline{\mathbb{C}}\setminus\Delta_{1}$ and
$\overline{\mathbb{C}}\setminus\Delta_{2}$, respectively. Using the maximum
principle for harmonic functions it follows that

$\max\left\\{\|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)\|_{\Delta_{2}},\|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)\|_{\Delta_{1}}\right\\}\\\
\leq\max\left\\{\sup_{\Delta_{1}}|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)|,\sup_{\Delta_{2}}|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)|\right\\}.$

Due to (1.16)

$\max\left\\{\sup_{\Delta_{1}}|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)|,\sup_{\Delta_{2}}|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)|\right\\}\\\
=\frac{1}{2}\max\left\\{\|\ln(f_{2}/g_{2})\|_{\Delta_{1}},\|\ln(f_{1}/g_{1})\|_{\Delta_{2}}\right\\}=\frac{1}{2}d(\vec{f},\vec{g}).$

Notice that in the quotient the weights cancel out and what remains is a
continuous function on the corresponding intervals and the supremum becomes a
maximum. The last equality is simply by definition. Adding up these relations
(1.17) follows with the value prescribed on $\gamma_{2}$. In the sequel, we
assume that $m\geq 3$ and, therefore, $\overline{m}>1$.

Now, let us see why $\vec{T}$ is not contractive when $m=3$. Proceeding as
above with
$\vec{f}=(f_{1},f_{2},f_{3}),\;\vec{g}=(g_{1},g_{2},g_{3})\in{C}^{+}({\vec{\Delta}})$,
and $\vec{\Delta}=(\Delta_{1},\Delta_{2},\Delta_{3})$, we get

$\displaystyle d(\vec{T}\vec{f},\vec{T}\vec{g})=$
$\displaystyle\max\left\\{\|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)\|_{\Delta_{2}},\|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)\|_{\Delta_{1}\cup\Delta_{3}},\|\ln(|T_{3}\vec{f}|/|T_{3}\vec{g}|)\|_{\Delta_{2}}\right\\}$
$\displaystyle\leq$
$\displaystyle\max\left\\{\|\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|)\|_{\Delta_{1}},\|\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|)\|_{\Delta_{2}},\|\ln(|T_{3}\vec{f}|/|T_{3}\vec{g}|)\|_{\Delta_{3}}\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\max\left\\{\|\ln(f_{2}/g_{2})\|_{\Delta_{1}},\|\ln(f_{1}/g_{1})+\ln(f_{3}/g_{3})\|_{\Delta_{2}},\|\ln(f_{2}/g_{2})\|_{\Delta_{3}}\right\\}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\max\left\\{d(\vec{f},\vec{g}),\|\ln(f_{1}/g_{1})+\ln(f_{3}/g_{3})\|_{\Delta_{2}},d(\vec{f},\vec{g})\right\\}.$

The problem comes with the norm appearing in the last line because the natural
way to bound it is to use the triangle inequality obtaining

$\|\ln(f_{1}/g_{1})+\ln(f_{3}/g_{3})\|_{\Delta_{2}}\leq\|\ln(f_{1}/g_{1})\|_{\Delta_{2}}+\|\ln(f_{3}/g_{3})\|_{\Delta_{2}}\leq
2d(\vec{f},\vec{g}).$

Putting everything together, we get that $T$ is non expansive; that is,

(3.1) $d(\vec{T}\vec{f},\vec{T}\vec{g})\leq d(\vec{f},\vec{g}).$

(The previous arguments allow to deduce that $\vec{T}$ is non expansive for
any $m\geq 3$.)

Let us see that equality is actually attained. Indeed, taking
$\vec{f}\equiv(e,e,e)$, $\vec{g}\equiv(1,1,1)$, where $e$ denotes Euler’s
constant, the vector of harmonic functions

$\left(\ln(|T_{1}\vec{f}|/|T_{1}\vec{g}|),\ln(|T_{2}\vec{f}|/|T_{2}\vec{g}|),\ln(|T_{3}\vec{f}|/|T_{3}\vec{g}|)\right)$

in $\prod_{k=1}^{3}(\overline{\mathbb{C}}\setminus\Delta_{k})$ reduces to the
constant vector function $(1/2,1,1/2)$ in which case

$d(\vec{T}\vec{f},\vec{T}\vec{g})=1=d(\vec{f},\vec{g}).$

Similarly, one can prove that $\vec{T}$ is not contractive for any $m\geq 3$.
Therefore, there is an error in the statement of [2, Proposition 1.1].
However, the conclusion that $\vec{T}$ has a unique fixed point is correct
since (1.17) is true as we show next.

We return to the general case. Take two functions
$\vec{f}=(f_{1},\ldots,f_{m}),\vec{g}=(g_{1},\ldots,g_{m})\in{C}^{+}(\vec{\Delta})$.
For simplicity in writing, in the sequel we use the notation

$\ln\vec{f}:=(\ln f_{1},\ldots,\ln
f_{m}),\qquad{\vec{f}}/{\vec{g}}:=(f_{1}/g_{1},\ldots,f_{m}/g_{m}),$

and

$|{\vec{f}}/{\vec{g}}|:=(|f_{1}/g_{1}|,\ldots,|f_{m}/g_{m}|),\qquad\|\ln({\vec{f}}/{\vec{g}})\|_{\vec{\Delta}}:=d(\vec{f},\vec{g}).$

According to the definition of the metric $d$, we have

(3.2)
$d(\vec{T}^{\overline{m}}\vec{f},\vec{T}^{\overline{m}}\vec{g})=\|\ln(|\vec{T}^{\overline{m}}\vec{f}|/|\vec{T}^{\overline{m}}\vec{g}|)\|_{\vec{\Delta}}.$

Indeed, the $k$-th component $T_{k}\vec{T}^{\overline{m}-1}\vec{f}$ of
$\vec{T}^{\overline{m}}\vec{f}$ is a Szegő function in
$\overline{\mathbb{C}}\setminus\Delta_{k}$ which is real and positive on
$\mathbb{R}\setminus\Delta_{k}$. Consequently,

$\ln
T_{k}\vec{T}^{\overline{m}-1}\vec{f}=\ln|T_{k}\vec{T}^{\overline{m}-1}\vec{f}|\qquad\mbox{on}\qquad{\mathbb{R}\setminus\Delta_{k}}.$

However, $\ln|T_{k}\vec{T}^{\overline{m}-1}\vec{f}|$ extends to a harmonic
function on $\overline{\mathbb{C}}\setminus\Delta_{k}$ and

$\|\ln(|T_{k}\vec{T}^{\overline{m}-1}\vec{f}|/|T_{k}\vec{T}^{\overline{m}-1}\vec{g}|)\|_{(\Delta_{k-1}\cup\Delta_{k+1})}=\|\ln|T_{k}\vec{T}^{\overline{m}-1}\vec{f}|-\ln|T_{k}\vec{T}^{\overline{m}-1}\vec{g}|\|_{(\Delta_{k-1}\cup\Delta_{k+1})}.$

Consequently, the norm $\|\cdot\|_{\vec{\Delta}}$ of
$\ln(\vec{T}^{\overline{m}}\vec{f}/\vec{T}^{\overline{m}}\vec{g})$ coincides
with the norm of
$\ln(|\vec{T}^{\overline{m}}\vec{f}|/|\vec{T}^{\overline{m}}\vec{g}|)$ whose
components are the harmonic functions
$\ln(|T_{k}\vec{T}^{\overline{m}-1}\vec{f}|/|T_{k}\vec{T}^{\overline{m}-1}\vec{g}|)$.
We must find these harmonic functions.

Fix $k=1,\ldots,m$. For each
$u_{k}\in\mathcal{C}({\Delta_{k-1}\cup\Delta_{k+1}})$ (the space of continuous
functions on ${\Delta_{k-1}\cup\Delta_{k+1}}$), we define $P_{j,k}(u_{k})$,
$j=k-1,k+1$, to be the harmonic function on
$\overline{\mathbb{C}}\setminus\Delta_{j}$ whose boundary values on
$\Delta_{j}$ coincide with the values of $u_{k}$ on $\Delta_{j}$ (recall that
$\Delta_{0}=\Delta_{m+1}=\varnothing$). Let us introduce the matrix map

(3.3) $P:=\left(\begin{array}[]{cccccc}0&P_{1,2}&0&\cdots&0&0\\\
P_{2,1}&0&P_{2,3}&\cdots&0&0\\\ 0&P_{3,2}&0&\ddots&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\ 0&0&0&\ddots&0&P_{m-1,m}\\\
0&0&0&\cdots&P_{m,m-1}&0\end{array}\right).$

(Only the entries in the sub-diagonal and supra-diagonal differ from $0$.) Set
$u_{k}=\ln(f_{k}/g_{k})$.

In the following, $(\cdot)^{t}$ denotes the transpose of the vector $(\cdot)$.
Using (1.16), we have

$(\ln|\vec{T}^{\overline{m}}\vec{f}|/|\vec{T}^{\overline{m}}\vec{g}|)^{t}=\frac{1}{2}P(\ln|\vec{T}^{\overline{m}-1}\vec{f}|/|\vec{T}^{\overline{m}-1}\vec{g}|)^{t}=\cdots=$
$\frac{1}{2^{\overline{m}-1}}P^{\overline{m}-1}(\ln|\vec{T}\vec{f}/\vec{T}\vec{g}|)^{t}=\frac{1}{2^{\overline{m}}}P^{\overline{m}}(u_{1},\ldots,u_{m})^{t}.$

This formula allows us to bound
$\|\ln(|\vec{T}^{\overline{m}}\vec{f}|/|\vec{T}^{\overline{m}}\vec{g}|)\|_{\vec{\Delta}}$.

To illustrate, let us consider first the cases when $m=3,4$ and thus
$\overline{m}=2$. We have

$\left(\begin{array}[]{ccc}0&P_{1,2}&0\\\ P_{2,1}&0&P_{2,3}\\\
0&P_{3,2}&0\end{array}\right)^{2}\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\end{array}\right)=\left(\begin{array}[]{c}P_{1,2}P_{2,1}u_{1}+P_{1,2}P_{2,3}u_{3}\\\
P_{2,1}P_{1,2}u_{2}+P_{2,3}P_{3,2}u_{2}\\\
P_{3,2}P_{2,1}u_{1}+P_{3,2}P_{2,3}u_{3}\end{array}\right)$

and

$\left(\begin{array}[]{cccc}0&P_{1,2}&0&0\\\ P_{2,1}&0&P_{2,3}&0\\\
0&P_{3,2}&0&P_{3,4}\\\
0&0&P_{4,3}&0\end{array}\right)^{2}\left(\begin{array}[]{c}u_{1}\\\ u_{2}\\\
u_{3}\\\
u_{4}\end{array}\right)=\left(\begin{array}[]{c}P_{1,2}P_{2,1}u_{1}+P_{1,2}P_{2,3}u_{3}\\\
P_{2,1}P_{1,2}u_{2}+P_{2,3}P_{3,2}u_{2}+P_{2,3}P_{3,4}u_{4}\\\
P_{3,2}P_{2,1}u_{1}+P_{3,2}P_{2,3}u_{3}+P_{3,4}P_{4,3}u_{3}\\\
P_{4,3}P_{3,2}u_{2}+P_{4,3}P_{3,4}u_{4}\end{array}\right).$

Using (3.2), the maximum principle for harmonic functions, and the triangle
inequality, when $m=3$ we get

$4d(\vec{T}^{2}\vec{f},\vec{T}^{2}\vec{g})=$
$\max\\{\|P_{1,2}P_{2,1}u_{1}+P_{1,2}P_{2,3}u_{3}\|_{\Delta_{2}},\|P_{2,1}P_{1,2}u_{2}+P_{2,3}P_{3,2}u_{2}\|_{\Delta_{1}\cup\Delta_{3}},\|P_{3,2}P_{2,1}u_{1}+P_{3,2}P_{2,3}u_{3}\|_{\Delta_{2}}\\}$
$\leq\max\\{\|P_{1,2}P_{2,1}u_{1}+P_{1,2}P_{2,3}u_{3}\|_{\Delta_{1}},\|P_{2,1}P_{1,2}u_{2}+P_{2,3}P_{3,2}u_{2}\|_{\Delta_{2}},\|P_{3,2}P_{2,1}u_{1}+P_{3,2}P_{2,3}u_{3}\|_{\Delta_{3}}\\}$
$\leq\max\\{\|P_{1,2}P_{2,1}u_{1}\|_{\Delta_{1}}+\|P_{1,2}P_{2,3}u_{3}\|_{\Delta_{1}},\|P_{2,1}P_{1,2}u_{2}\|_{\Delta_{2}}+\|P_{2,3}P_{3,2}u_{2}\|_{\Delta_{2}},$
$\|P_{3,2}P_{2,1}u_{1}\|_{\Delta_{3}}+\|P_{3,2}P_{2,3}u_{3}\|_{\Delta_{3}}\\}$

Using the maximum principle for harmonic functions it is easy to deduce that
any one of the norms appearing in the last two lines can be bounded by
$d(\vec{f},\vec{g})$. For example,

$\|P_{1,2}P_{2,3}u_{3}\|_{\Delta_{1}}\leq\|P_{2,3}u_{3}\|_{\Delta_{2}}\leq\|\ln(f_{3}/g_{3})\|_{\Delta_{3}}\leq
d(\vec{f},\vec{g})$

where the last equality follows from the fact that $\ln(f_{3}/g_{3})$ is one
of the components of the vector $(u_{1},u_{2},u_{3})$ whose norm coincides
with $d(\vec{f},\vec{g})$. Consequently,

$d(\vec{T}^{2}\vec{f},\vec{T}^{2}\vec{g})\leq\frac{1}{2}d(\vec{f},\vec{g})$

and $1/2$ is the value assigned to $\gamma_{3}$.

When $m=4$, reasoning as above you get

$4d(\vec{T}^{2}\vec{f},\vec{T}^{2}\vec{g})\leq 3d(\vec{f},\vec{g})$

which is equivalent to (1.17) with $\gamma_{4}=3/4$. The $3$ in the previous
inequality comes from the fact that the second and third entries of
$P^{2}(u_{1},\ldots,u_{4})^{t}$ have three terms while the first and last only
two.

Following the ideas expressed in the two examples, in order to estimate
$\|\ln(|\vec{T}^{\overline{m}}\vec{f}|/|\vec{T}^{\overline{m}}\vec{g}|)\|_{\vec{\Delta}}$
all we have to do is to count the number of terms in each component of the
vector $P^{\overline{m}}(u_{1},\ldots,u_{m})^{t}$ and divide the largest
number by $2^{\overline{m}}$ to get $\gamma_{m}$.

The component $r$ of $P^{\overline{m}}(u_{1},\ldots,u_{m})^{t}$ is the sum of
all the expressions of the form

$P_{\alpha_{1},\beta_{1}}P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}$

satisfying the rules:

  * •

$\alpha_{1}=r$,

  * •

$\beta_{i}=\alpha_{i}\pm 1,1\leq\beta_{i}\leq m,i=1,\ldots,{\overline{m}}$,

  * •

$\alpha_{i+1}=\beta_{i},i=1,\ldots,{\overline{m}}-1$.

If this is so, notice that using the maximum principle for harmonic functions
and (1.16) we have

$\|P_{\alpha_{1},\beta_{1}}P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{r-1}\cup\Delta_{r+1}}\leq\|P_{\alpha_{1},\beta_{1}}P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{\alpha_{2}}}\\\
=\|P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{\alpha_{2}}}\leq\|P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{\beta_{2}}}\leq\cdots\leq\\\
\|P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{\alpha_{\overline{m}}}}\leq\|P_{\alpha_{\overline{m}},\beta_{\overline{m}}}u_{\beta_{\overline{m}}}\|_{\Delta_{\beta_{\overline{m}}}}=\|u_{\beta_{\overline{m}}}\|_{\Delta_{\beta_{\overline{m}}}}\leq
d(\vec{f},\vec{g}).$

The assertion of the previous sentence is a consequence of the definition of
the product of two matrices and the structure of the matrix $P$. To prove it
we introduce some terminology. We say that an expression of the form
$P_{\alpha_{1},\beta_{1}}P_{\alpha_{2},\beta_{2}}\cdots
P_{\alpha_{k},\beta_{k}}$ is a chain with $k$ links. Only the entries of the
form $P_{k,k\pm 1}$, where $k+1\leq m,k-1\geq 1$, of $P$ are different from
zero. The first subindex indicates the row and the second subindex the column
where they are located. When we multiply $P$ by itself the entry $P_{k,k+1}$
with $k+1\leq m$ gets multiplied by all the entries of $P$ in row $k+1$, but
the only products that survive (are different from zero) are the chains with
two links $P_{k,k+1}P_{k+1,k+2}$ (provided $k+2\leq m$) and
$P_{k,k+1}P_{k+1,k}$ which are located in $P^{2}$ in the positions $(k,k+2)$
and $(k,k)$, respectively. The entry $P_{k,k-1}$ with $k-1\geq 1$ gets
multiplied by all the entries of $P$ in row $k-1$, but the only products that
survive are the chains with two links $P_{k,k-1}P_{k-1,k}$ and
$P_{k,k-1}P_{k-1,k-2}$ (provided $k-2\geq 1$) which are located in $P^{2}$ in
the positions $(k,k)$ and $(k,k-2)$, respectively. We can go on in the same
manner arriving to the rules stated above.

Summarizing, whenever $2\leq\beta_{i}\leq{m}-1$, the link
$P_{\alpha_{i},\beta_{i}}$ may be followed by $P_{\beta_{i},\beta_{i}+1}$ or
$P_{\beta_{i},\beta_{i}-1}$; otherwise, there is only one option, either
$P_{1,2}$ when $\beta_{i}=1$ or $P_{m,m-1}$ when $\beta_{i}=m$. Consequently,
the central components of $P^{\overline{m}}(u_{1},\ldots,u_{m})^{t}$ are the
ones with the largest number of terms since $1$ and $m$ are furthest away from
the initial value $r$ determined by the component in question.

When $m$ is odd, the central component is the ${\overline{m}}$-th one. In that
case, the number of terms in the ${\overline{m}}$-th component of
$P^{\overline{m}}(u_{1},\ldots,u_{m})^{t}$ equals $2^{\overline{m}}-2$ because
there are only two chains with ${\overline{m}}-1$ links (namely
$P_{{\overline{m}},{\overline{m}}-1}P_{{\overline{m}}-1,{\overline{m}}-2}\cdots
P_{2,1}$ and
$P_{{\overline{m}},{\overline{m}}+1}P_{{\overline{m}}+1,{\overline{m}}+2}\cdots
P_{m-1,m}$), which can be extended with only one link. Therefore,
$\gamma_{m}=(2^{\overline{m}}-2)/2^{\overline{m}}$.

If $m$ is even the central components are the ${\overline{m}}$ and
${\overline{m}}+1$ ones. In the first case, only one chain with
${\overline{m}}-1$ links can be extended with only one link, namely
$P_{{\overline{m}},{\overline{m}}-1}P_{{\overline{m}}-1,{\overline{m}}-2}\cdots
P_{2,1}$. In the second case, the only chain with ${\overline{m}}-1$ links
which can be extended with only one link is
$P_{{\overline{m}}+1,{\overline{m}}+2}P_{{\overline{m}}+2,{\overline{m}}+3}\cdots
P_{m-1,m}$. Thus, the total number of terms in these two components is
$2^{\overline{m}}-1$. This implies that
$\gamma_{m}=(2^{\overline{m}}-1)/2^{\overline{m}}$. With this we conclude the
proof. $\Box$

Incidentally, for any fixed $m$ the maps
$T^{{\overline{m}}^{\prime}},{\overline{m}}^{\prime}\geq 2,$ can be defined.
They are contractive if and only if
${\overline{m}}^{\prime}\geq{\overline{m}}$. The constants of contraction are
(not necessarily strictly) decreasing with ${\overline{m}}^{\prime}$ and tend
to zero as ${\overline{m}}^{\prime}\to\infty$.

###  3.2. Proof of Theorem 1.4

First, we will deal with the asymptotics of the polynomials $Q_{\vec{n},j}$,
$j=1,\ldots,m$ and then we will obtain the asymptotics of the ML Hermite-Padé
polynomials $a_{\vec{n},j}$.

In order to obtain the strong asymptotics of the polynomials $Q_{\vec{n},j}$
we consider the particular case of the operator $\vec{T}_{\vec{w}}$ for the
vector weight

(3.4)
$\vec{w}(x)=(\sqrt{(b_{1}-x)(x-a_{1})}\tilde{h}_{1}(x)\sigma_{1}^{\prime}(x),\ldots,\sqrt{(b_{m}-x)(x-a_{m})}\tilde{h}_{m}(x)\sigma_{m}^{\prime}(x))$

where $\tilde{h}_{m}\equiv 1$ and the functions $\tilde{h}_{j}$,
$j=1,\ldots,m-1$ are given by (2.19). Since $\vec{w}$ is fixed throughout this
subsection, to simplify the notation, from now on $\vec{T}_{\vec{w}}=\vec{T}$.

Let $\vec{\mathsf{G}}=(\mathsf{G}_{1},\ldots,\mathsf{G}_{m})$ be the fixed
point of the operator $\vec{T}=(T_{1},\ldots,T_{m})$. As above, let
$\mathcal{P}_{\vec{n},k}$ be the set of all monic polynomials of degree
$\eta_{\vec{n},k}$ with real coefficients whose zeros lie in
$\mathbb{C}\setminus(\Delta_{k-1}\cup\Delta_{k+1})$. As usual
$\Delta_{0}=\Delta_{m+1}\equiv\varnothing$. Set

$H^{+,\vec{n}}:=\left\\{\left(\frac{\mathsf{G}_{1}(\infty)P_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)P_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right):P_{\vec{n},k}\in\mathcal{P}_{\vec{n},k},\,k=1,\ldots,m\right\\}.$

Notice that
$\frac{\mathsf{G}_{k}(\infty)P_{\vec{n},k}(x)}{\Phi_{k}^{\eta_{\vec{n},k}}(x)}>0,x\in\Delta_{k-1}\cup\Delta_{k+1}$.

Let $\tilde{T}_{\vec{n},k}$, $k=1,\ldots,m$, be the operators defined on
$\mathcal{P}_{\vec{n}}=\mathcal{P}_{\vec{n},1}\times\cdots\times\mathcal{P}_{\vec{n},m}$
which determine the components of $\tilde{T}_{\vec{n}}$; namely,

$\tilde{T}_{\vec{n}}=(\tilde{T}_{\vec{n},1},\ldots,\tilde{T}_{\vec{n},m}).$

Define $T_{\vec{n}}:H^{+,\vec{n}}\to H^{+,\vec{n}}$ where

$T_{\vec{n}}\left(\frac{\mathsf{G}_{1}(\infty)P_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)P_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)=\\\
\left(\mathsf{G}_{1}(\infty)\frac{\tilde{T}_{\vec{n},1}(P_{\vec{n},1},\ldots,P_{\vec{n},m})}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\mathsf{G}_{m}(\infty)\frac{\tilde{T}_{\vec{n},m}(P_{\vec{n},1},\ldots,P_{\vec{n},m})}{\Phi_{m}^{\eta_{\vec{n},m}}}\right).$

Notice that any fixed point of $T_{\vec{n}}$ generates a fixed point of
$\tilde{T}_{\vec{n}}$ and reciprocally any fixed point of
$\tilde{T}_{\vec{n}}$ generates a fixed point of ${T}_{\vec{n}}$. The
continuity of $T_{\vec{n}}$ follows from the continuity of
$\tilde{T}_{\vec{n}}$.

###### Proof.

Let $\vec{g}_{0}^{(\vec{n})}$ be the fixed point of $T_{\vec{n}}$. This means
that

$\vec{g}_{0}^{(\vec{n})}=\left(\frac{\mathsf{G}_{1}(\infty)Q_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)Q_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)$

where $(Q_{\vec{n},1},\ldots,Q_{\vec{n},m})$ is the unique fixed point of the
operator $\tilde{T}_{\vec{n}}$, see Lemma 2.2. We will prove that

(3.5)
$\lim_{\vec{n}\in\Lambda}\|\vec{g}_{0}^{(\vec{n})}-\vec{\mathsf{G}}\|_{\vec{\Delta}}=0.$

Recall that the components of $\vec{\mathsf{G}}$ are Szegő functions on
$\Omega_{k}=\overline{\mathbb{C}}\setminus\Delta_{k}$, $k=1,\ldots,m$.

In order to prove this, we will show that any neighborhood of
$\vec{\mathsf{G}}$ in the $\|\cdot\|_{\vec{\Delta}}$ norm contains a fixed
point of the operator $T_{\vec{n}}$ for all $|\vec{n}|\geq n_{0}$,
$\vec{n}\in\Lambda$. This is done using Brouwer’s fixed point theorem.

Fix $\delta>0$. For each $k=1,\ldots,m$ set

$\Delta_{k,\delta}:=\\{z\in\mathbb{C}:\min_{x\in\Delta_{k}}|z-x|\leq\delta\\}.$

Take $\delta$ sufficiently small so that

$\Delta_{k,\delta}\cap(\Delta_{k-1}\cup\Delta_{k+1})=\varnothing.$

Recall that $\Delta_{0}=\varnothing=\Delta_{m+1}$. Set

$\Omega_{k,\delta}:=\overline{\mathbb{C}}\setminus\Delta_{k,\delta},$

which obviously contains $\Delta_{k-1}\cup\Delta_{k+1}$. Denote

$\vec{\Omega}_{\delta}:=(\Omega_{1,\delta},\ldots,\Omega_{m,\delta}).$

Let $H^{+}(\vec{\Omega}_{\delta})$ denote the cone of all $m$-tuples
$(g_{1},\ldots,g_{m})$ of vector functions such that $g_{k}$, $k=1,\ldots,m,$
is holomorphic in $\Omega_{k,\delta}$, symmetric (with respect to the real
line), and positive on $\Omega_{k,\delta}\cap\mathbb{R}$. For
$\vec{g}=(g_{1},\ldots,g_{m})\in H^{+}(\vec{\Omega}_{\delta})$, we define

$\|\vec{g}\|_{\vec{\Omega}_{\delta}}:=\max\\{\sup\\{|g_{k}(z)|:z\in\Omega_{k,\delta}\\}:k=1,\ldots,m\\}\,\,(\leq\infty)$

and

$\min_{\vec{\Delta}}\vec{g}:=\min\\{\min\\{g_{k}(z):z\in\Delta_{k-1}\cup\Delta_{k+1}\\}:k=1,\ldots,m\\},$

where $\Delta_{0}=\Delta_{m+1}=\varnothing$.

Fix a constant $C\geq 1$ so that $C\geq
2\|\vec{\mathsf{G}}\|_{\vec{\Omega}_{\delta}}$ and
$C^{-1}\leq\frac{1}{2}\min_{\vec{\Delta}}\vec{\mathsf{G}}$. Define

$H^{+}(\vec{\Omega}_{\delta},C):=\\{\vec{g}\in
H^{+}(\vec{\Omega}_{\delta}):\|\vec{g}\|_{\vec{\Omega}_{\delta}}\leq
C,\;\min_{\vec{\Delta}}\vec{g}\geq C^{-1}\\}.$

Obviously, $\vec{\mathsf{G}}\in H^{+}(\vec{\Omega}_{\delta},C)$.

Let $(\vec{\rho}_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence of elements
in $H^{+}(\vec{\Omega}_{\delta},C)$. For each $k=1,\ldots,m$ the sequence of
functions made up by the $k$-th component of $\vec{\rho}_{k}$ is uniformly
bounded on $\Omega_{k,\delta}$. Therefore, there exists
$\mathbb{I}\subset\mathbb{N}$ such that $(\vec{\rho}_{n})_{n\in\mathbb{I}}$
converges componentwise to some vector function $\vec{\rho}$ uniformly on each
compact subset of $\vec{\Omega}_{\delta}$, The $k$-th component of
$\vec{\rho}$ is, therefore, holomorphic and symmetric on $\Omega_{k,\delta}$,
and non-negative on $\Omega_{k,\delta}\cap\mathbb{R},k=1,\ldots,m$.
Additionally,

$\min_{\vec{\Delta}}\vec{\rho}=\lim_{n\in\mathbb{I}}\min_{\vec{\Delta}}\vec{\rho}_{n}\geq
C^{-1},$

and

$\|\vec{\rho}\|_{\vec{\Omega}_{\delta}}=\lim_{n\in\mathbb{I}}\|\vec{\rho}_{n}\|_{\vec{\Omega}_{\delta}}\leq
C.$

Consequently, $\vec{\rho}\in H^{+}(\vec{\Omega}_{\delta},C)$. We conclude that
$H^{+}(\vec{\Omega}_{\delta},C)$ is compact with the topology of uniform
convergence on compact subsets on the space $H(\vec{\Omega}_{\delta})$ of
vector analytic functions on $\vec{\Omega}_{\delta}$.

Fix an arbitrary $\theta>0$. Let

$\omega(\theta)=\\{\vec{g}\in
H^{+}(\vec{\Omega}_{\delta},C):\|\vec{g}-\vec{\mathsf{G}}\|_{\vec{\Delta}}\leq\theta\\}.$

Analogously, for every $\varepsilon>0$, set

$\omega_{\varepsilon}:=\\{\vec{g}\in
H^{+}(\vec{\Omega}_{\delta},C):d(\vec{g},\vec{\mathsf{G}})\leq\varepsilon\\}.$

There exists $\varepsilon_{0}$ such that

$\omega_{\varepsilon}\subset\omega(\theta),\qquad
0<\varepsilon\leq\varepsilon_{0},$

for, otherwise, we could find a sequence of vector functions in
$H^{+}(\vec{\Omega}_{\delta},C)\subset\mathcal{C}^{+}(\vec{\Delta})$ which
converges to $\vec{\mathsf{G}}$ in the $d$ metric but not in the
$\|\cdot\|_{\vec{\Delta}}$ norm which would contradict (1.15).

Consider

$\omega_{\varepsilon,\vec{n}}:=\omega_{\varepsilon}\cap H^{+,\vec{n}}.$

Let $\mu_{k}$ be the representing measure of $\mathsf{G}_{k}$; that is,
$\mathsf{G}_{k}(z)=\mathsf{G}_{k}(\mu_{k},z)$, $z\in\Omega_{k}$. Taking into
account Lemma 2.6, in particular (2.28) and (1.15), we get

$\lim_{\vec{n}\in\Lambda}d(\vec{\tilde{g}}_{1}^{(\vec{n})},\vec{\mathsf{G}})=0,$

where

$\vec{\tilde{g}}_{1}^{(\vec{n})}:=\left(\frac{\mathsf{G}_{1}(\infty)\tilde{Q}_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)\tilde{Q}_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)$

and the $\tilde{Q}_{\vec{n},j}$, $j=1,\ldots,m,$ are the orthonormal
polynomials $\tilde{q}_{\vec{n},j}$ given in (2.26) re-normalized to be monic.
Because of (2.28) we conclude that, for every $\varepsilon$,
$0<\varepsilon<\varepsilon_{0},$ there exists $n_{0}$ such that
$\omega_{\varepsilon,\vec{n}}\neq\varnothing$ for $\vec{n}\in\Lambda$,
$|\vec{n}|>n_{0}$.

Let $H(\vec{\Omega})$ be the locally convex space of all vector analytic
functions on $\vec{\Omega}=(\Omega_{1},\ldots,\Omega_{m})$ with the topology
of uniform convergence on compact subsets of $\vec{\Omega}$. Let us show that
for each $\vec{n}\in\Lambda$ fixed, $\omega_{\varepsilon,\vec{n}}$ is a
compact subset of $H(\vec{\Omega})$ with this topology.

For $\vec{n}\in\Lambda$ fixed, consider an arbitrary sequence

$(\rho_{n,1},\ldots,\rho_{n,m})_{n\in\mathbb{N}}:=\left(\frac{\mathsf{G}_{1}(\infty){P}_{n,1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty){P}_{n,m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)_{n\in\mathbb{N}}\subset\omega_{\varepsilon,\vec{n}}.$

We must show that it has a subsequence which converges to an element of
$\omega_{\varepsilon,\vec{n}}$.

Fix $k\in\\{1,\ldots,m\\}$. Consider the sequence of monic polynomials with
real coefficients $(P_{n,k})_{n\in\mathbb{N}}$ of degree $\eta_{\vec{n},k}$
whose zeros lie in $\mathbb{C}\setminus(\Delta_{k-1}\cup\Delta_{k+1})$. Since
$\omega_{\epsilon,\vec{n}}\subset H^{+}(\vec{\Omega}_{\delta},C)$ it readily
follows that the zeros of the polynomials $P_{n,k}$ remain uniformly bounded
away from $\Delta_{k-1}\cup\Delta_{k+1}\cup\\{\infty\\}$ in $n\in\mathbb{N}$.
Therefore, there exists a subsequence of indices $I\subset\mathbb{N}$ and a
monic polynomial $P_{k}$ with real coefficients of degree $\eta_{\vec{n},k}$,
whose zeros lie on $\mathbb{C}\setminus(\Delta_{k-1}\cup\Delta_{k+1})$, such
that $\lim_{n\in I}P_{n,k}=P_{k}$ uniformly on every compact subset of
$\mathbb{C}$. Since this is true for every $k=1,\ldots,m$, it readily follows
that there exists a subsequence $I^{\prime}\subset\mathbb{N}$ such that

$\lim_{n\in
I^{\prime}}(\rho_{n,1},\ldots,\rho_{n,m})=\left(\frac{\mathsf{G}_{1}(\infty){P}_{1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty){P}_{m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)\in\omega_{\varepsilon,\vec{n}},$

uniformly on every compact subset of $\vec{\Omega}$. With this we conclude the
proof that $\omega_{\varepsilon,\vec{n}}$ is compact as claimed.

Fix $\vec{n}\in\Lambda$. Let us show that $\omega_{\varepsilon,\vec{n}}$ is
convex. Take two arbitrary vector functions in $\omega_{\varepsilon,\vec{n}}$

$\vec{p}_{j}=\left(\frac{\mathsf{G}_{1}(\infty){P}_{j,1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty){P}_{j,m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right),\qquad
j=1,2.$

Let us show that for every $\beta,0\leq\beta\leq 1$

$\beta\vec{p}_{1}+(1-\beta)\vec{p}_{2}\in\omega_{\varepsilon,\vec{n}}.$

Obviously, $\beta\vec{p}_{1}+(1-\beta)\vec{p}_{2}\in H^{+,\vec{n}}\cap
H^{+}(\vec{\Omega}_{\delta},C)$. In particular, for each $k=1,\ldots,m,$

(3.6)
${\mathsf{G}_{k}(\infty)(\beta{P}_{1,k}(x)+(1-\beta)P_{2,k}(x))}{\Phi_{k}^{-\eta_{\vec{n},k}}(x)}>0,\qquad
x\in\Delta_{k-1}\cup\Delta_{k+1}.$

It remains to prove that

(3.7)
$\left\|\ln\left(\frac{\mathsf{G}_{k}(\infty)(\beta{P}_{1,k}+(1-\beta)P_{2,k})}{\Phi_{k}^{\eta_{\vec{n},k}}G_{k}}\right)\right\|_{\Delta_{k-1}\cup\Delta_{k+1}}\leq\varepsilon.$

For each $x\in\Delta_{k-1}\cup\Delta_{k+1}$ fixed,
$\beta{P}_{1,k}(x)+(1-\beta)P_{2,k}(x)$ is an intermediate point between
${P}_{1,k}(x)$ and ${P}_{2,k}(x)$. This fact, together with (3.6), implies
that for $x\in\Delta_{k-1}\cup\Delta_{k+1}$

$0<\min\left\\{\frac{\mathsf{G}_{k}(\infty)({P}_{1,k}(x))}{\Phi_{k}^{\eta_{\vec{n},k}(x)}G_{k}(x)},\frac{\mathsf{G}_{k}(\infty)({P}_{2,k}(x))}{\Phi_{k}^{\eta_{\vec{n},k}(x)}G_{k}(x)}\right\\}\leq$
$\frac{\mathsf{G}_{k}(\infty)(\beta{P}_{1,k}(x)+(1-\beta)P_{2,k}(x))}{\Phi_{k}^{\eta_{\vec{n},k}(x)}G_{k}(x)}\leq\max\left\\{\frac{\mathsf{G}_{k}(\infty)({P}_{1,k}(x))}{\Phi_{k}^{\eta_{\vec{n},k}(x)}G_{k}(x)},\frac{\mathsf{G}_{k}(\infty)({P}_{2,k}(x))}{\Phi_{k}^{\eta_{\vec{n},k}(x)}G_{k}(x)}\right\\}.$

Using the monotonicity of the logarithm (3.7) readily follows.

Now we need to show that
$T_{\vec{n}}(\omega_{\varepsilon,\vec{n}})\subset\omega_{\varepsilon,\vec{n}}$
for all $\vec{n}\in\Lambda$ with $|\vec{n}|$ large enough. Indeed, there exist
$0<\varepsilon^{\prime}<\varepsilon$ and $n_{0}$ such that for all
$\vec{n}\in\Lambda,|\vec{n}|\geq n_{0}$, we have

(3.8)
$T_{\vec{n}}(\omega_{\varepsilon^{\prime},\vec{n}})\subset\omega_{\varepsilon,\vec{n}}.$

If we assume the contrary, we could find a sequence
$(\vec{g}^{(\vec{n})})_{\vec{n}\in\Lambda^{\prime}}$,
$\Lambda^{\prime}\subset\Lambda$,

$\vec{g}^{(\vec{n})}=\left(\frac{\mathsf{G}_{1}(\infty)\tilde{Q}_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)\tilde{Q}_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)\in\omega_{\varepsilon,\vec{n}},$

such that

$\lim_{\vec{n}\in\Lambda^{\prime}}d(\vec{\mathsf{G}},\vec{g}^{(\vec{n})})=0,\qquad\mbox{and}\qquad
d(\vec{\mathsf{G}},T_{\vec{n}}(\vec{g}^{(\vec{n})}))=d(T(\vec{\mathsf{G}}),T_{\vec{n}}(\vec{g}^{(\vec{n})}))\geq\varepsilon.$

Since $(\vec{g}^{(\vec{n})})_{\vec{n}\in\Lambda^{\prime}}\subset
H^{+}(\vec{\Omega}_{\delta},C)$ it follows that for every $j=1,\ldots,m$ the
zeros of $(\tilde{Q}_{\vec{n},j})_{\vec{n}\in\Lambda^{\prime}}$ remain
uniformly bounded away from $\Delta_{j-1}\cup\Delta_{j+1}$. Indeed, if we
assume the contrary, taking a convenient subsequence, we would conclude that
$\mathsf{G}_{j}$ must have some zero in $\Delta_{j-1}\cup\Delta_{j+1}$.
Therefore, we can apply Lemma 2.5 and from (2.25) and (1.15) it follows that

$\lim_{\vec{n}\in\Lambda^{\prime}}d(T(\vec{\mathsf{G}}),{T}_{\vec{n}}(\vec{g}^{(\vec{n})}))=0.$

which is not possible because the limit value should have to be
$\geq\varepsilon$. So, such $\varepsilon^{\prime}>0$ exists.

It remains to prove that

(3.9)
$T_{\vec{n}}(\omega_{\varepsilon,\vec{n}}\setminus\omega_{\varepsilon^{\prime},\vec{n}})\subset\omega_{\varepsilon,\vec{n}}.$

for all $\vec{n}\in\Lambda$ with $|\vec{n}|$ sufficiently large. An inmediate
consequence of the maximum principle is that the operator $\vec{T}$ is non-
expansive, see [2, Proposition 1.1] or (3.1) for the special case $m=3$.
Therefore,

$d(\vec{\mathsf{G}},\vec{T}(\vec{g}))=d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))\leq
d(\vec{\mathsf{G}},\vec{g}),\qquad\vec{g}\in\omega_{\varepsilon}.$

Consequently,

$\sup_{\vec{g}\in\omega_{\varepsilon}\setminus\omega_{\varepsilon^{\prime}}}\frac{d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))}{d(\vec{\mathsf{G}},\vec{g})}\leq
1.$

Consider

$\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}=\\{\vec{g}\in\omega_{\varepsilon}\setminus\omega_{\varepsilon^{\prime}}:g_{k}(\infty)=\mathsf{G}_{k}(\infty),k=1,\ldots,m\\}.$

Suppose that

$\sup_{\vec{g}\in\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}}\frac{d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))}{d(\vec{\mathsf{G}},\vec{g})}=1.$

Then, there exist a sequence
$(\vec{g}_{n})_{n\in\mathbb{N}}\subset\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}$
such that

$\lim_{n\to\infty}\frac{d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}_{n}))}{d(\vec{\mathsf{G}},\vec{g}_{n})}=1,$

a subsequence $(\vec{g}_{n})_{n\in\mathbb{I}\subset\mathbb{N}}$, and
$\vec{g}\in H^{+}(\vec{\Omega}_{\delta},C)$ such that

$\lim_{n\in\mathbb{I}}\|\vec{g}_{n}-\vec{g}\|_{\vec{\Omega}_{\delta}}=0,\qquad\lim_{n\in\mathbb{I}}d(\vec{T}(\vec{g}_{n}),\vec{T}(\vec{g}))=0,$

and

${d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))}={d(\vec{\mathsf{G}},\vec{g})}.$

That is

(3.10)
$\|\ln(\vec{T}(\vec{g})/\vec{T}(\vec{\mathsf{G}}))\|_{\vec{\Delta}}=\|\ln(\vec{g}/\vec{\mathsf{G}})\|_{\vec{\Delta}}.$

We will show that this implies that $\vec{g}=\vec{\mathsf{G}}$ which is not
possible because $d(\vec{g},\vec{\mathsf{G}})\geq\varepsilon^{\prime}>0$.

Let
$\vec{u}:=\ln(\vec{g}/\vec{\mathsf{G}})=(\ln(g_{1}/\mathsf{G}_{1}),\ldots,\ln(g_{m}/\mathsf{G}_{m}))$
be the vector function whose $k$-th component $u_{k}$ is the restriction to
$\Delta_{k-1}\cup\Delta_{k+1}$ of $\ln(g_{k}/G_{k})$. By the definition of the
operator $\vec{T}$, we have
$\ln(|\vec{T}(\vec{g})|/|\vec{T}(\vec{\mathsf{G}})|)$ is the vector function
whose $j$-th component is the harmonic function on $\Omega_{j}$ whose boundary
values on $\Delta_{j}$ coincide with the restriction on $\Delta_{j}$ of
$(u_{j-1}+u_{j+1})/2$. If (3.10) takes place, there exists $k$, $1\leq k\leq
m$ such that (recall the definition of the operators $P_{j,k}$ just before the
introduction of the matrix (3.3))

$\displaystyle\|\vec{u}\|_{\vec{\Delta}}=$
$\displaystyle\frac{1}{2}\|P_{k,k-1}u_{k-1}+P_{k,k+1}u_{k+1}\|_{\Delta_{k-1}\cup\Delta_{k+1}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(\|P_{k,k-1}u_{k-1}\|_{\Delta_{k-1}\cup\Delta_{k+1}}+\|P_{k,k+1}u_{k+1}\|_{\Delta_{k-1}\cup\Delta_{k+1}}\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\left(\|P_{k,k-1}u_{k-1}\|_{\Delta_{k}}+\|P_{k,k+1}u_{k+1}\|_{\Delta_{k}}\right)\leq\|\vec{u}\|_{\vec{\Delta}}.$

If $k=1$ or $k=m$, you actually get
$\|\vec{u}\|_{\vec{\Delta}}\leq\frac{1}{2}\|\vec{u}\|_{\vec{\Delta}}$ because
in all the intermediate steps there is only one term, so
$\|u\|_{\vec{\Delta}}=0$ and we would be done.

Now, for any other value of $k$ the chain of inequalities implies that there
is equality on each step; in particular

$\|P_{k,k-1}u_{k-1}\|_{\Delta_{k-1}\cup\Delta_{k+1}}=\|P_{k,k-1}u_{k-1}\|_{\Delta_{k}}=\|\vec{u}\|_{\vec{\Delta}},\qquad$

and

$\|P_{k,k+1}u_{k+1}\|_{\Delta_{k-1}\cup\Delta_{k+1}}=\|P_{k,k+1}u_{k+1}\|_{\Delta_{k}}=\|\vec{u}\|_{\vec{\Delta}}.$

We conclude that $P_{k,k-1}u_{k-1}$ and $P_{k,k+1}u_{k+1}$ are constant
because they are harmonic functions in $\Omega_{k}$ which attain their maximum
at interior points. The constant value they take is
$\|\vec{u}\|_{\vec{\Delta}}$ (or $-\|\vec{u}\|_{\vec{\Delta}}$). In turn this
implies that $u_{k+1}$ and $u_{k-1}$ restricted to $\Delta_{k}$ are constantly
equal to $\|\vec{u}\|_{\vec{\Delta}}$ (or $-\|\vec{u}\|_{\vec{\Delta}}$).
Consequently,

$\ln(g_{k+1}(x)/\mathsf{G}_{k+1}(x))=\ln(g_{k-1}(x)/\mathsf{G}_{k-1}(x))=\|\vec{u}\|_{\vec{\Delta}},\qquad
x\in\Delta_{k}.$

However, $\ln(g_{k+1}/\mathsf{G}_{k+1})\in
H^{+}(\Omega_{\delta,k+1}),\ln(g_{k-1}/\mathsf{G}_{k-1})\in
H^{+}(\Omega_{\delta,k-1})$. Since they are constant on the interval
$\Delta_{k}$ of their regions of holomorphy, they are constantly equal to
$\|u\|_{\vec{\Delta}}$. Now,

$g_{k+1}(\infty)=\mathsf{G}_{k+1}(\infty)\qquad\mbox{and}\qquad
g_{k-1}(\infty)=\mathsf{G}_{k-1}(\infty).$

It follows that

$\|\vec{u}\|_{\vec{\Delta}}=\ln(g_{k+1}(\infty)/\mathsf{G}_{k+1}(\infty))=\ln(g_{k-1}(\infty)/\mathsf{G}_{k-1}(\infty))=0$

Therefore, $\vec{g}\equiv\vec{\mathsf{G}}$, which is impossible as pointed out
above. This means that

$\sup_{\vec{g}\in\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}}\frac{d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))}{d(\mathsf{G},\vec{g})}=\gamma<1.$

In other words,

(3.11) $d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))\leq\gamma
d(\vec{\mathsf{G}},\vec{g}),\qquad\vec{g}\in\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}.$

Now, we are going to show that there exists $n_{0}$ such that for all
$\vec{n}\in\Lambda$ with $|\vec{n}|>n_{0}$ and
$\vec{g}\in\omega_{\varepsilon,\vec{n}}$, we have

(3.12) $d(T_{\vec{n}}(\vec{g}),\vec{T}(\vec{g}))<(1-\gamma)\varepsilon.$

Indeed, if we assume the contrary, we can find
$(\vec{g}^{(\vec{n})})_{\vec{n}\in\Lambda^{\prime}}$,
$\Lambda^{\prime}\subset\Lambda$,
$\vec{g}^{(\vec{n})}\in\omega_{\varepsilon,\vec{n}}$ such that

(3.13)
$d(T_{\vec{n}}(\vec{g}^{(\vec{n})}),\vec{T}(\vec{g}^{(\vec{n})}))\geq(1-\gamma)\varepsilon.$

Since $\omega_{\varepsilon,\vec{n}}\subset H^{+}(\vec{\Omega}_{\delta},C)$ the
sequence $(\vec{g}^{(\vec{n})})_{\vec{n}\in\Lambda^{\prime}}$ is uniformly
bounded in the norm $\|\cdot\|_{\vec{\Omega}_{\delta}}$. Therefore, there
exists a function $\vec{g}\in H^{+}(\vec{\Omega}_{\delta},C)$ and a
subsequence $\mathbf{\Lambda}^{\prime\prime}\subset\Lambda^{\prime}$ such that

$\lim_{\vec{n}\in\Lambda^{\prime\prime}}\|\vec{g}^{(\vec{n})}-\vec{g}\|_{\vec{\Omega}_{\delta}}=0.$

In particular,

$\lim_{\vec{n}\in\Lambda^{\prime\prime}}\|\vec{g}^{(\vec{n})}-\vec{g}\|_{\vec{\Delta}}=0,$

and (2.24) in Lemma 2.5 takes place with
$\vec{f}_{\vec{n}}=\vec{g}^{(\vec{n})}$, $\vec{f}=\vec{g}$, and
$\Lambda=\Lambda^{\prime\prime}$. For each $j=1,\ldots,m$, the zeros of the
$j$-th component of the vector functions $\vec{g}^{(\vec{n})}$ remain
uniformly bounded away from $\Delta_{j-1}\cup\Delta_{j+1}$ for
$\vec{n}\in\Lambda^{\prime\prime}$ . Indeed, the contrary implies that the
$j$-th component of $\vec{g}$ has a zero on $\Delta_{j-1}\cup\Delta_{j+1}$
which is not the case. According to (2.25)

$\lim_{\vec{n}\in\mathbf{\Lambda}^{\prime\prime}}\|T_{\vec{n}}(\vec{g}^{(\vec{n})})-\vec{T}(\vec{g})\|_{\vec{\Delta}}=0$

which contradicts (3.13) due to (1.15) and the continuity of the operator
$\vec{T}$.

For
$\vec{g}\in\omega_{\varepsilon,\vec{n}}\setminus\omega_{\varepsilon^{\prime},\vec{n}}\subset\tilde{\omega}_{\varepsilon,\varepsilon^{\prime}}$,
on account of (3.11) and (3.12), we obtain

$d(\vec{\mathsf{G}},T_{\vec{n}}(\vec{g}))\leq
d(\vec{T}(\vec{\mathsf{G}}),\vec{T}(\vec{g}))+d(\vec{T}(\vec{g}),T_{\vec{n}}(\vec{g}))\leq\gamma
d(\vec{\mathsf{G}},\vec{g})+(1-\gamma)\varepsilon<\varepsilon.$

This implies (3.9) which together with (3.8) give that there exists $n_{0}$
such that

$T_{\vec{n}}(\omega_{\varepsilon,\vec{n}})\subset\omega_{\varepsilon,\vec{n}},\qquad\vec{n}\in\Lambda,\qquad|\vec{n}|\geq
n_{0}.$

We have shown that for each fixed $\vec{n}\in\Lambda$, $|\vec{n}|>n_{0}$,
$\omega_{\varepsilon,\vec{n}}$ is a non-empty convex, compact subset of the
locally convex space $H(\vec{\Omega})$ of vector analytic functions on
$\vec{\Omega}=(\Omega_{1},\ldots,\Omega_{m})$ with the topology of uniform
convergence on compact subsets of $\vec{\Omega}$, and $T_{\vec{n}}$ is a
continuous operator which maps $\omega_{\varepsilon,\vec{n}}\subset
H(\vec{\Omega})$ into itself; therefore, Brouwer’s fixed point theorem asserts
that $T_{\vec{n}}$ has at least one fixed point in
$\omega_{\varepsilon,\vec{n}}$ (see [22, Theorem V.19]). This combined with
Lemma 2.2 gives us that the fixed point is unique and equals

$\left(\frac{\mathsf{G}_{1}(\infty)Q_{\vec{n},1}}{\Phi_{1}^{\eta_{\vec{n},1}}},\cdots,\frac{\mathsf{G}_{m}(\infty)Q_{\vec{n},m}}{\Phi_{m}^{\eta_{\vec{n},m}}}\right)\in\omega_{\varepsilon,\vec{n}},$

where $(Q_{\vec{n},1},\ldots,Q_{\vec{n},m})$ is the vector whose components
are the polynomials whose roots coincide with the zeros of the forms
$\mathcal{A}_{\vec{n},j}$, $j=1,\ldots,m$. Since $\varepsilon$ can be taken
arbitrarily small (3.5) follows and the statements (1.12) and (1.13) of
Theorem 1.4 hold using again Lemma 2.5.

From [18, Prop. 1.2] and [12, Th 1.1], we know that for $j=0,\ldots,m-1$

$\lim_{\vec{n}\in\Lambda}\frac{a_{\vec{n},j}(z)}{a_{\vec{n},m}(z)}=\widehat{s}_{m,j+1}(z)$

uniformly on compact subsets of $\Omega_{m}$. Since $a_{\vec{n},m}$ coincides
with $Q_{\vec{n},m}$ we have

$\lim_{\vec{n}\in\Lambda}\frac{a_{\vec{n},j}(z)}{\Phi_{m}^{|\vec{n}|}(z)}\frac{\Phi_{m}^{|\vec{n}|}(z)}{Q_{\vec{n},m}(z)}=\widehat{s}_{m,j+1}(z).$

Now, using (1.12), (1.14) follows and we conclude the proof of Theorem 1.4. ∎

With the asymptotic behavior of the vector sequence
$(Q_{\vec{n},1},\ldots,Q_{\vec{n},m}),\vec{n}\in\Lambda,$ it is easy to derive
the asymptotics of the linear forms $\mathcal{A}_{\vec{n},j}$,
$j=0,\ldots,m-1$.

######  Corollary 3.1.

Suppose that the assumptions of Theorem 1.4 hold and the
$\mathcal{A}_{\vec{n},j}$, $j=0,\ldots,m-1$, are defined by (1.4). Let
$\varepsilon_{\vec{n},j+1}$, $j=0,\ldots,m-1$ denote the constant sign which
the varying measure
$\frac{\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)}$
adopts on $\Delta_{j+1}$. Then,

(3.14)
$\lim_{\vec{n}\in\Lambda}{\varepsilon_{\vec{n},1}K_{\vec{n},0}^{2}\Phi_{1}^{\eta_{\vec{n},1}}(z)\mathcal{A}_{\vec{n},0}(z)}=\frac{\mathsf{G}_{1}(\infty)}{\mathsf{G}_{1}(z)}\frac{1}{\sqrt{(z-a_{1})(z-b_{1})}},$

uniformly on compact subsets of $\overline{\mathbb{C}}\setminus\Delta_{1}$,
and for $j=1,\ldots,m-1$

(3.15)
$\lim_{\vec{n}\in\Lambda}\frac{\varepsilon_{\vec{n},j+1}K_{\vec{n},j}^{2}\Phi_{j+1}^{\eta_{\vec{n},j+1}}(z)\mathcal{A}_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}_{j}(z)}{\mathsf{G}_{j}(\infty)}\frac{\mathsf{G}_{j+1}(\infty)}{\mathsf{G}_{j+1}(z)}\frac{1}{\sqrt{(z-a_{j+1})(z-b_{j+1})}},$

uniformly on compact subsets of
$\overline{\mathbb{C}}\setminus(\Delta_{j}\cup\Delta_{j+1})$.

###### Proof.

From (2.5) and the definition of $\mathcal{H}_{\vec{n},j}$ we have

$\mathcal{A}_{\vec{n},j}(z)=\frac{Q_{\vec{n},j}(z)}{Q_{\vec{n},j+1}(z)}\int\frac{Q_{\vec{n},j+1}^{2}(x)}{z-x}\frac{\mathcal{H}_{\vec{n},j+1}(x)\operatorname{d}\sigma_{j+1}(x)}{Q_{\vec{n},j}(x)Q_{\vec{n},j+2}(x)},\qquad
j=0,\ldots,m-1,$

where the equality holds in $\Omega_{j+1}$. Due to (2.8),

$\frac{\varepsilon_{\vec{n},j+1}K_{\vec{n},j}^{2}\mathcal{A}_{\vec{n},j}(z)}{(\Phi_{j}^{\eta_{\vec{n},j}}/\Phi_{j+1}^{\eta_{\vec{n},j+1}})(z)}=\frac{Q_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}\frac{\Phi_{j+1}^{\eta_{\vec{n},j+1}}(z)}{Q_{\vec{n},j+1}(z)}h_{\vec{n},j}(z).$

Now, from Lemma 2.3 and Theorem 1.4, (3.15) is immediate. For (3.14) take into
account that $Q_{\vec{n},0}\equiv 1,\Phi_{0}\equiv 1,$ and
$\Delta_{0}=\varnothing$. ∎

One may be tempted to think that condition (1.5) on the sequence of multi-
indices $\Lambda$ in Theorem 1.4 is too restrictive. The fact is that the
strong asymptotics of multilevel Hermite-Padé polynomials is very sensitive to
the way the sequence of multi-indices is taken. For example, if $\Lambda$
verifies (1.5) and we define ${\Lambda}^{l}$ as

${\Lambda}^{l}:=\\{{\vec{n}}^{l}\in(\mathbb{Z}_{+}^{m})^{*}:{\vec{n}}^{l}=\vec{n}+\vec{e}_{l},\vec{n}\in\Lambda\\},$

where $\vec{e}_{l}$ is the unitary vector with $1$ in $l$-th component, then
for each $l$ fixed there exists

(3.16)
$\lim_{\vec{n}^{l}\in\Lambda^{l}}\frac{Q_{\vec{n}^{l},j}(z)}{\Phi_{j}^{\eta_{\vec{n}^{l},j}}(z)},\qquad
j=1,\ldots,m,$

but the limiting functions will depend on $l$. To prove this, write

$\frac{Q_{\vec{n}^{l},j}(z)}{\Phi_{j}^{\eta_{\vec{n}^{l},j}}(z)}=\frac{Q_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}\frac{\Phi_{j}^{\eta_{\vec{n},j}}(z)}{\Phi_{j}^{\eta_{\vec{n}^{l},j}}(z)}\frac{Q_{\vec{n}^{l},j}(z)}{Q_{\vec{n},j}(z)}.$

Since $\Lambda$ verifies (1.5)

$\lim_{\vec{n}\in\Lambda}\frac{Q_{\vec{n},j}(z)}{\Phi_{j}^{\eta_{\vec{n},j}}(z)}=\frac{\mathsf{G}_{j}(z)}{\mathsf{G}_{j}(\infty)},$

uniformly on each compact subset of
$\overline{\mathbb{C}}\setminus\Delta_{j}$. The second factor satisfies

$\frac{\Phi_{j}^{\eta_{\vec{n},j}}(z)}{\Phi_{j}^{\eta_{\vec{n}^{l},j}}(z)}=\left\\{\begin{array}[]{cc}1,&j=1,\ldots,l-1,\\\
1/\Phi_{j}(z),&j=l,\ldots,m.\end{array}\right.$

Finally, using [12, Theorem 3.4]

$\lim_{\vec{n}\in\Lambda}\frac{Q_{\vec{n}^{l},j}(z)}{Q_{\vec{n},j}(z)}=\frac{{F}_{j}^{(l)}(z)}{{F}_{j}^{(l)}(\infty)},$

uniformly on each compact subset of
$\overline{\mathbb{C}}\setminus\Delta_{j}$, where
${F}_{j}^{(l)},j=1,\ldots,m,$ is the solution of a system of boundary value
problems defined in [12, Lemma 3.3]. Consequently, the limit in (3.16) can be
given explicitly. We will not dwell into more details and simply mention that
using [12, Theorem 3.4] one can extend Theorem 1.4 to sequences of multi-
indices of the form

$\Lambda^{\vec{v}}:=\\{\vec{n}+\vec{v}:\vec{n}\in\Lambda\\},$

where $\Lambda\subset(\mathbb{Z}_{+}^{m})^{*}$ verifies (1.5) and
$\vec{v}\in\mathbb{Z}_{+}^{m}$ is fixed.

###  3.3. Rate of convergence of ML Hermite-Padé approximants

In [18], the author obtains the logarithmic asymptotics of the linear forms
$\mathcal{A}_{\vec{n},j}$, $j=0,\ldots,m-1,$ and with it estimates the rate of
convergence of the rational functions $\frac{a_{\vec{n},j}}{a_{\vec{n},m}}$,
$j=0,\ldots,m-1,$ as $|\vec{n}|\to\infty$, along ray sequences of multi-
indices. For straight ray sequences and Nikishin systems with Szegő’s
condition, using the results on strong asymptotics, we are able to give the
exact rate of convergence.

######  Corollary 3.2.

Under the assumptions of Theorem 1.4, we have

(3.17)
$\lim_{\vec{n}\in\Lambda}\frac{\varepsilon_{\vec{n},m}K_{\vec{n},m-1}^{2}\Phi_{m}^{2|\vec{n}|}(z)}{\Phi_{m-1}^{\eta_{\vec{n},m-1}}(z)}\left(\frac{a_{\vec{n},j}}{a_{\vec{n},m}}-\widehat{s}_{m,j+1}\right)(z)=\frac{\mathsf{G}_{m-1}(z)\mathsf{G}_{m}^{2}(\infty)}{\mathsf{G}_{m}^{2}(z)\mathsf{G}_{m-1}(\infty)}\frac{(-1)^{m-1}\widehat{s}_{m-1,j+1}(z)}{\sqrt{(z-a_{m})(z-b_{m})}},$

uniformly on compact subsets of
$\overline{\mathbb{C}}\setminus\cup_{i=j+1}^{m}\Delta_{i}$.

###### Proof.

Taking into account [15, Lemma 2.1] with
$\mathcal{L}_{j}=\mathcal{A}_{\vec{n},j}$, $j=0,1,\ldots,m-2$ and $r=m-1$, we
have (for all points where the expression is meaningful)

$\mathcal{A}_{\vec{n},j}+\sum_{k=j+1}^{m-1}(-1)^{k-j}\widehat{s}_{k,j+1}\mathcal{A}_{\vec{n},k}=(-1)^{j}(a_{\vec{n},j}-a_{\vec{n},m}\widehat{s}_{m,j+1}).$

Though [15] is dedicated to a special case of ML Hermite-Padé approximation
the forms $\mathcal{L}_{j}$ have a general character and that lemma is also
applicable in the present situation. Dividing the above equality by
$Q_{\vec{n},m}=a_{\vec{n},m}$, we obtain

$\frac{\mathcal{A}_{\vec{n},j}}{Q_{\vec{n},m}}+\sum_{k=j+1}^{m-1}(-1)^{k-j}\widehat{s}_{k,j+1}\frac{\mathcal{A}_{\vec{n},k}}{Q_{\vec{n},m}}=(-1)^{j}\left(\frac{a_{\vec{n},j}}{a_{\vec{n},m}}-\widehat{s}_{m,j+1}\right).$

This is equivalent to

(3.18)
$\frac{\mathcal{A}_{\vec{n},m-1}}{Q_{\vec{n},m}}\left(\frac{\mathcal{A}_{\vec{n},j}}{\mathcal{A}_{\vec{n},m-1}}+\sum_{k=j+1}^{m-1}(-1)^{k-j}\widehat{s}_{k,j+1}\frac{\mathcal{A}_{\vec{n},k}}{\mathcal{A}_{\vec{n},m-1}}\right)=(-1)^{j}\left(\frac{a_{\vec{n},j}}{a_{\vec{n},m}}-\widehat{s}_{m,j+1}\right).$

From formula (1.8) in [18, Proposition 1.2], it follows that the ratios
$\mathcal{A}_{\vec{n},k}/\mathcal{A}_{\vec{n},m-1}$, $k=j,\ldots,m-2$,
converge uniformly to zero on compact subsets of
$\overline{\mathbb{C}}\setminus\cup_{i=j+1}^{m}\Delta_{i}$; consequently,

(3.19)
$\lim_{\vec{n}\in\Lambda}\frac{\mathcal{A}_{\vec{n},j}}{\mathcal{A}_{\vec{n},m-1}}+\sum_{k=j+1}^{m-1}(-1)^{k-j}\widehat{s}_{k,j+1}\frac{\mathcal{A}_{\vec{n},k}}{\mathcal{A}_{\vec{n},m-1}}=(-1)^{m-1-j}\widehat{s}_{m-1,j+1},$

on compact subsets of
$\overline{\mathbb{C}}\setminus\cup_{i=j+1}^{m}\Delta_{i}$. On the other hand,
from Corollary 3.1 and (1.12) we have

(3.20)
$\lim_{n\in\Lambda}\frac{\varepsilon_{\vec{n},m}K_{\vec{n},m-1}^{2}\Phi_{m}^{2|\vec{n}|}(z)}{\Phi_{m-1}^{\eta_{\vec{n},m-1}}(z)}\frac{\mathcal{A}_{\vec{n},m-1}(z)}{Q_{\vec{n},m}(z)}=\frac{\mathsf{G}_{m-1}(z)\mathsf{G}_{m}^{2}(\infty)}{\mathsf{G}_{m}^{2}(z)\mathsf{G}_{m-1}(\infty)}\frac{1}{\sqrt{(z-a_{m})(z-b_{m})}},$

uniformly on compact subsets of
$\overline{\mathbb{C}}\setminus(\Delta_{m-1}\cup\Delta_{m})$.

Therefore, from (3.19) and (3.20), we get

$\lim_{\vec{n}\in\Lambda}\frac{\varepsilon_{\vec{n},m}K_{\vec{n},m-1}^{2}\Phi_{m}^{2|\vec{n}|}(z)}{\Phi_{m-1}^{\eta_{\vec{n},m-1}}(z)}\frac{\mathcal{A}_{\vec{n},m-1}(z)}{Q_{\vec{n},m}(z)}\left(\frac{\mathcal{A}_{\vec{n},j}}{\mathcal{A}_{\vec{n},m-1}}+\sum_{k=j+1}^{m-1}(-1)^{k-j}\widehat{s}_{k,j+1}\frac{\mathcal{A}_{\vec{n},k}}{\mathcal{A}_{\vec{n},m-1}}\right)(z)=$
$\frac{\mathsf{G}_{m-1}(z)\mathsf{G}_{m}^{2}(\infty)}{\mathsf{G}_{m}^{2}(z)\mathsf{G}_{m-1}(\infty)}\frac{(-1)^{m-1-j}\widehat{s}_{m-1,j+1}(z)}{\sqrt{(z-a_{m})(z-b_{m})}},$

on compact subsets of $\overline{\mathbb{C}}\setminus\cup_{i=j+1}^{m}$ which
together with (3.18) give us (3.17). ∎

###  3.4. Strong asymptotics of Cauchy biorthogonal polynomials

In this subsection, we apply Theorem 1.4 to obtain the strong asymptotics of
Cauchy biorthogonal polynomials.

As above consider the Nikishin system
$\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$ where
$\operatorname{supp}\sigma_{k}\subset\Delta_{k}$ and the intervals
$\Delta_{k}$ are bounded. Define the kernel function

$K(x_{1},x_{m})=\int_{\Delta_{2}}\int_{\Delta_{3}}\cdots\int_{\Delta_{m-1}}\frac{\,\operatorname{d}\sigma_{m-1}(x_{m-1})\cdots\,\operatorname{d}\sigma_{3}(x_{3})\operatorname{d}\sigma_{2}(x_{2})}{(x_{m-1}-x_{m})(x_{m-2}-x_{m-1})\cdots\,(x_{2}-x_{3})(x_{1}-x_{2})}.$

When $m=2$, take

$K(x_{1},x_{2})\equiv 1.$

There exist two sequences of monic polynomials
$(P_{n}),(Q_{n}),n\in\mathbb{Z}_{+},$ such that for each $n$,
$\deg(P_{n})=\deg(Q_{n})=n$ and

$\int_{\Delta_{1}\times\Delta_{m}}P_{k}(x_{1})K(x_{1},x_{m})Q_{n}(x_{m})\,\mbox{d}\sigma_{1}(x_{1})\,\mbox{d}\sigma_{m}(x_{m})=C_{n}\delta_{k,n},\qquad
C_{n}\neq 0.$

As usual, $\delta_{k,n}=0$, $k\neq n$, $\delta_{n,n}=1$. The polynomials
$P_{n}$ and $Q_{n}$ are called Cauchy biorthogonal polynomials with respect to
$(\sigma_{1},\ldots,\sigma_{m})$.

When $m=2$ biorthogonal polynomials appear in the analysis of the two matrix
model [4, 6] and in the search of discrete solutions of the Degasperis-Procesi
equation [5] through a Hermite-Padé approximation problem for two discrete
measures. In [6] a result on the strong asymptotics of Cauchy biorthogonal
polynomials was obtained for a pair of Laguerre type weights with unbounded
support and exponential decay at infinity. Using Theorem 1.4 we can give the
strong asymptotics of general Cauchy biorthogonal polynomials with respect to
a system of measures with compact support.

In [10, Lemma 2.4] it was proved that the $n$-th biorthogonal polynomial
$Q_{n}$ with respect to $(\sigma_{1},\ldots,\sigma_{m})$ coincides with the ML
Hermite-Padé polynomial $a_{\vec{n},m}$ of the Nikishin system
$\mathcal{N}(\sigma_{1},\ldots,\sigma_{m})$ associated with the multi-index
$\vec{n}=(n,0,\ldots,0)$. Because of the symmetry of the kernel it turns out
that the biorthogonal polynomial $P_{n}$ coincides with the corresponding
$a_{\vec{n},m}$ constructed for the reversed Nikishin system
$\mathcal{N}(\sigma_{m},\ldots,\sigma_{1})$ and the multi-index
$\vec{n}=(n,0,\ldots,0)$ (see [10, Subsection 2.3]. Let us focus on $Q_{n}$.
From Theorem 1.4, we obtain

######  Corollary 3.3.

Assume that $\vec{\sigma}\in S(\vec{\Delta})$. Then

$\lim_{n\to\infty}\frac{Q_{n}(z)}{\Phi_{m}^{n}(z)}=\frac{\mathsf{G}_{m}(z)}{\mathsf{G}_{m}(\infty)},$

where $\Phi_{m}$ is given by (1.10) for the vector equilibrium problem on the
system of intervals $\vec{\Delta}$ with interaction matrix

$\left(\begin{array}[]{r r r r r}1&-1/2&0&\cdots&0\\\ -1/2&1&-1/2&\ddots&0\\\
0&-1/2&1&\ddots&0\\\ \vdots&\ddots&\ddots&\ddots&\vdots\\\
0&0&0&\cdots&1\end{array}\right)_{m\times m}$

and $\mathsf{G}_{m}$ is the last component of the fixed point of the operator
$\vec{T}_{\vec{w}}$ for the vector weight (3.4) where $\tilde{h}_{m}\equiv 1$
and the functions $\tilde{h}_{j}$, $j=1,\ldots,m-1$ are given by (2.19).

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