Rate of convergence of row sequences of multipoint Padé approximants

Journal of Computational and Applied Mathematics 284 (2015) 155–170

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Rate of convergence of row sequences of multipoint Padé approximants✩ B. de la Calle Ysern a,∗ , J. Mínguez Ceniceros b a

Dpto. de Matemática Aplicada, E.T.S. de Ingenieros Industriales, Universidad Politécnica de Madrid, José G. Abascal 2, 28006 Madrid, Spain b

Departamento de Matemáticas y Computación, Universidad de La Rioja, c/ Luis de Ulloa s/n, 26004 Logroño, La Rioja, Spain

article

abstract

info

Article history: Received 16 June 2014 Received in revised form 10 October 2014 MSC: 30C15 41A21 41A25

As a consequence of studying the exact rate of convergence of row sequences of multipoint Padé approximants, we prove that their zero limit distribution is a generalized balayage measure determined by the table of interpolation points and the region of meromorphy of the function being approximated, provided the configuration of these sets satisfies mild topological restrictions. Should there exist a subsequence of approximants converging at a faster rate on a given continuum that does not reduce to a single point, we prove that such a subsequence is overconvergent in the sense of the Hausdorff content. © 2014 Elsevier B.V. All rights reserved.

Keywords: Distribution of zeros Jentzsch–Szegő theorem Rows of the Padé table Overconvergence Rate of convergence Multipoint Padé approximation

1. Introduction The purpose of this article is to prove certain results connected with the geometric rate of convergence of interpolatory rational functions that have a bounded number of poles and whose associated interpolation points may adopt an arbitrary distribution. The broached subject matter is inspired by some classic results on the Taylor series that date back to the earliest decades of the past century and which have been profusely extended over the years. Let f (z ) =

∞ 

an z n ,

an ∈ C ,

(1)

n =0

denote a Taylor expansion about the origin with a finite radius of convergence R > 0. The Jentzsch–Szegő theorem [1,2] deals with the limiting behavior of the zeros of the partial sums sn (z ) =

n 

ak z k ,

n ∈ N.

k=0

✩ Dedicated to our colleague and friend Pablo González-Vera.

∗

Corresponding author. Tel.: +34 91 336 3102; fax: +34 91 336 3001. E-mail addresses: [email protected] (B. de la Calle Ysern), [email protected] (J. Mínguez Ceniceros).

http://dx.doi.org/10.1016/j.cam.2014.10.007 0377-0427/© 2014 Elsevier B.V. All rights reserved.

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For each r > 0, set Dr = {z ∈ C : |z | < r }, Cr = {z ∈ C : |z | = r }, and Dr = {z ∈ C : |z | ≤ r }. Also, set Sα,β = {z ∈ C : α < Arg z < β} ,

0 < β − α < 2π .

Given a set U ⊂ C and n ∈ N, let Zn (U ) be the number of zeros of sn in the set U. Jentzsch–Szegő Theorem. Let Λ ⊂ N be a subsequence with lim |an |1/n = 1/R > 0.

n∈Λ

Then, for each r > R and for any sector Sα,β , it holds that lim

Zn (Dr )

n∈Λ

n

= 1 and

lim

n∈Λ

Zn (Sα,β ) n

=

β −α . 2π

For a proof, see [3, Section 2.1]. Together with the Hurwitz theorem, the Jentzsch–Szegő theorem implies that the limiting distribution of zeros of {sn }n∈Λ is the uniform distribution on the circle of convergence CR . Setting aside the extensions to different kinds of polynomial approximants (see [4] and the references therein), results analogous to the Jentzsch–Szegő theorem have been proved for rational functions of best uniform approximation [5] and Padé approximants [6]. The presence of poles of the approximants may impair the convergence and make results about zero behavior of the approximants harder to prove than in the polynomial case, or even impossible [7]. Nevertheless, see [8] for Jentzsch–Szegő-type theorems (with additional hypotheses) for Padé approximants and best rational approximations with unbounded number of poles. For results on angular distribution of zeros of Padé approximants of entire functions, see [9]. Most of these polynomial and rational extensions strongly rely on the extremal properties satisfied by the corresponding approximants and the resulting zero limit distribution turns out to be an equilibrium measure in all cases. Recently, the first author [4] has extended the Jentzsch–Szegő theorem showing, without consideration of extremal properties, that the zero limit distribution of general interpolating polynomials is a generalized balayage (sweeping out) measure depending on the interpolation points and the region of analyticity of the function f . The first part of the present work shows that an analog is valid for row sequences of multipoint Padé approximants and thus constitutes a natural continuation of [4]. To handle the poles of the rational interpolants, we make use of a technique developed by Gonchar in [10] that consists in avoiding in our reasonings certain neighborhoods of the poles of the approximants. Such sets are small in the sense of the Hausdorff content (see Section 2), which, together with some technicalities, allows us to reduce the proof to that of the polynomial case. As in [4], certain natural topological restrictions concerning the configuration of the set of convergence in Hausdorff content of the approximants and the set where the interpolation is carried out are required. The Jentzsch–Szegő theorem essentially depends on the fact that lim sup ∥f − sn ∥ n→∞

1/n Dr

=

r R

(2)

for any positive r < R. That is, the exact rate of geometric convergence of the whole sequence of approximants can be determined. If the circle of convergence is not the natural boundary of analyticity of f and, for some subsequence, the rate of convergence is faster than the one appearing in (2), then the phenomenon of overconvergence arises. The Taylor series diverges pointwise outside the circle of convergence, but Porter [11] discovered that certain subsequences of partial sums may converge outside that circle. In such a case, we say that the subsequence overconverges or that the Taylor series has overconvergent subsequences. The nature of this phenomenon and its connection with gaps (groups of consecutive small coefficients) was studied and fully cleared up by Ostrowski. See, for instance, the survey [12]. Ostrowski Theorem. Let f be a Taylor series as in (1), and suppose that there exist ρ > R, θ > 0, and a pair of subsequences {nk }k∈N , {n′k }k∈N , such that, for all k ∈ N, we have (i) nk < n′k ≤ nk+1 , (ii) n′k − nk ≥ θ nk , and, for nk ≤ m ≤ n′k , we have (iii) |am | ≤ ρ −m . Then the subsequence of partial sums snk of the Taylor expansion converges uniformly on a neighborhood of every regular point of the circle of convergence CR . We say that a Taylor series (1) has Ostrowski gaps if it verifies the hypotheses of Ostrowski’s theorem. Ostrowski [12] also proved that if a Taylor series has overconvergent subsequences, then it necessarily has Ostrowski gaps. See the books [13,14] and the monograph [15] for more information on the overconvergence of the Taylor series, and [16,17] for related questions and open problems concerning the domain of overconvergence. Ostrowski’s results have been extended to more general kinds of series [18] and to Padé approximants [19], among others.

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It is not difficult to prove that the presence of Ostrowski gaps is equivalent to the fact that the asymptotic inequality r 1/n lim sup ∥f − snk ∥ k < Dr R k→∞ holds for some subsequence {nk } and a positive r < R (cf. (2)). Walsh [20, Theorem 5], using the notion of harmonic majorant of a sequence of subharmonic functions (see Section 2), proved that a fast convergent subsequence of analytic functions is overconvergent provided the exact rate of convergence of the whole sequence is established. In the second part of our work we follow this latter approach and [19], where some refinements were obtained, to prove an analog of Ostrowski’s theorem for row sequences of multipoint Padé approximants. It is clear that uniform convergence on compact subsets is no longer possible in general for rational functions due to the presence of poles. Instead, we prove that if a subsequence of approximants converges faster than the whole sequence inside the domain of convergence, then the subsequence overconverges outside such a domain near regular points or poles in the sense of the Hausdorff content. The paper is structured as follows. In Section 2 we give some definitions and preliminary results. Section 3 is dedicated first to studying the exact rate of geometric convergence of row sequences of multipoint Padé approximants. After that, we prove a generalized Jentzsch–Szegő theorem for them. Section 4 deals with overconvergent subsequences. 2. Definitions and preliminary results 2.1. Potential theory Let Σ be a compact set of the complex domain C with connected complement. Let α be a positive unit Borel measure whose support S (α) is contained in Σ . The logarithmic potential of α is denoted by P (α; z ) and is equal to

 −

log |z − ζ | dα(ζ ).

Set r0 = inf exp{−P (α; z )} ≥ 0 z ∈Σ

and Eα (r ) = {z ∈ C : exp{−P (α; z )} < r },

r > r0 .

(3)

Since the function exp{−P (α; ·)} is upper semi-continuous on C, for each r > r0 , Eα (r ) is a nonempty bounded open set that, in general, does not contain the whole of Σ . By the same token, the function exp{−P (α; ·)} attains its maximum on compact sets. So, for each compact set K ⊂ C let us write

ρα (K ) = max exp{−P (α; z )}. z ∈K

As the function exp{−P (α; ·)} is subharmonic on C, each connected component of Eα (r ) is simply connected. It may be proved (see the arguments given in [21, p. 18]) that every component of Eα (r ) has a nonempty intersection with Σ . Let K be a compact set of the complex domain C and G the unbounded component of C \ K . We say that K is regular if the domain G is regular with respect to the Dirichlet problem. By cap (K ), we mean the logarithmic capacity of K . If cap (K ) > 0, for each a ∈ G, there exists the generalized Green function of G with pole at z = a, which we denote by gG (z , a), see [22, Section II.4]. The fact that K is regular implies cap (K ) > 0. If cap (K ) > 0, we have gG (z , ∞) = − log cap (K ) − P (µK ; z ),

z ∈ C,

where µK is the equilibrium measure of the set K . 2.2. Convergence in σ -content Let A be a subset of the complex plane C. By U(A) we denote the class of all coverings of A by at most a numerable set of disks. Set

σ (A) = inf

 

 |Ui | : {Ui }i∈I ∈ U(A) ,

i∈I

where |Ui | stands for the radius of the disk Ui . The quantity σ (A) is called the 1-dimensional Hausdorff content of the set A. This set function is not a measure, but fulfills some good properties like countable semiadditivity which, for instance, is not satisfied by the logarithmic capacity. Let {ϕn }n∈N be a sequence of functions defined on a domain D and ϕ another function also defined on D with values in C. We say that the sequence {ϕn }n∈N converges in σ -content to the function ϕ inside D if for each compact subset K of D and for each ε > 0 we have lim σ ({z ∈ K : |ϕn (z ) − ϕ(z )| > ε}) = 0.

n→∞

The next lemma was proved by Gonchar in [23].

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Gonchar’s Lemma. Suppose that the sequence {ϕn } of functions defined on a domain D ⊂ C converges in σ -content to a function ϕ inside D. If each of the functions ϕn is meromorphic in D and has no more than k < +∞ poles in this domain, then the limit function ϕ is also meromorphic (more precisely, it is equal to a meromorphic function in D except on a set of σ -content zero) and has no more than k poles in D. 2.3. Multipoint Padé approximants Let f be an holomorphic function on a neighborhood V of the compact set Σ ⊂ C. Let us fix a family of monic polynomials

wk (z ) =

k  (z − αk,i ),

k ∈ N,

i=1

whose zeros are contained in Σ . It is easy to verify that for each pair of nonnegative integers n and m, there exist polynomials Pn,m and Qn,m satisfying – deg Pn,m ≤ n, deg Qn,m ≤ m, and Qn,m ̸≡ 0. – (Qn,m f − Pn,m )/wn+m+1 ∈ H (V ), where H (V ) denotes the space of analytic functions in V . Any pair of such polynomials Pn,m and Qn,m defines a unique rational function Pn,m /Qn,m , which is called the multipoint Padé approximant of type (n, m) of f and denoted by Πn,m for short. Throughout the paper all the limit processes are taken as n grows to infinity and m remains fixed. Let Πn,m = Pn,m /Qn,m , where Qn,m is normalized so that Qn,m (z ) =

 

z − ζn,k

  

|ζn,k |≤1

|ζn,k |>1

1−

z



ζn,k

.

(4)

Let Rα,m , m ∈ Z+ , be the supremum of the numbers r > r0 such that f admits meromorphic continuation with at most m poles on Eα (r ), see (3). Lemma 2.1 of [24] proves that Rα,m > r0 . We define the set of m-meromorphy of f relative to α as the set Eα (Rα,m ) and we denote it by Dα,m . It is easy to see that f admits meromorphic continuation with at most m poles on Dα,m . To properly handle the poles of the approximants, we need some additional definitions. Take an arbitrary ε > 0 and define the open set Jε as follows. Let Jn,ε , n ≥ m, denote the ε/6mn2 -neighborhood of the set of zeros of Qn,m and let Jm−1,ε denote the ε/6m-neighborhood of the set of poles of f . Set Jε = ∪n≥m−1 Jn,ε . We have σ (Jε ) < ε and Jε1 ⊂ Jε2 for ε1 < ε2 . For any set B ⊂ C we write B(ε) = B \ Jε . From these properties it readily follows that if {ϕn }n∈N converges uniformly to the function ϕ on K (ε) for every compact K ⊂ D and for each ε > 0, then {ϕn }n∈N converges in σ -content to ϕ inside D. Due to the normalization (4), for any compact set K of C and for every ε > 0, there exist positive constants C1 , C2 , independent of n, such that

∥Qn,m ∥K < C1 ,

min |Qn,m (z )| > C2 n−2m ,

z ∈K (ε)

(5)

where the second inequality is meaningful when K (ε) is a nonempty set. The following technical lemma is key for the subsequent reasonings in Section 3. We denote by int γ the interior of the Jordan curve γ . Lemma 2.1. Let Γ1 , Γ2 be two Jordan curves with Γ1 ⊂ int Γ2 . Then there exists an analytic Jordan curve Γ with Γ ⊂ int Γ2 and Γ1 ⊂ int Γ such that

Γ (ε) = Γ

for all ε ≤ ε0 ,

with ε0 > 0 depending on Γ1 and Γ2 . Proof. Let B be the doubly connected domain limited by the curves Γ1 and Γ2 . Then it is known (see, for instance, [25, Chapter V, Section 1, Theorem 1]) that there exists a conformal mapping h that maps B onto a circular annulus C centered at the origin. If we take the preimage of two concentric circles in C , we can suppose that the original curves Γ1 and Γ2 are analytic, the mapping function h is analytic in B, with h′ (z ) ̸= 0 for all z ∈ B, and the inverse function h−1 is analytic in the closed annulus. Therefore, h and h−1 have bounded derivatives in B and C , respectively and, consequently, they are Lipschitz functions. On the other hand, the circular projection p onto a radial segment I ⊂ C is also a Lipschitz function because it does not increase distances. Then the function g = h−1 ◦ p ◦ h is also a Lipschitz function. We denote the Lipschitz constant of the function g by L. We choose now two other concentric circles in C : γ1 , γ2 , and we consider β1 = h−1 (γ1 ), β2 = h−1 (γ2 ). We denote by B1 and C1 the doubly connected domains limited by β1 and β2 and γ1 and γ2 , respectively. Set K = h−1 (I ∩ C 1 ),

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159

which is a compact connected set because it is the continuous image of a compact connected set. As K is a continuum that does not reduce to a single point, it is a regular set, see [26, Theorem 3.8.3]. Write cap K = d > 0. Fix ε0 < d/2L and take ε ≤ ε0 , ε > 0. We additionally choose ε0 sufficiently small so that any open disk with radius ε0 centered at a point of B1 is contained in B. We know that Jε = ∪n≥m−1 Jn,ε , where Jn,ε is an open disk with radius less than ε/6, n ≥ m − 1. Moreover, the sum of all these radii is less than ε , whence σ (Jε ) < ε. Set J˜ε = Jε ∩ B1 and build a covering of J˜ε in the following way. Set J˜n,ε =



Jn,ε ,

if Jn,ε ∩ B1 ̸= ∅; if Jn,ε ∩ B1 = ∅.

∅,

Then it is obvious that J˜ε ⊂ ∪n≥m−1 J˜n,ε , and each J˜n,ε is contained in B for n ≥ m − 1. Now, let {D(zi , δi )}i∈H denote such a covering of J˜ε , where D(zi , δi ) = {z ∈ C : |z − zi | < δi } ⊂ B,

i ∈ H.

Then, for each i ∈ H, we obtain |g (z ) − g (zi )| ≤ L|z − zi | < Lδi , z ∈ D(zi , δi ), and thus {D(g (zi ), Lδi )}i∈H is also a covering of g (J˜ε ). We consequently have σ (g (J˜ε )) < Lε ≤ d/2. Therefore, K \ g (J˜ε ) ̸= ∅, and we can choose z0 ∈ K \ g (J˜ε ). Then h−1 ({z ∈ C1 : |z | = |h(z0 )|}) is the analytic curve Γ we sought. It is clear that the choice of Γ is independent of ε since J˜ε1 ⊂ J˜ε2 if ε1 < ε2 .



∗

Let αn and α be finite positive Borel measures on C. By αn −→ α, n → ∞, we denote the weak∗ convergence of αn to α as n tends to infinity. This means that for every continuous function f on C it holds that

 lim

n→∞

f (x) dαn (x) =



f (x) dα(x).

For a given polynomial p, we denote by Θp the normalized zero counting measure of p. That is,

Θp =

1



δξ .

deg p ξ :p(ξ )=0

The sum is taken over all the zeros of p and δξ denotes the Dirac measure concentrated at ξ . It is said that the sequence of interpolation points given by the polynomials {wn }n∈N has the measure α as its asymptotic zero distribution if ∗

Θwn −→ α,

n → ∞.

The following lemma has essentially the same proof as Lemma 2.3 of [24]. Lemma 2.2. Let the measure α be the asymptotic zero distribution of the sequence of interpolation points given by {wn }n∈N . Then 1/n lim sup Qn,m (z ) f (z ) − Πn,m (z )  ≤







n→∞



e−P (α;z ) Rα,m

,



locally uniformly on Dα,m ∪ V \ Pf , where Pf stands for the set of distinct poles of the function f in Dα,m . Let us denote by R′α,m the supremum of the numbers r > r0 such that there exists a subsequence of {Πn,m }n≥m that converges in σ -content inside Eα (r ). Due to Lemma 2.2, we have 0 < Rα,m ≤ R′α,m . On the other hand, from Gonchar’s Lemma it follows that f admits meromorphic continuation with at most m poles in the region where there exists a subsequence of {Πn,m }n≥m converging in σ -content. Therefore, it is clear that Rα,m = R′α,m , and we could have used this property for an alternative definition of Rα,m . 2.4. Harmonic majorant of a sequence of functions A compact set Q ⊂ C is called a nondegenerate continuum if Q is connected and consists of more than a single point. Every nondegenerate continuum is a regular set, see [26, Theorem 3.8.3]. Let {un }n∈N be a sequence of subharmonic functions in a domain D ⊂ C and let h be a harmonic function in D. We say that h is a harmonic majorant of the sequence {un }n∈N if for any nondegenerate continuum Q ⊂ D, we have





lim sup max un (z ) z ∈Q

n→∞

≤ max h(z ). z ∈Q

The harmonic majorant h of {un }n∈N is called exact if for any nondegenerate continuum Q ⊂ D, we have



lim sup max un (z ) n→∞

z ∈Q



= max h(z ). z ∈Q

The sequence {un } need not be defined for every n ∈ N; an analogous definition is given if the index n runs along a subsequence Λ of N.

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The concept of an exact harmonic majorant of a sequence of subharmonic functions was introduced and studied by Walsh in [20]. The next result appears as Theorem 1 in [20]. Proposition 2.3. Let the function h be a harmonic majorant of {un }n∈Λ in the domain D and suppose that there exists a compact set K ⊂ D such that





lim sup max un (z ) n∈Λ

z ∈K

= max h(z ). z ∈K

Then h is an exact harmonic majorant of {un }n∈Λ in D. The following corollary is easily obtained. Corollary 2.4. Let the function h be a harmonic majorant of {un }n∈Λ in the domain D and suppose that there exists a nondegenerate continuum Q ⊂ D such that





lim sup max un (z ) n∈Λ

z ∈Q

< max h(z ). z ∈Q

Then this strict inequality holds for any compact subset K of D. 2.5. Balayage measures Let G be a bounded open set. Given any finite positive Borel measure µ with compact support in C and verifying µ(C \ G) = 0, we denote by  µ the sweeping out or balayage of µ onto ∂ G, see [22, Chapter II.4]. Among other properties, the balayage measure  µ satisfies P ( µ; z ) = P (µ; z )

(6)

for all z ̸∈ G. If the measure µ does not fulfill the condition µ(C \ G) = 0, we write

µ = µ|G + µ|C\G , and only the part of µ lying on G is swept out. That is, we define the sweeping out or balayage of the measure µ onto ∂ G as the measure  µ given by

 µ=µ  |G + µ|C\G . Now, if α is the asymptotic zero distribution of the interpolation points given by {wn }n∈N , we define the balayage of α onto ∂ Dα,m as

 α = α |Dα,m + α|C\Dα,m .

(7)

So defined,  α is a probability measure with support S ( α ) ⊂ (Dα,m ∪ Σ ) \ Dα,m that verifies P (α; z ) = P ( α ; z ),

z ∈ C \ (Dα,m ∪ Σ ),

due to (6). It may be proved (see [4, Proposition 2.5]) that if Σ ⊂ Dα,m , then the balayage of the measure α onto ∂ Dα,m is the equilibrium measure of the compact set ∂ Dα,m . 3. Jentzsch–Szegő theorem Throughout the rest of the paper we maintain the notations introduced above, i.e., Σ is a compact set with connected complement and f is an analytic function on a neighborhood V of Σ . For each pair of nonnegative integers n, m, Πn,m stands for the multipoint Padé approximant of type (n, m) of f with interpolation points given by wn+m+1 . Such points belong to Σ and have the measure α as their asymptotic zero distribution. Recall that S (α) denotes the support of α, ρα (K ) = maxz ∈K exp{−P (α; z )}, and Dα,m is the set of m-meromorphy of f relative to α . In correspondence with the fact that the series (1) has a finite radius of convergence, we also assume henceforth that f is not a meromorphic function on C with at most m poles, whence Rα,m < +∞. Set Ω = C \ (Dα,m ∪ Σ ). Denote by Ω∞ the unbounded connected component of Ω and by Tα,m the polynomial convex hull of Dα,m ∪ Σ , i.e., Tα,m = C \ Ω∞ . Fix a subsequence Λ = {nk }k∈N ⊂ N. For each nk ∈ Λ, define snk (Λ; z ) = Qnk ,m (z ) Qnk−1 ,m (z ) Πnk ,m (z ) − Πnk−1 ,m (z )



and Gnk (Λ; z ) =

1 nk

log |snk (Λ; z )| + P (α; z ) + log Rα,m .



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161

Write sn (z ) = sn (N; z ) and Gn (z ) = Gn (N; z ), n ∈ N, for short. For each n ∈ N, we also define Hn ( z ) =

1 n

log |Pn,m (z )| + P (α; z ) + log Rα,m .

Notice that the functions Gnk and Hn are subharmonic on C \ Σ . Obviously, the functions just defined depend on m as well, but, as this quantity remains constant throughout the paper, this notation will not be misleading. Lemma 3.1. Let Υ = {nk } ⊂ N be a subsequence verifying lim

nk+1

k→∞

nk

= 1.

(8)

For any compact set K ⊂ C \ Σ , it holds that





lim sup max Gnk (Υ ; z ) z ∈K

k→∞

≤ 0.

(9)

For each compact set K ⊂ Ω , we have





lim sup max Hn (z ) z ∈K

n→∞

≤ 0.

(10)

Proof. Let K ⊂ C \ Σ be a compact set. Let W be a bounded open set such that C \ W is a connected regular set, Σ ⊂ W ⊂ W ⊂ V , and W ∩ K = ∅. The existence of such a set follows from the Hilbert lemniscate theorem [26, Theorem 5.5.8]. As the functions Gn (Υ ; ·) are subharmonic in C \ Σ , in order to obtain (9) it is enough to prove that



lim sup max Gnk (Υ ; z )



z ∈∂ W

k→∞

≤ 0.

(11)

As ∂ W is a compact subset of V \ Σ , we can apply Lemma 2.2 to obtain 1/n lim sup Qn,m (z ) f (z ) − Πn,m (z )  ≤







e−P (α;z )

n→∞

Rα,m

,

uniformly on ∂ W . This, together with (5) and (8), implies 1/nk ≤ lim sup Qnk ,m (z )Qnk−1 ,m (z ) Πnk ,m (z ) − Πnk−1 ,m (z ) 







k→∞

e−P (α;z ) Rα,m

,

uniformly on ∂ W , whence (11), and thus (9), follows. Now, let K be an arbitrary compact subset of Ω and fix ε > 0. Then min e−P (α;z ) > Rα,m .

(12)

z ∈K

From (9) with Υ = N, given any δ > 1, for all j ≥ Nδ , and all z ∈ K , it holds that

 −P (α;z ) j e . |sj (z )| ≤ δ Rα,m

Thus, with the aid of (5), we arrive at

   −P (α;z ) j  j4m  e sj (z )   |Πj,m (z ) − Πj−1,m (z )| =  ≤ 2 δ Qj,m (z )Qj−1,m (z )  Rα,m C2 for z ∈ K (ε). For n ≥ n0 + 1 ≥ Nδ + 1, we have

|Πn,m (z )| ≤ |Πn0 ,m (z )| +

n 

|Πj,m (z ) − Πj−1,m (z )|.

j=n0 +1

Choose n0 sufficiently large such that j4m < δ j for all j ≥ n0 . Thus, taking (12) into account, we obtain

|Πn,m (z )|1/n ≤ M 1/n δ 2

e−P (α;z ) Rα,m

,

z ∈ K (ε),

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for sufficiently large n with M > 1. This implies, again using (5), max Hn (z ) ≤

log(MC1 )

z ∈K (ε)

n

+ 2 log δ

for sufficiently large n. So, taking limits in the expression above as n tends to infinity and then making δ go to 1, we prove that

 lim sup

max Hn (z )



z ∈K (ε)

n→∞

≤0

(13)

for each compact set K of Ω and each ε > 0. Now, again fix a compact subset K of Ω . From the generalized Hilbert lemniscate theorem (see, for instance, [27]) it follows that there exists a lemniscate Γ1 that satisfies (Dα,m ∪ Σ ) ⊂ int Γ1 ⊂ int Γ 1 ⊂ U, where U = C \ K . If Γ1 has double points, we can slightly modify it to obtain a lemniscate made up by a finite number of analytic curves where each curve is exterior to the rest of them. Let Γ1 be given by the expression

Γ1 = {z ∈ C : |z − τ1 | |z − τ2 | · · · |z − τν | = M }, where τi ∈ C, i = 1, . . . , ν , and M > 0. Let Γ2 be given by

Γ2 = {z ∈ C : |z − τ1 | |z − τ2 | · · · |z − τν | = M + ϵ},

ϵ > 0,

with the same properties of Γ1 and Γ1 ⊂ int Γ2 ⊂ int Γ 2 ⊂ U. Then, from Lemma 2.1, there exist an analytic cycle Γ and a constant ε0 > 0 such that Γ ⊂ int Γ2 , Γ1 ⊂ int Γ , and Γ = Γ (ε) for all ε ≤ ε0 . From (13) applied to the compact set Γ , we have



lim sup max Hn (z ) n→∞

z ∈Γ



≤ 0.

As the functions Hn are subharmonic in Ω , we readily obtain (10).



Our next goal is to prove that the function constantly zero is an exact harmonic majorant of the functions Hn on each component of Ω . This is achieved for the most part with the following result. Recall that Pf stands for the set of distinct poles of the function f in Dα,m . Proposition 3.2. Let the measure α be the asymptotic zero distribution of the sequence of interpolation points given   by {wn }n∈N . Let Υ = {nk } ⊂ N be a subsequence verifying (8). Then, for each nondegenerate continuum Q ⊂ Dα,m \ Σ ∪ Pf , it holds that 1/nk

lim sup ∥Qnk ,m f − Πnk ,m ∥Q





=

k→∞

ρα (Q ) Rα,m

< 1,

where ∥ · ∥B stands for the supremum norm on the set B. Proof. We may suppose that ρα (Q ) > 0, otherwise the result is trivial on account of Lemma 2.2. From Lemma 2.2 it follows that 1/nk

lim sup ∥Qnk ,m (f − Πnk ,m )∥Q

≤

ρα (Q )

=

ρα (Q )

k→∞

Rα,m

< 1.

Let R ≥ Rα,m such that 1/nk

lim sup ∥Qnk ,m (f − Πnk ,m )∥Q k→∞

R

≤

ρα (Q ) Rα,m

< 1.

(14)

We need to prove that R ≤ Rα,m . Let us suppose that Rα,m < R, hence we will come into contradiction. We can take R < +∞ since, from hypotheses, f is not a meromorphic function on C with at most m poles. Write, for short, snk (Υ ; z ) = sk (z ) and Gnk (Υ ; z ) = Gk (z ). To reach a contradiction it is enough to prove that the functions Gk are uniformly bounded by a negative constant on a neighborhood U of the set

{z ∈ C \ V : −P (α; z ) = log Rα,m } ̸= ∅. In that case, there exist constants q < 1 and C3 > 0 verifying

|sk (z )| < C3 qnk ,

k ≥ k1 , z ∈ U .

(15)

Fix ε > 0 and L a compact subset of U. Then, because of (5) and (15), we have

  |Πnk ,m (z ) − Πnk−1 ,m (z )| =  Q

sk (z )

  C3 n 4m k ≤  C 2 q nk ( z ) Q ( z ) n k ,m nk−1 ,m 2

B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

163

< +∞, {Πnk ,m }k converges uniformly on L(ε). Since ε > 0 and L are arbitrary, for all k ≥ k1 and z ∈ L(ε). As k=1 qnk n4m k Gonchar’s Lemma proves that the subsequence {Πnk ,m }k converges in σ -content inside U to a meromorphic continuation of f with at most m poles, which contradicts the definition of Rα,m . It follows from (14), the upper bound of (5), and (8) that ∞

1/nk

lim sup ∥sk ∥Q

ρα (Q )

≤

< 1.

R

k→∞

Then there exists R′ with Rα,m < R′ < R such that 1 nk



log |sk (z )| ≤ log

ρα (Q )



R′

for all z ∈ Q and sufficiently large k. Equivalently,

Gk (z ) ≤ P (α; z ) + log ρα (Q ) − log

R′





Rα,m

(16)

for all z ∈ Q and sufficiently large k. On the other hand, let z0 ∈ Q such that P (α; z0 ) = − log ρα (Q ). As the potential P (α; ·) is a continuous function on Dα,m \ Σ , there exists an open disk B centered at z = z0 such that B ⊂ Dα,m \ Σ and P (α; z ) < − log ρα (Q ) +

1 2

 log

R′



Rα,m

(17)

for all z ∈ B. Then inequalities (16) and (17) give

Gk (z ) ≤ −

R′



1

log

2



Rα,m

<0

for all z ∈ B ∩ Q and sufficiently large k. Therefore, if Q0 denotes the connected component of B ∩ Q that contains z0 , we have

  ′  1 R lim sup max Gk (z ) ≤ − log < 0. 

2

z ∈Q0

k→∞

Rα,m

This asymptotic relation, together with (9) and Corollary 2.4, proves that





lim sup max Gk (z ) z ∈F

k→∞

<0

(18)

for any compact subset F of C \ Σ . In particular, (18) is valid for the set U considered above, which gives the desired contradiction.  Lemma 3.3. Let Υ = {nk } ⊂ N be a subsequence verifying (8). For any nondegenerate continuum Q ⊂ C \ Σ , it holds that



 lim sup max Gnk (Υ ; z ) = 0.

(19)

z ∈Q

k→∞

Let Q be a nondegenerate continuum of Ω . Then we have



 lim sup max Hnk (z ) = 0.

(20)

z ∈Q

k→∞

Proof. Let ε > 0 and let γ be a circle contained in Dα,m \ Σ such that γ = γ (ε). This circle exists because of Lemma 2.1. Again, write snk (Υ ; z ) = sk (z ) and Gnk (Υ ; z ) = Gk (z ). Suppose that



lim sup max Gk (z ) k→∞



z ∈γ

< 0.

Then there exist k1 ∈ N and δ < 1 such that

∥sk ∥1γ/nk ≤ δ

ρα (γ ) Rα,m

,

k ≥ k1 .

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B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

Then, using (5), for k ≥ k1 we have

 nk  )nk 4m  ≤ δ ρα (γ nk . nk  2 C2 Rα,m nk ,m Qnk−1 ,m γ

  ∥Πnk ,m − Πnk−1 ,m ∥γ =  Q

sk

As ∞ 

f (z ) − Πnk ,m (z ) =

(Πnj ,m (z ) − Πnj−1 ,m (z )),

j=k+1

we obtain lim sup ∥f − Πnk ,m ∥1γ/nk ≤ δ k→∞

ρα (γ ) Rα,m

.

This last expression and (5) give lim sup ∥Qnk ,m (f − Πnk ,m )∥1γ/nk ≤ δ k→∞

ρα (γ ) Rα,m

<

ρα (γ ) Rα,m

,

which contradicts Proposition 3.2. Necessarily





lim sup max Gk (z ) z ∈γ

k→∞

= 0.

Then relation (19) follows from (9) and Proposition 2.3. Fix a nondegenerate continuum Q of Ω . To obtain (20), we will reason by contradiction. Let us suppose that





lim sup max Hnk (z ) z ∈Q

k→∞

< 0.

(21)

Notice that min e−P (α;z ) > Rα,m .

(22)

z ∈Q

Formulas (21) and (22) imply that there exists δ < 1 with

 min

δ e−P (α;z )



Rα,m

z ∈Q

>1

(23)

such that

|Pnk ,m (z )|1/nk ≤ δ

e−P (α;z ) Rα,m

for all z ∈ Q and k ≥ k0 . Then, using (5) and (23), we obtain



 lim sup max Gk (z ) < 0, z ∈Q

k→∞

which, in light of (19), is absurd.



For each n ∈ N, set dn = deg Pn,m ≤ n and define Fn (z ) =

1 dn

log |Pn,m (z )| + P (α; z ) + log Rα,m ,

which is a subharmonic function on C \ Σ . Lemmas 3.1 and 3.3 allow us to prove the following lemma with the same arguments as those of [4, Lemma 3.4]. Lemma 3.4. Let Υ = {nk } ⊂ N be a subsequence verifying (8). Then there exists a subsequence Λ ⊂ Υ such that lim

n∈Λ

dn n

=1

(24)

and



lim max Fn (z )

n∈Λ

z ∈Q



= 0,

where Q is any nondegenerate continuum of Ω∞ .

(25)

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165

Given any set U ⊂ C and n ∈ N, let us define Ψn (U ) as the number of zeros of Pn,m in U. Recall that Tα,m stands for the polynomial convex hull of the compact set Dα,m ∪ Σ . Now, we are ready to obtain a first result on the location of the zeros of the approximants, which is essentially a consequence of Lemma 3.4. Although the arguments employed are similar to those used in Theorem 16 of [20], we include the proof for completeness given the relevant role played by this result in Theorem 3.7. Proposition 3.5. Let Υ = {nk } ⊂ N be a subsequence verifying (8). Then there exists a subsequence Λ ⊂ Υ such that lim

Ψn ( U )

n∈Λ

n

= 1,

where U is any open neighborhood of Tα,m . Proof. Let Λ be the subsequence of Υ given by Lemma 3.4. Fix an arbitrary open neighborhood U of Tα,m . From the Hilbert lemniscate theorem it follows that there exists a bounded open set W such that D = C \ W is connected and regular, and Tα,m ⊂ W ⊂ W ⊂ U. Let W ′ be another bounded open set with the same features as W and W ⊂ W ′ ⊂ W ′ ⊂ U. As the boundaries ∂ W and ∂ W ′ are lemniscates formed by a finite number of nondegenerate continua, we can use (25) to obtain



lim max Fn (z )

n∈Λ



  = lim max Fn (z ) = 0. n∈Λ

z ∈∂ W

(26)

z ∈∂ W ′

Set K = C \ U and, for each n ∈ N, let zn,1 , . . . , zn,kn be the zeros of Pn,m in K . Due to (24), it is enough to prove that kn

= 0. n Consider the functions lim

(27)

n∈Λ

fn (z ) = Fn (z ) +

kn 1

n j =1

gD (z ; zn,j ),

n ∈ Λ.

For z ∈ ∂ D = ∂ W , we have fn (z ) = Fn (z ), n ∈ N. Since the functions fn are subharmonic on D, it holds that max fn (z ) ≤ max fn (z ) = max Fn (z ),

z ∈∂ W ′

z ∈∂ W

n ∈ Λ.

z ∈∂ W

(28)

On the other hand, for all z ∈ ∂ W ′ and n ∈ Λ, we have fn (z ) − Fn (z ) =

kn 1

n j =1

gD (z ; zn,j ) ≥ δ

kn n

,

where δ = inf gD (z , ω), z ∈ ∂ W ′ , ω ∈ K > 0. Then, taking account of (28), for all z ∈ ∂ W ′ and n ∈ Λ, we obtain



0≤δ

kn n



≤ max fn (z ) − Fn (z ) ≤ max Fn (z ) − Fn (z ). z ∈∂ W ′

z ∈∂ W

Therefore, 0≤δ

kn n

≤ max Fn (z ) − max Fn (z ), z ∈∂ W

z ∈∂ W ′

n ∈ Λ.

Taking limits in the expression above as n ∈ Λ and using (26) gives (27) since δ > 0.



Before addressing the main result, we still need to prove an additional lemma. Lemma 3.6. Let f be an analytic function on an open neighborhood V of Σ that is not constantly zero in any component of V . Let K be a compact subset of Dα,m . Then Ψn (K ) is uniformly bounded by a constant. Proof. Fix a compact subset K of Dα,m . As −P (α; z ) is a subharmonic function on C, every connected component of Dα,m is simply connected. Then the polynomial convex hull of K , denoted by Pc(K ), is a compact subset of Dα,m that has a connected complement. Therefore, there exists a Jordan curve (lemniscate) Γ1 ⊂ Dα,m that contains Pc(K ) in its interior. By virtue of Lemma 2.1, there exists an analytic curve Γ ⊂ Dα,m which contains Γ1 in its interior and verifying Γ = Γ (ε) for ε sufficiently small. We may additionally suppose that there are no zeros of f on Γ since each component of Dα,m intersects Σ ⊂ V and, consequently, f cannot be constantly zero in any component of Dα,m . Denote by qm the monic polynomial whose zeros, counting multiplicities, are the poles of f in Dα,m . It follows from Lemma 2.2 that

 ∥qm Qn,m f − qm Pn,m ∥Γ ≤

ρα (Γ ) + ε Rα,m

n

<1

for sufficiently large n and ε > 0 arbitrarily small.

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On the other hand, from the way Γ was selected and (5), we obtain min |qm (z )Qn,m (z )f (z )| ≥ z ∈Γ

C n2m

.

Then, for sufficiently large n, it holds that

  ∥qm Qn,m f − qm Pn,m ∥Γ < min qm (z )Qn,m (z )f (z ) . z ∈Γ

Using Rouché’s theorem, we deduce that in the interior of Γ and, therefore, on K the number of zeros of qm Pn,m is equal to the number of zeros of qm Qn,m f for sufficiently large n, which concludes the proof.  Now most of the work has already been done and the proof below follows the same lines as that of the polynomial case in [4]. Recall that Θp stands for the normalized zero counting measure of the polynomial p. Theorem 3.7. Let f be an analytic function on an open neighborhood V of Σ that is not constantly zero in any component of V and it is not a meromorphic function on C with at most m poles. Let the measure α be the asymptotic zero distribution of the sequence of interpolation points given by {wn }n∈N , and let {Πn,m }n∈N be the corresponding sequence of multipoint Padé approximants of f . Let Υ = {nk } ⊂ N be a subsequence verifying (8). Suppose that the open set Ω = C \ (Dα,m ∪ Σ ) is connected with ∂ Ω = (Dα,m ∪ Σ ) \ Dα,m . Then there exists a subsequence Λ ⊂ Υ such that ∗

ΘPn,m −→  α,

n ∈ Λ,

where  α is the balayage of α onto ∂ Dα,m as defined by (7). Proof. Let Λ ⊂ Υ be the subsequence given by Lemma 3.4. Write ΘPn,m = βn , n ∈ Λ, for short. It is enough to prove that any limit point of the sequence {βn }n∈Λ in the weak∗ topology is equal to  α . Thus, suppose that there exist a measure β and a subsequence Λ1 ⊂ Λ such that ∗

βn −→ β,

n ∈ Λ1 .

Our task is to prove that β =  α . Notice that both of them are probability measures. Besides, from Proposition 3.5, the support S (β) of the measure β is contained in Tα,m = Dα,m ∪ Σ . On the other hand, from Lemma 3.6 it follows that S (β) ∩ Dα,m = ∅. Therefore, S (β) ⊂ ∂ Ω due to the hypothesis ∂ Ω = (Dα,m ∪ Σ ) \ Dα,m . By the same token, S ( α ) ⊂ ∂ Ω as well. Thus, in order to see that β =  α , it is enough to prove P (β; z ) = P ( α ; z ) for all z ∈ Ω , due to Carleson’s unicity theorem [22, Theorem II.4.13]. From here, the proof follows step by step the proof of [4, Theorem 3.1].  4. Overconvergence We maintain the notations previously introduced, see the first paragraphs of Section 3. As the following results deal with the meromorphic continuation of the function f , we require f to be determined by a unique analytic germ. Consequently, we assume henceforth that the open set V where f is analytic is connected. Since every component of Dα,m has a nonempty intersection with Σ , we conclude that Dα,m ∪ V and (Dα,m ∪ V ) \ (Σ ∪ Pf ) are connected sets. In this final section we study what can be said about a subsequence of approximants whose convergence rate is faster than the one specified in Proposition 3.2. We first prove that this fact does not depend on the particular continuum considered. Lemma 4.1. Let {nk } ⊂ N be a subsequence of natural numbers and let τ ∈ [0, 1). Then the following statements are equivalent: (a) There exists a nondegenerate continuum Q0 ⊂ (Dα,m ∪ V ) \ (Σ ∪ Pf ) such that 1/nk

lim sup ∥Qnk ,m (f − Πnk ,m )∥Q0

≤τ

k→∞

ρα (Q0 ) Rα,m

.

(b) For each compact subset K of (Dα,m ∪ V ) \ (Σ ∪ Pf ), it holds that 1/nk

lim sup ∥Qnk ,m (f − Πnk ,m )∥K

≤τ

k→∞

ρα (K ) Rα,m

.

Proof. It is enough to prove that (a) implies (b), since the reciprocal statement is trivial. Denote by qm the monic polynomial whose zeros, counting multiplicities, are the poles of f in Dα,m . Obviously, we have 1/nk

lim sup ∥qm Qnk ,m (f − Πnk ,m )∥Q0 k→∞

≤τ

ρα (Q0 ) Rα,m

.

Let z0 ∈ Q0 such that P (α; z0 ) = − log ρα (Q0 ) and fix δ > 0 arbitrarily small. As the potential P (α; ·) is a continuous function on C \ Σ , there exists an open disk B centered at z = z0 such that B ⊂ (Dα,m ∪ V ) \ (Σ ∪ Pf ) and P (α; z ) < − log ρα (Q0 ) + δ

B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

167

for all z ∈ B. We denote by Q1 the connected component of B ∩ Q0 that contains z0 . It is clear that Q1 is a nondegenerate continuum that verifies



lim sup max hk (z )



z ∈Q1

k→∞

≤ log τ + δ,

(29)

where 1

hk (z ) =

log qm (z )Qnk ,m (z )[f (z ) − Πnk ,m (z )] + P (α; z ) + log Rα,m .



nk



It is very easy to see that the functions hk are subharmonic in (Dα,m ∪ V ) \ Σ . Given an arbitrary ε > 0, it follows from (29) that there exists kε ∈ N such that max hk (z ) ≤ log τ + δ + ε

(30)

z ∈Q1

for all k ≥ kε . Now, choose a bounded open set W such that C \ W is a connected regular set, Σ ⊂ W ⊂ W ⊂ V , and W ∩ Q1 = ∅. The existence of such a set follows from the Hilbert lemniscate theorem [26, Theorem 5.5.8]. We can consider that ∂ W is made up of a finite number of analytic Jordan curves, each one exterior to the others. Let us denote by W1 a set with the same features as W such that Σ ⊂ W1 ⊂ W 1 ⊂ W . Our intermediate goal is to prove that



lim sup max hk (z )



z ∈∂ W

k→∞

≤ log τ .

(31)

Note that the disk B, and thus Q1 , depends on δ while W may remain fixed as δ tends to zero. In order to prove (31), take η > 0 small enough so that there exists a cycle Γ contained in



 



Dα,m ∪ V \ Eα (Rα,m − η) ∪ W ,

homologous to 0 in Dα,m ∪ V with winding number equal to 1 for all the points in Q1 ∪ W . This can be done because the compact set Eα (Rα,m − η) ∪ W is included in the domain Dα,m ∪ V . We can choose Γ with the additional properties that no pole of f belongs to Γ and that the open set D made up of the points with winding number equal to 1 for Γ is connected, since Dα,m ∪ V is connected. On account of Lemma 2.2, we have max hk (z ) ≤ ε

(32)

z ∈Γ ∪∂ W1

for all k ≥ kε , taking a larger number kε if necessary. Let us denote the polynomial convex hull of Q1 by K1 . Therefore, the open set D1 = D \ (W 1 ∪ K1 ) is connected and we can construct, for each ε > 0, a harmonic function ψε as the solution on D1 of the Dirichlet problem given by the boundary values

ψε (z ) =



ε,

log τ + δ + ε,

for z ∈ Γ ∪ ∂ W1 ; for z ∈ ∂ext K1 .

Since the functions hk , k ∈ N, are subharmonic on D1 and taking account of (30) and (32), we see that ψε is a harmonic majorant of hk for all k ≥ kε . From the two-constant theorem (see [26, Theorem 4.3.7]) it follows that there exist positive constants m1 and m2 depending on ∂ W ⊂ D1 such that

ψε (z ) ≤ ε m1 + (log τ + δ + ε) m2 ,

z ∈ ∂ W , ε > 0.

As the constants m1 and m2 are less than 1, for ε > 0 sufficiently small, the function ψε is bounded from above on ∂ W by log τ + δ . The same property is then satisfied by the functions hk , k ≥ kε . Therefore, we obtain





lim sup max hk (z ) k→∞

z ∈∂ W

≤ log τ + δ,

which gives (31) by making δ tend to 0. Fix now an arbitrary compact set K ⊂ (Dα,m ∪ V ) \ (Σ ∪ Pf ). As before, we consider a bounded open set W such that C \ W is a connected regular set, Σ ⊂ W ⊂ W ⊂ V , and W ∩ K = ∅. Such a set verifies (31). On the other hand, we take η > 0 small enough so that there exists a cycle Γ contained in



 



Dα,m ∪ V \ Eα (Rα,m − η) ∪ W ,

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B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

homologous to 0 in Dα,m ∪ V with winding number equal to 1 for all the points in K ∪ W . Again using Lemma 2.2, we have max hk (z ) ≤ ε

(33)

z ∈Γ

for all k ≥ kε . Taking account of (31) and (33), another application of the two-constant theorem to the domain D1 = D \ W , where D is the set of points with winding number equal to 1 for Γ , gives



lim sup max hk (z )



≤ log τ ,

z ∈K

k→∞

which, in turn, implies 1/nk

lim sup ∥qm Qnk ,m (f − Πnk ,m )∥K

≤τ

ρα (K )

k→∞

Rα,m

.

As K ∩ Pf = ∅, the above expression is equivalent to what we wanted to prove.



Given a subsequence {nk } ⊂ N and τ ∈ [0, 1), we say that {nk } is an Ostrowski subsequence of f of order at least τ if condition (a) in Lemma 4.1 holds true for such subsequence and τ . We denote by G the largest domain on which f admits meromorphic continuation with any amount of poles. Obviously, Dα,m ∪ V ⊂ G. Also, we can consider that V is equal to G minus the set of poles of f in G. The consideration of the parameter τ allows us to specify how far away from ∂ Dα,m the phenomenon of overconvergence takes place. This statement is made precise in the next result. Theorem 4.2. Suppose that {nk } ⊂ N is an Ostrowski subsequence of the function f of order at least τ ∈ [0, 1). Let z0 ∈ G \ Σ be such that ρα (z0 ) < Rα,m /τ . Then the sequence {Πnk ,m }k∈N converges in σ -content to f inside a neighborhood of z0 . Proof. Fix a point z1 ∈ Dα,m \ (Σ ∪ Pf ). Then there exists a path completely contained in G \ Σ connecting z0 with z1 along which the function f can be continued as a meromorphic function. Consequently, we can construct a Jordan curve γ whose interior D contains the points z0 and z1 and it is contained in G \ Σ . We can assume that no pole of f , except possibly z = z0 , belongs to D ∪ γ . By B0 and B1 , we denote open disks centered at z = z0 and z = z1 , respectively, with radii sufficiently small so that B0 ∪ B1 ⊂ D and B1 ⊂ Dα,m \ (Σ ∪ Pf ). Let q(z ) = (z − z0 )µ , where µ is the order of z0 as a pole of f . For each k ∈ N, set 1

hk (z ) =

nk

log |qk (z )| + P (α; z ) + log Rα,m ,

where

qk (z ) = q(z )Qnk ,m (z )[f (z ) − Πnk ,m (z )]. It is clear that the functions hk , k ∈ N, are subharmonic on D. From Lemma 2.2 it follows that 1/n lim sup q(z )Qn,m (z ) f (z ) − Πn,m (z )  ≤







n→∞

e−P (α;z ) Rα,m

,

uniformly on the curve γ . Note that we can consider that γ ⊂ V , since γ is contained in the domain of analyticity of f . Therefore,



lim sup max hk (z )



z ∈γ

k→∞

≤ 0.

(34)

On the other hand, from hypotheses and the proof of Lemma 4.1 we know that



lim sup max hk (z ) k→∞



z ∈B1

≤ log τ .

(35)

Taking account of (34) and (35), given any ε > 0, there exists kε ∈ N such that max hk (z ) ≤ ε, z ∈γ

max hk (z ) ≤ log τ + ε,

(36)

z ∈B1

for all k ≥ kε . Now, we construct, for each ε > 0, a harmonic function ψε as the solution on D1 = D \ B1 of the Dirichlet problem given by the boundary values

ψε (z ) =

 ε,

log τ + ε,

for z ∈ γ ; for z ∈ ∂ B1 .

B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

169

Since the functions hk , k ∈ N, are subharmonic on D1 and taking account of (36), we see that ψε is a harmonic majorant of hk for all k ≥ kε . From the two-constant theorem it follows that there exist positive constants m1 and m2 depending on B0 such that

ψε (z ) ≤ ε m1 + (log τ + ε) m2 ,

z ∈ B0 , ε > 0.

As the constants m1 and m2 are less than 1, for ε > 0 sufficiently small, the function ψε is bounded from above on B0 by log τ . The same property is then satisfied by the functions hk , k ≥ kε . So, we have obtained



lim sup max hk (z ) k→∞

z ∈B0



≤ log τ .

Then, for each η > 0, we have 1 nk

log |qk (z )| ≤ −P (α; z ) − log Rα,m + log τ + η

(37)

for all z ∈ B0 and k ≥ kη . From hypotheses it follows that

−P (α; z0 ) ≤ log Rα,m − log τ − 3δ for fixed δ > 0 sufficiently small. Therefore, there exists an open disk B centered at z = z0 such that B ⊂ B0 and

− P (α; z ) ≤ log Rα,m − log τ − 2δ

(38)

for all z ∈ B. Combining (37), with η < δ , and (38), we obtain 1 nk

log |qk (z )| ≤ −δ < 0

for all z ∈ B and k ≥ kη . Finally, using (5), we arrive at lim sup ∥q (f − Πnk ,m )∥ k→∞

1/nk B(ε)

≤ exp(−δ) < 1

for each ε > 0. This last expression implies that the sequence {Πnk ,m }k∈N converges in σ -content to f inside B, which is what we wanted to prove.  In particular, we obtain the next corollary. Corollary 4.3. Suppose that {nk } ⊂ N is an Ostrowski subsequence of the function f and let z0 ∈ ∂ Dα,m \ Σ be either a regular point or a pole of f . Then the sequence {Πnk ,m }k∈N converges in σ -content to f inside a neighborhood of z0 . Note that, as a consequence of Corollary 4.3 and Gonchar’s Lemma, if the orders of the poles of f on the closure of Dα,m add up to an amount greater than m, then f cannot have Ostrowski subsequences. Corollary 4.4. Suppose that the function f has an Ostrowski subsequence of order τ = 0. Then the largest domain G of meromorphic continuation of f is simply connected, and f has at most m poles in G. Proof. Let {nk } ⊂ N be an Ostrowski subsequence of f of order τ = 0. From Theorem 4.2 it follows that the subsequence {Πnk ,m } converges in σ -content to f inside G \ Σ . Then, by virtue of Gonchar’s Lemma, the function f has at most m poles on G \ Σ . As f has no poles on Σ , the second assertion follows straightforwardly. Now, suppose that G is not simply connected. Then there exists a bounded connected component A of the complement of G. Take a Jordan curve Γ contained in G surrounding A. We may assume that no pole of f belongs to Γ and thus Γ ⊂ V . Therefore, from Lemma 4.1, we obtain 1/nk

lim sup ∥Qnk ,m (f − Πnk ,m )∥Γ

= 0.

k→∞

Take a subsequence Λ ⊂ N such that lim Qnk ,m = Q , k∈Λ

where Q is a polynomial of degree at most m. Using the last two formulas, we have lim Qnk ,m Πnk ,m = Qf , k∈Λ

uniformly on Γ . The functions Qnk ,m Πnk ,m , k ∈ Λ, are analytic on the domain bounded by Γ . Using the maximum principle, it holds that the sequence of polynomials {Qnk ,m Πnk ,m }k∈Λ converges to an analytic function g on such a domain. Since g = Qf on G, we conclude that the function f can be extended as a meromorphic function onto the connected set A. This contradicts our definition of G as the largest set of meromorphic continuation of f , which, in turn, proves that G is simply connected. 

170

B. de la Calle Ysern, J. Mínguez Ceniceros / Journal of Computational and Applied Mathematics 284 (2015) 155–170

If m = 0 and f has an Ostrowski subsequence of order τ = 0, Corollary 4.4 implies that the domains of analytic and meromorphic continuation of f coincide. Acknowledgments The first author received support from Dirección General de Investigación, Ministerio de Educación y Ciencia (MINCINN) under grant MTM2009-14668-C02-02 and from Universidad Politécnica de Madrid through Research Group ‘‘Constructive Approximation Theory and Applications’’ with grant GI 110550208. The second author received support from MINCINN under grant MTM2012-36732-C03-02. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

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