On overconvergent subsequencs of closed to rows classical Pade' approximants

TL;DR

This paper investigates the overconvergence behavior of classical Pade' approximants for power series with positive radius of convergence, extending classical results and analyzing subsequences with specific growth conditions.

Contribution

It extends classical overconvergence results to broader classes of Pade' approximants and subsequences with specific growth constraints on m(n).

Findings

Overconvergence occurs for certain subsequences of Pade' approximants.

Extended classical results by Hadamard and Ostrowski to new contexts.

Provided conditions under which overconvergence is observed.

Abstract

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/\log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lo'pez Lagomasino and A. Ferna'ndes Infante concerning overconvergent subsequences of a fixed row of the Pade' table.

Equations

$$f(z):=\sum _{j=0}^{\infty }{f}_j{z}^j$$

$$(fq-p)(z)=0(z^{n+m+1}).$$

$${\pi }_{n,m}:=P_{n,m}/Q_{n,m},$$

$$\sigma (e):=\inf \left\{\sum _{\nu }|V_{\nu }|\right\}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\Vert f-{\pi }_{n,m}{\Vert }_{K(\epsilon )}^{1/n}=\frac{{\max }_K|z|}{R_m}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\Vert f-{\pi }_{n,m_n}{\Vert }_{K(\epsilon )}^{1/n}\le \frac{{\max }_K|z|}{R(f)}$$

$$\underset{n\to \infty }{\mathrm{limsup}}\Vert f-{\pi }_{n,m_n}{\Vert }_{K(\epsilon )}^{1/n}=\frac{{\max }_{z\in K}|z|}{R(f)}$$

$${\pi }_{n,m_n}=\frac{P_{n,m_n}}{Q_{n,m_n}}:={\pi }_n=P_n/Q_n,$$

$$f(z)-{\pi }_n(z)=0(z^{n+m_n+1-{\tau }_n}).$$

$$Q_n(z)=\prod (z-{\zeta }_{n,l}^{\ast })\prod (1-\frac{z}{{\tilde{\zeta }}_{n,l}})$$

$$P_n(z)=a_n{z}^{deg{P}_n}+\cdots .$$

$${\pi }_{n+1}(z)-{\pi }_n(z)=A_n\frac{z^{n+m_n+1-{\tau }_n}}{Q_n(z)Q_{n+1}(z)},$$

$$A_{n}=\left\{\begin{array}[]{ll}a_{n+1}(\prod\frac{-1}{\tilde{\zeta}_{n,k}})-a_{n}(\prod\frac{-1}{\tilde{\zeta}_{n+1}}),&m_{n+1}=m_{n}+1\\ a_{n+1}(\prod\frac{-1}{\tilde{\zeta}_{n,k}}),&m_{n+1}=m_{n}\\ \end{array}\right.$$

$$f(z)-{\pi }_{n,m}(z)=\sum _{k=n}^{\infty }\frac{A_k{z}^{k+m+1-{\tau }_k}}{Q_k(z)Q_{k+1}(z)},$$

$$f(z)-{\pi }_n(z)=\sum _{k=n}^{\infty }\frac{A_k{z}^{k+m_k+1-{\tau }_k}}{Q_k(z)Q_{k+1}(z)}$$

$$\underset{n\to \infty }{\mathrm{limsup}}|A_n{|}^{1/n}=1/R(f).$$

$$\underset{n\to \infty }{\mathrm{limsup}}|a_n{|}^{1/n}=1/R(f)$$

$$f_n=0for{n}_k<n\le n_{k^{\prime }}$$

$$n_k/n_k^{\prime }\to 0,k\to \infty ,k\to \infty .$$

$$\mathrm{limsup}{n}_k/n_k^{\prime }<1$$

$$\underset{n\in {\bigcup }_k(n_k,n_{k^{\prime }}]}{\mathrm{limsup}}|f_n{|}^{1/n}<1/R_0.$$

$${\pi }_{n,m}={\pi }_{n_k,m}for{n}_k<n\le n_{k^{\prime }}$$

$$\underset{k}{\lim }\frac{n_k}{n_k^{\prime }}=0,k\to \infty .$$

$$\underset{k}{\mathrm{limsup}}\frac{n_k}{n_k^{\prime }}<1,k\to \infty .$$

$$\underset{n\in \Lambda }{\mathrm{limsup}}|a_n{|}^{1/n}<1/R(f)$$

$$\underset{n\in \Lambda }{\mathrm{limsup}}|A_{n-1}{|}^{1/n}<1/R(f).$$

$$\underset{k}{\mathrm{liminf}}\frac{n_k}{n_k^{\prime }}>0.$$

$${\pi }_n={\pi }_{n_k}asn\in \bigcup _k(n_k,n_k^{\prime }].$$

$$n_k/n_{k^{\prime }}\to 0,ask\to \infty$$

$$\underset{k\to \infty }{\mathrm{limsup}}{n}_k/n_{k^{\prime }}<1,ask\to \infty$$

Full text

On overconvergent subsequences of closed to rows classical Padé approximants

Ralitza K.Kovacheva

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Bonchev str. 8, 1113 Sofia, Bulgaria, rkovach@math.bas.bg

Abstract: Let $f(z):=\sum f_{\nu}z^{\nu}$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Padé approximants $\{\pi_{n,m_{n}}\}$ associated with $f,$ where $m_{n}\leq m_{n+1}\leq m_{n}+1$ and $m_{n}=o(n/\log n),$ resp. $m_{n}=0(n)$ as $n\to\infty.$ We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. López Lagomasino and A. Fernándes Infante concerning overconvergent subsequences of a fixed row of the Padé table.

MSC: 41A21, 41A25, 30B30

Key words: *Padé approximants, overconvergence, meromorphic continuation, convergence in $\sigma$-content. *

Introduction

Let

[TABLE]

be a power series with positive radius of convergence $R_{0}(f):=R_{0},\,R_{0}>0$. By $f$ we will denote not only the sum of $f$ in $D_{R_{0}}:=\{z,|z|<R_{0}\}$ but also the holomorphic (analytic and single valued) function determined by the element $(f,D_{R_{0}}).$ Fix a nonnegative integer $m\,(m\in\mathbb{N})$ and denote by $R_{m}(f):=R_{m}$ the *radius of $m-$meromorphy * of $f$: that is the radius of the largest disk centered at the zero into which the power series $f$ admits a continuation as a meromorphic function with no more than $m$ poles (counted with regard to their multiplicities). As it is known (see [1]), $R_{m}>0$ iff $R_{0}>0.$ Analogously, we define the radius of meromorphy $R(f)$ as the radius of the greatest disk $D_{R}$ into which $f$ can be extended as a meromorphic function in $\mathbb{C}.$ Apparently, $R(f)\geq R_{m}\geq R_{0}$. We denote the meromorphic continuations again by $f.$

Given a pair $(n,m),\,n,m\in\mathbb{N}$, let $\pi_{n,m}$ be the classical Padé approximant of $f$ of order $(n,m)$. Recall that (see [2]) ${\pi}_{n,m}=p/q$, where $p,q$ are polynomials of degree $\leq n,m$ respectively and satisfy

[TABLE]

As it is well known ([2]), the Padé approximant $\pi_{n,m}$ always exists and is uniquely determined by the conditions above.

Set

[TABLE]

where $P_{n,m}$ and $Q_{n,m}$ are relatively prime polynomials (we write $(P_{n,m},Q_{n,m})=1).$

We recall the concept of *convergence in $\sigma$-content * (cf. [3]). Given a set $e\subset\mathbb{C},$ we put

[TABLE]

where the infimum is taken over all coverings $\{\bigcup V_{\nu}\}$ of $e$ by disks and $|V_{\nu}|$ is the diameter of the disk $V_{\nu}$. Let $\Omega$ be an open set in $\mathbb{C}$ and $\varphi$ a function defined in $\Omega$ with values in $\overline{\mathbb{C}}$. The sequence of functions $\{\varphi_{n}\}$, rational in $\Omega$, is said to converge in $\sigma-$content to a function $\phi$ inside $\Omega,$ if for each compact set $K\subset\Omega$ and $\varepsilon>0$ $\sigma\{z,z\in K,|\varphi_{n}(z)-\varphi(z)|>\varepsilon\}\to 0$ as $n\to 0.$ The sequence $\{\varphi_{n}\}$ converges to $\varphi,$ as $n\to\infty,$ $\sigma-$almost uniformly inside $\Omega$, if for any compact set $K\subset\Omega$ and every $\varepsilon>0$ $\{\varphi_{n}\}$ converges to $\varphi$ uniformly in the $max-$ norm on a set of the form $K\setminus K_{\varepsilon},$ where $\sigma(K_{\varepsilon})<\varepsilon$. Analogously, we define *convergence in Green’s capacity * and convergence almost uniformly in Green’s capacity inside $\Omega$. It follows from Cartan’s inequality $\hbox{cap}(e)\geq C\sigma(e)$ (see [16], Chp.3) that convergence in capacity implies $\sigma-$convergence. The reader is referred for details to [3].

The next result may be found in [1].

Theorem 1, ([1]): Given a power series (1) and a fixed integer $m\in\mathbb{N},$ suppose that $0<R_{m}<\infty.$

Then the sequence $\{\pi_{n,m}\},\,n\to\infty,m-$fixed converges $\sigma-$almost uniformly to $f$ inside $D_{R_{m}}$ and

[TABLE]

*for any compact subset $K$ of $D_{R_{m}}$ and $\varepsilon>0.$ *

(here $\|...\|_{K}$ stands for the $\max-$ norm on $K$.)

Theorem 1 generalizes the classical result of Montessus de Ballore about rows in the Padé table ([4]).

In the present paper, we will be concentrating on the case $\limsup_{n\to\infty}m_{n}=\infty$. If the sequence $\{m_{n}\}$ increases ”slowly enough”, i,e, if $m_{n}=o(n),$ (resp. $m_{n}=o(n/\log n)$ as $n\to\infty,$) then the following result is valid:

Theorem 2, ([3], Chpt.3): Given $f$ with $0<R(f)<\infty,$ let $m_{n}=o(n/logn),\,n\to\infty$. Then the sequence $\{\pi_{n,m_{n}}\}$ converges $\sigma-$almost uniformly to $f$ inside $D_{R(f)}.$

In case $m_{n}=o(n),\,n\to\infty$, the sequence $\{\pi_{n,m_{n}}\}$ converges to $f$ in Green’s capacity inside $D_{R(f)}.$

For any compact set $K\in D_{R(f)}$ and any $\varepsilon>0$

[TABLE]

In [5], the question about specifying the speed of convergence above was posed. It was shown that for a class of functions the following result is valid:

Theorem 3, ([5]): Given $f$ with $0<R(f)<\infty,$ let $m_{n}\leq n,m_{n}\leq m_{n+1}\leq m_{n}+1,\,m_{n}=o(n/\log n),\,n\to\infty.$ Suppose that $f$ has a multivalued singularity on $\partial{D_{R(f)}}.$

Then the sequence $\{\pi_{n,m_{n}}\}$ converges $\sigma-$almost uniformly to $f$ inside the disk $D_{R(f)}$ and

[TABLE]

for every compact set $K\subset D_{R(f)}$ and every $\varepsilon>0$.

Research devoted to imposing weaker conditions on the growth of the sequence $\{m_{n}\}$ as $n\to\infty$ was carried out by H.P. Blatt. It follows from his results that the statement of Theorem 2 remains valid if $m_{n}=o(n)$ as $n\to\infty.$ Furthermore, the sequence $\{\pi_{n,m_{n}}\}$ converges almost uniformly to $f$ in capacity inside $D_{R(f)}$ ( see the comprehensive paper [7]).

Let now the sequence $\{m_{n}\}$ of positive integers satisfy the conditions $m_{n}\leq n,\,m_{n}\leq m_{n+1}\leq m_{n}+1$. Set

[TABLE]

where $(P_{n},Q_{n})=1;$ $\hbox{deg}P_{n}\leq n,\,\hbox{deg}Q_{n}\leq m_{n}.$

Denote by $\tau_{n,m_{n}}:=\tau_{n}$ the defect of $\pi_{n};$ that is $\hbox{min}(n-\hbox{deg}P_{n},m_{n}-\hbox{deg}Q_{n}).$ Then the order of the zero of $f(z)-\pi_{n}$ at $z=0$ is not less than $n+m_{n}+1-\tau_{n}$ (see [2]); in other words

[TABLE]

Following the terminology of G. A. Baker, Jr. and P. Gr. Morris (see [9], p. 31), we say that the rational function $\pi_{n}$ exists iff $\tau_{n}=0.$

The zeros $\zeta_{n,l},0\leq l\leq m_{n}$ of the polynomial $Q_{n}$ are called free poles of the rational function $\pi_{n}.$ Let $\mu_{n}$ be the exact degree of $Q_{n},\,\mu_{n}\leq m_{n}.$ We shall always normalize $Q_{n}$ by the condition

[TABLE]

where $|\zeta_{n,l}^{*}|<2R(f)$ and $|\tilde{\zeta}_{n,l}|\geq 2R(f).$

Set

[TABLE]

Suppose that $\tau_{n}>0$ for some $n\in\mathbb{N}$ (comp. (3)). Then, by the block structure of the Padé table (see [2]) $\pi_{n-l,m-k}\equiv\pi_{n,m}$ if $\max(k,l)\leq\tau_{n}$. Suppose that $f(z)-\pi_{n}(z)=B_{n}z^{n+m_{n}+1-\tau_{n}}$ with $B_{n}\not=0$. Then $\tau_{n+1}=0$ and $\pi_{n}\not=\pi_{n+1}.$ The definition of Padé approximants leads to

[TABLE]

where

[TABLE]

It was shown in [1], Eq. 33 (see also [10]) that for a fixed $m\in\mathbb{N}$ the Padé approximant $\pi_{n,m},m-$ fixed converges, as $n\to\infty$, together with the series $\sum_{n=1}^{\infty}\frac{A_{n}z^{n+m+1-\tau_{n}}}{Q_{n}(z)Q_{n+1}(z)}$, i.e.,

[TABLE]

where $\limsup|A_{n}|^{1/n}=1/R_{m}$.

It is easy to check that under the conditions of Theorem 3 an analogous result holds also for sequences $\{\pi_{n,m_{n}}\},\,\{m_{n}\}-$ as in Theorem 3 (compare with (16) below). In other words,

[TABLE]

and

[TABLE]

It follows from (7) that, under the above conditions on the growth of the sequence $\{m_{n}\}$ as $n\to\infty$

[TABLE]

Basing on the block structure of the Padé table (see [2]), we will be assuming throughout the paper, that $\tau_{n}=0$ for all $n\in\mathbb{N}.$ Also, for the sake of simplicity, we assume that $\hbox{deg}P_{n}=n$ for all $n\in\mathbb{N}$.

Let $f(z)=\sum_{n=0}^{\infty}f_{n}z^{n}$ be given and suppose that $0<R(f)<\infty.$ Let $m_{n}=o(n/\ln n),\,m_{n}\leq m_{n+1}\leq m_{n}+1,\,n\to\infty.$ Set, as before, $\pi_{n}:=\pi_{n,m_{n}}.$ Suppose now that a subsequence $\{\pi_{n_{k}}\},\,n_{k}\in\Lambda\subset\mathbb{N}$ converges $\sigma-$almost uniformly inside some domain $U$ such that $U\supset D_{R(f)}$ and $\partial U\bigcap\partial{D_{R(f}}\not\equiv\emptyset.$ Following the classical terminology related to power series ([11]), we say that $\{\pi_{n_{k}}\}_{n_{k}\in\Lambda}$ is overconvergent. The original definitions and results, given for overconvergent sequences of Taylor polynomials, may be found in [11].

Theorem 4, [11], [12]: Given a power series $f=\sum f_{n}z^{n}$ with radius of holomorphy $R_{0},0<R_{0}<\infty$ and sequences $\{n_{k}\}$ and $\{n_{k}^{\prime}\}$ with $n_{k}<n_{k}^{\prime}\leq n_{k+1},\,k=1,2....,$ suppose that

either

a)

[TABLE]

and

[TABLE]

or

b)

[TABLE]

and

[TABLE]

Then

*a) the sequence of Taylor sums $\{S_{n_{k}}\}$ converges to $f$ , as $n_{k}\to\infty$ uniformly in the *$\max-$norm inside the largest domain in $\mathbb{C}$ into which $f$ is analytically continuable.

or

b) $\{S_{n_{k}}\}$ converges uniformly to $f$ inside neighborhoods of all regular points of $f$ on $\Gamma_{R_{0}}$.

(here $S_{n}(z)=\sum_{\nu=0}^{n}f_{\nu}z^{\nu}.$)

Ostrowski’s theorem was extended to Fourier series associated with orthogonal polynomials in [17] and to infinite series of Bessel and of multi-index Mittag-Lefler functions in [18].

Before presenting the next result, we introduce the term $G(f)$ as the largest domain in $\mathbb{C}$ into which $(f,D_{R_{0}})$ given by (1) admits a meromorphic continuation. More exactly, $G(f)$ is made up by the analytic continuation of the element $(f,D_{R_{O}})$ plus the points which are poles of the corresponding analytic function. Obviously, $D_{R(f)}\subseteq G(f).$ Further, we say that the point $z_{0}\in\partial D_{m}$ resp. $z_{0}\in\partial D_{R(F)}$ is regular, if $f$ is either holomorphic, or meromorphic in a neighborhood of $z_{0}.$

Theorem 5, [13]: Let $f(z)$ be a power series with positive radius of convergence and $m\in\mathbb{N}$ be a fixed number. Suppose that $R_{m}<\infty.$ Suppose that there are infinite sequences $\{n_{k}\}$ and $\{n_{k}^{\prime}\},n_{k}<n_{k}^{\prime}\leq n_{k+1},k=1,2,...$ such that

[TABLE]

Suppose, further, that either

a)

[TABLE]

or

b)

[TABLE]

Then

a) The sequence $\{\pi_{n_{k},m}\}$ converges to $f$, as $n_{k}\to\infty$, $\sigma-$ almost uniformly inside $G(f);$

or

*b) $\{\pi_{n_{k},m}\}$ converges to $f$, as $n_{k}\to\infty$, $\sigma-$ almost uniformly in a neighborhood of each point $z_{0}\in\Gamma_{R(f)}$ at which $f$ is regular *.

The results of [13] have been extended in [14] to the $m-$th row of a large class of multipoint Padé approximants, associated with regular compact sets $E$ in $\mathbb{C}$ and regular Borel measures supported by $E.$

2. Statement of the new results

In the present paper, we prove

Theorem 6: Given a power series $f$ with $R(f)\in(0,\infty)$ and a sequence of integers $\{m_{n}\}$ such that $m_{n}\leq n,m_{n}\leq m_{n+1}\leq m_{n}+1,\,m_{n}=o(n),\,n\to\infty$ assume that the subsequence $\{\pi_{n}\}_{n\in\Lambda},\,\Lambda\subset\mathbb{N}$ converges to a holomorphic, resp., meromorphic function in $\sigma-$content inside some domain $W$ such that $W\bigcap D_{R_{m}}^{c}\not\equiv\emptyset$.

Then

[TABLE]

Remark: If $m_{n}=m$ for all $n\in\mathbb{N}$, then under the conditions of Theorem 6

[TABLE]

Theorem 7: Given the power series $f$ with $0<R(f)<\infty$ and a sequence of integers $\{m_{n}\},\,m_{n}\leq n,m_{n}\leq m_{n+1}\leq m_{n}+1,m_{n}=o(n/\log n)$ as $n\to\infty$, suppose, that $f$ is regular at the point $z_{0}\in\Gamma_{R(f)}$. Suppose, also that there exist increasing sequences $\{n_{k}\}$ and $\{n_{k}^{\prime}\},$ $n_{k}<n_{k}^{\prime}\leq n_{k+1},$ such that $\limsup_{k\to\infty}\frac{n_{k}}{n_{k}^{\prime}}<1$ and $\limsup_{n\in\bigcup_{k\to\infty}[n_{k},n_{k}^{\prime}]}|a_{n}|^{1/n}<1/R(f).$ Let

[TABLE]

Then there is a neighborhood $U$ of $z_{0}$ such that the sequence $\{\pi_{n_{k}^{\prime}}\}$ converges to the function $f$ $\sigma-$almost uniformly inside $D_{R(f)}\bigcup U.$

The next result extends Theorem 5 to closed to row sequences of classical Padé approximants.

Theorem 8: Let $f$ be given by (1), $0<R(f)<\infty$ and $\{m_{n}\}$ be as in Theorem 7. Assume that $n_{k}<n_{k}^{\prime}\leq n_{k+1},\,k=1,2,...$ and

[TABLE]

Assume, further, that either

a)

[TABLE]

or

b)

[TABLE]

and $f$ is regular at the point $z_{0}\in\Gamma_{R(f)}$.

Then

*a) the sequence $\{\pi_{n_{k}}\}$ converges to $f$ *$\sigma-$almost uniformly inside $G(f)$

or

*b) there exists a neighborhood $U$ of $z_{0}$ such that the sequence $\{\pi_{n_{k}}\}$ converges to the function $f$ *$\sigma-$almost uniformly inside $D_{R(f)}\bigcup U.$

At the end, we provide a result dealing with overconvergent subsequences of the $m$th row of the classical Padé table.

Theorem 9: Let $f$ be given, $m\in\mathbb{N}$ be fixed and $R_{m}(f):=R_{m}\in(0,\infty).$ Suppose that the subsequence $\{\pi_{n_{k},m}\},\,m-$ fixed, converges, as $n_{k}\to\infty$, $\sigma-$alsmost uniformly inside a domain $U\supset D_{R_{m}},\,\partial U\bigcap\Gamma_{R_{m}}\not\equiv 0.$

Then there exists a sequence $\{l_{k}\},\,l_{k}\in\mathbb{N},\,0\leq l_{k}<n_{k}$ such that for $n_{k}-l_{k}\leq\nu\leq n_{k}$

[TABLE]

3. Proofs

Auxiliary

Given an open set $B$ in $\mathbb{C},$ we denote by ${\cal A}(B)$ the class of analytic and single valued functions in $B.$ We recall that a function $g$ is meromorphic at some point $z_{0}$, if there is a neighborhood $U$ of $z_{0}$ where $g$ is meromorphic, i.e. $g=\frac{G}{q}$ as $z\in U$, where $G\in{\cal A}(U),\,G(z_{0})\not=0$ and $q$ is a polynomial with $q(z_{0})=0.$ We will use the notation $g\in{\cal M}(U).$

In the sequel, $D_{R},\,R>0$ stands for the open disk $\{z,|z|<R\};\,\Gamma_{R}:=\partial D_{R}$, respectively; $D_{1}:=D,\,\Gamma:=\partial D$.

With the normalization (4) we have

[TABLE]

for every compact set $K\subset\mathbb{C}$, where $C=C(K)$ is independent on $n,\,0<C<\infty$. Under the condition $m_{n}=o(n),\,n\to\infty$ we have, for every $\Theta>0$ and $n$ large enough

[TABLE]

In what follows, we will denote by $C$ positive constants, independent on $n$ and different at different occurrences (they may depend on all other parameters that are involved) The same convention applies to $C_{i},i=1,2,....$

We take an arbitrary $\varepsilon$ an define the open sets

[TABLE]

We have $\sigma(\Omega(\varepsilon))<\varepsilon$ and $\Omega_{\varepsilon_{1}}\subset\Omega_{\varepsilon_{2}}$ for $\varepsilon_{1}<\varepsilon_{2}.$ For any set $K\subset\mathbb{C}$ we put $K(\varepsilon):=K\setminus\Omega(\varepsilon).$

Let $m_{n}=o(n/\ln n)$ as $n\to\infty$ and $\Theta$ be a fixed positive number. Then, as it is easy to check

[TABLE]

for any compact set $K\in\mathbb{C}$ and $\varepsilon>0$. If $m_{n}=m$ for every $n$, then

[TABLE]

We recall in brief the properties of the convergence in $\sigma-$content. Let $\Omega$ be a domain and $\{\varphi_{n}\}$ a sequence of rational functions, converging uniformly in $\sigma-$content to a function $\varphi$ inside $\Omega$. If $\{\varphi_{n}\}\in{\cal A}(\Omega),$ then $\{\varphi_{n}\}$ converges uniformly in the $\max-$norm inside $\Omega.$ If $\varphi$ has $m$ poles in $D$, then each $\varphi_{n}$ has at least $m$ poles in $\Omega$; if each $\varphi_{n}$ has no more than $m$ poles in $\Omega$, then so does the function $\varphi$. For details, the reader is referred to [3].

Proof of Theorem 6

As it follows directly from (14) and from Theorem 2,

[TABLE]

for every compact set $K\subset D_{R(f)}$. Set

[TABLE]

Let $\Theta$ be a fixed positive number with $e^{\Theta}<R(f).$ The functions $v_{n}$ are subharmonic in $D_{R(f)}^{c}$; hence, by the maximum principle (see [15]) and by (17)

[TABLE]

Let now $U_{j},U_{j}\subset W,j=1,2$ be concentric open disks of radii $0<r_{1}<r_{2},$ respectively, and not intersecting the closed disk $\overline{D_{R(f)}}.$

The proof will be based on the contrary to the assumption that

[TABLE]

Then there is a subsequence of $\Lambda$ which we denote again by $\Lambda$ such that

[TABLE]

Fix an $\varepsilon,r_{2}-r_{1}>4\varepsilon>0.$ Under the conditions of the theorem, the sequence $\{\pi_{n}\}_{n\in\Lambda}$ converges in $\sigma-$content inside $U_{2}$. Set $g$ for the limit function. Select a subsequence $\tilde{\Lambda}\subset\Lambda$ such that

[TABLE]

Set $B_{n_{k}^{\prime}}:=\{z\in U_{2}\setminus U_{1},|\pi_{n_{k}^{\prime}}(z)-g(z)|>\varepsilon\}$ and $B^{\prime}:=\bigcup_{n_{k}^{\prime}\in\tilde{\Lambda}}B_{n_{k}^{\prime}}.$ We have

[TABLE]

By the principle of the circular projection ([8], p. 293, Theorem 2), there is a circle $F$, lying in the annulus $U_{2}\setminus U_{1}$ and concentric with $\partial U_{j},j=1,2$ such that $F\bigcap B^{\prime}=\emptyset.$ Hence,

[TABLE]

which yields

[TABLE]

Select now a number $r$ in such a way that the disk $D_{r}$ intersects the disk $U_{1}$ and set $\gamma:=\overline{U}_{1}\bigcap\Gamma_{r}.$ By construction, $\gamma$ is an analytic curve lying in the disk $U_{1}$. Applying the maximum principle to the last inequality, we get

[TABLE]

Therefore,

[TABLE]

Fix now a number $\rho,R(f)<\rho$ and such that the circle $\Gamma_{\rho}$ does not intersect the closed disk $\overline{U}_{2}.$ By the two constants theorem ([8], p. 331) applied to the domain $D_{R(f)}^{c}-\gamma$ there is a positive constant $\alpha=\alpha(\rho),\,\alpha<1$ such that

[TABLE]

From (18) and (20), it follows that

[TABLE]

Hence,

[TABLE]

After letting $\Theta$ tend to zero, we get

[TABLE]

The last inequality contradicts (19), since

[TABLE]

On this, Theorem 6 is proved. Q.E.D.

Proof of Theorem 7

As known, the Padé approximants are invariant under linear transformation, therefore without loss of generality, we may assume that $R(f)=1$ and $z_{0}=1.$ Under the conditions of the theorem, there is a neighborhood of $1$, say $V$, such that $f\in{\cal M}(V).$

Set, as before,

[TABLE]

where $(P_{n},Q_{n})=1$ and $Q_{n}$ are normalized as in (4).

Fix a positive number $\alpha>1$ such that

[TABLE]

In view of the conditions of the theorem, there is a number $\tau>0$ such that

[TABLE]

Hence (see (7)),

[TABLE]

Introduce the circles $C(\rho):=C_{1/2}(\rho):=\{|z-1/2|=\rho\},\,\rho>0$ and set $D(\rho):=\{|z-1/2|<\rho\}.$ By our previous convention, $D_{\rho}:=\{z,|z|<\rho\};\,\Gamma_{\rho}:=\partial D_{\rho};D_{1}:=D,\Gamma_{1}:=\Gamma$.

Consider the function

[TABLE]

It is easy to verify that there is a positive number $\delta_{0}$, such that

[TABLE]

Fix a number $\delta,\,0<\delta<\delta_{0}$ such that $\overline{D(\delta)}\subset D\bigcup V$ and $\delta<e^{\tau}-1.$

Select now a positive $\varepsilon<\delta/4$ and introduce, as above, the sets $\Omega_{n}(\varepsilon)$ and $\Omega(\varepsilon).$ By the principle of the circular projection, there is a number $R,1/2<R<1/2+\delta$ such that $C(R)=C(R,\varepsilon):=C(R)\setminus\Omega(\varepsilon).$ Set

[TABLE]

Denote by $\omega$ the monic polynomial of smallest degree such that $F:=f\omega\in{\cal A}(\overline{D_{r+1/2}\bigcup D(R)});\omega(z)=\prod_{k=1}^{\mu}(z-a_{k}),\,a_{k}\in\overline{D_{r+1/2}\bigcup D(R)}.$

In what follows we will estimate the terms $\|FQ_{n_{k}^{\prime}}-\omega P_{n_{k}^{\prime}}\|_{C(r)}$ and $\|FQ_{n_{k}^{\prime}}-\omega P_{n_{k}^{\prime}}\|_{C(R)}.$ For this purpose, we select a number $\Theta>0$ such that $\Theta<\tau,\,\,e^{\Theta}(r+1/2)<1$ and $e^{\Theta-\tau}(R+1/2)<1.$

By the maximum principle

[TABLE]

We obtain from Theorem 2, after keeping in mind (7), (21) and the choice of $\Theta,$

[TABLE]

Thus,

[TABLE]

Estimate now $\|FQ_{n_{k}^{\prime}}-\tilde{\omega}P_{n_{k}^{\prime}}\|_{C(R)}.$

Clearly,

[TABLE]

From (17), we have

[TABLE]

On the other hand,

[TABLE]

Combining the latter and the former, we get

[TABLE]

From here, we obtain (see (16)

[TABLE]

Let now $n_{l}\in[n_{k},n_{k^{\prime}}-1].$ By (6),

[TABLE]

which leads, thanks (23) and (16), to

[TABLE]

Finally, the choice of $R$ and $\Theta$ and the conditions of the theorem imply

[TABLE]

[TABLE]

From the last inequality, combined with (27) and (28), we derive

[TABLE]

Hence, after utilization (14), we get

[TABLE]

We now apply Hadamard’s three circles theorem ([8], p. 333, pp. 337 – 348) to $\frac{1}{n_{k}^{\prime}}\log|FQ_{n_{k}^{\prime}}(z)-\omega P_{n_{k}^{\prime}}(z)|_{C(1/2)}$ and the annulus $\{z,r\leq|z-1/2|\leq R\}$. Recall that by our convention $Rr=1/4$. Using now (26), (29), (24), we get

[TABLE]

Hence,

[TABLE]

Viewing (9) we get, after letting $\Theta\to 0$

[TABLE]

The last inequality is strong. Hence, we may choose a number $\rho,1/2<\rho<R$ and close enough to $1/2$ such that the inequality preserves the sign; in other words, there are numbers $\rho\in(1/2,R)$ and $q=q(\rho)<1$ such that

[TABLE]

From here, the $\sigma-$ almost uniform convergence inside the disk $D(R)$ immediately follows (see [3], Eq. (23).) Indeed, fix an appropriate number $\rho.$ In view of the last inequality,

[TABLE]

Take $\varepsilon<1/4(\rho-1/2)$ and introduce the sets $\Omega_{n_{k}^{\prime}}(\varepsilon),\,n_{k}^{\prime}>n_{0}$ with $\Omega_{n_{0}}(\varepsilon)$ covering the zeros of the polynomial $\omega$ (see (15). As shown above, $\sigma(\Omega(\varepsilon)<\varepsilon;$ thus $\|f-\pi_{n_{k}^{\prime}}P_{n_{k}^{\prime}}\|_{K(\varepsilon)}\leq C^{"}q^{n_{k}^{\prime}},\,n_{k},\geq N.$

On this, the $\sigma-$almost uniform convergence is established and Theorem 7 is proved. Q.E.D.

Proof of Theorem 8

As in the previous proof, we suppose that $R(f)=1.$ With this convention,

[TABLE]

Fix a compact set $K\subset G(f)$. Our purpose is to show that $\pi_{n_{k}}$ converges, as $n_{k}\to\infty\,\sigma$-almost unformloy on $K.$ We exclude the case $K\subset D.$ In the further considerations, we assume, that $K\nsupseteq D.$ Apparently, the generality will be not lost.

Take a curve $\gamma_{1}$ such that $\gamma_{1}\bigcap D\not=\emptyset$, the compact set $K$ lies in the interior $B_{1}$ of $\gamma_{1}$ and $\gamma_{1}\subset G(f).$ Suppose that $f\in{\cal A}(\gamma_{1})$ and denote by $Q$ the monic polynomial whit zeros at the poles of $f$ in $B_{1}$ (poles are counted with their multiplicities). Set $F:=fQ;\,\hbox{deg}Q:=\mu$. Choose a disk $B_{2},\,B_{2}\subset D\bigcap B_{1}$ and not intersecting $K;\gamma_{2}:=\partial B_{2}.$ In what follows, we will be estimating $\|FQ_{n_{k}}-QP_{n_{k}}\|_{\gamma_{1}}$ and $\|FQ_{n_{k}}-QP_{n_{k}}\|_{\gamma_{2}}.$

Take a number $r_{2}<1$ such that $B_{2}\subset D_{r_{1}}$.

Fix $\Theta>0$ such that $r_{2}e^{\Theta}<1$ Then, for every $n$ great enough there holds

[TABLE]

Hence, by (10) and the choice of $r_{2}$ and $\Theta$

[TABLE]

In order to estimate $\|FQ_{n_{k}}-QP_{n_{k}}\|_{\gamma_{1}},$ we proceed as follows: fix a number $\varepsilon,0<\varepsilon<\hbox{dist}(\gamma_{1},\partial G(f))/4$ and take $r_{1}>1$ such that the circle $\Gamma_{r_{1}}$ does not intersect the set $\Omega(\varepsilon)$ and surrounds $\gamma_{1}.$

Relying on (6), on (16) and (30), we get

[TABLE]

Using now ( 13) and following the same argumentation as in the proof of Theorem 7, we obtain

[TABLE]

The application of Hadamard’s two constants theorem leads to

[TABLE]

with $\alpha:=\alpha(K)<1.$

We get, thanks the choice of $\Theta$

[TABLE]

The statement of the theorem follows now after using standard arguments. On this, the proof of the first part of Theorem 8 is completed.

b) The proof of the second part is based on the arguments provided in the proof of Theorem 7. As in Theorem 7, we introduce the number $\alpha$ (21), the function $\phi(R)$ (24), the circles $C(r)$ and $C(R)$ (25) and the polynomial $\omega$. Let $R$ and $r$ be a in THeorem 7 and set and $F:=f\omega.$

Fix a positive number $\Theta$ such that $e^{\Theta}(r+1/2)<1,\Theta<-\frac{1}{2}\frac{\log\phi(R)}{2+\alpha}.$

We get, first, thanks (10)

[TABLE]

and, then, following the same way of considerations,

[TABLE]

Applying the tree circles theorem, we get

[TABLE]

By the choice of $\Theta$,

[TABLE]

In what follows, we use standard arguments to complete the proof of (b), Theorem 8. Q.E.D.

Proof of Theorem 9

Without losing the generality, we assume that $R_{m}=1$ and $\tau_{n}=0$ for all $n.$ Normalize the polynomials $Q_{n}$ as it was done in (4) with $R(f)$ replaced by $R_{m}$. Fix a positive number $\varepsilon,\,\varepsilon<1/2$ and introduce the set $\Omega(\varepsilon).$ Select a number $R,\,R>1$ such that $\Gamma_{R}\bigcap\Omega(\varepsilon)=\emptyset$. Recall that (see (6)) there are positive constants $C_{j}(\varepsilon):=C_{j},j=1,2$ such that

[TABLE]

In the sequel, we assume that $C_{1},\,C_{2}>1.$

By Theorem 1, there is a positive number $\tau=\tau(R)>0$ such that

[TABLE]

and (by the maximum principle for subharmonic functions),

[TABLE]

Without losing the generality, we suppose that

[TABLE]

We will prove that for every $l,0\leq l\leq n_{k}$ and for $n_{k}$ great enough

[TABLE]

From the last inequality, it follows directly that

[TABLE]

We prove first (36) for $l=1.$ For this purpose, we introduce the polynomial

[TABLE]

By definitions of Padé approximants (see (6)),

[TABLE]

where, according to (7),

[TABLE]

(recall that by presumption the defect $\tau_{n}=0$ for all $n.$) Viewing (34) and (36), we get

[TABLE]

Keeping now track of (34) and (33), we arrive at

[TABLE]

which yields

[TABLE]

[TABLE]

We further get

[TABLE]

Suppose now that (37) is true for $l-1,l\geq 2.$ In other words,

[TABLE]

and

[TABLE]

Introducing into considerations the polynomial ${\cal P}_{n_{k}-l+1}:=P_{n_{k}-1}Q_{n_{k}-l+1}-P_{n_{k}-l+1}Q_{n_{k}-1}$ and following the same arguments as before, we see that (37) and (38) are true also for $l.$

Equipped with inequality (38), we complete the proof of the theorem. We will be looking for numbers $l$ such that

[TABLE]

Set $2RC_{2}C_{1}:=C4.$ We check that

[TABLE]

where

[TABLE]

For $n_{k}$ large enough, say $n_{k}\geq n_{6}$, $\psi$ is strongly increasing, and $\psi(0)<0.$ Hence, there is a number $x_{k}\in(0,n_{k})$ such that $\psi_{n_{k}}(x)<\psi_{n_{k}}(x_{k})<-\tau/2$ every time when $0<x<x_{k}.$ Set $l_{k}:=x_{k}.$ Therefore

[TABLE]

Q.E.D.