Distribution of points of interpolation and of zeros of exact maximally convergent multipoint Pad\'e approximants

Given a regular compact set $E$ in the complex plane, a unit measure $\mu$ supported by $\partial E,$ a triangular point set $\beta := \{\{\beta_{n,k}\}_{k=1}^n\}_{n=1}^{\infty},\beta\subset \partial E$ and a function $f$, holomorphic on $E$, let $\pi_{n,m}^{\beta,f}$ be the associated multipoint $\beta-$ Pad\'e approximant of order $(n,m)$. We show that if the sequence $\pi_{n,m}^{\beta,f}, n\in\Lambda, m-$ fixed, converges exact maximally to $f$, as $n\to\infty,n\in\Lambda$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $\mu,$ then the points $\beta_{n,k}$ are uniformly distributed on $\partial E$ with respect to the measure $\mu$ as $ n\in\Lambda$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $\Lambda$ is provided, under the condition that $\Lambda$ is"dense enough."


Introduction
We first introduce some needed notations.
Let Π n , n ∈ N be the class of the polynomials of degree ≤ n and R n,m := {r = p/q, p ∈ Π n , q ∈ Π m , q ≡ 0}.
Given a compact set E, we say that E is regular, if the unbounded component of the complement E c := C \ E is solvable with respect to Dirichlet problem. We will assume throughout the paper that E possesses a connected complement E c . In what follows, we will be working with the max− norm ||...|| E on E; that is ||...|| E := max z∈E |...|(z).
Finally, we associate with a polynomial p ∈ Π n the normalized counting measure µ p of p, that is µ p (F ) := number of zeros of p on F deg p , where F is a point set in C.
Given a domain B ⊂ C, a function g and a number m ∈ N, we say that g is m−meromorphic in B (g ∈ M m (B)) if it has no more than m poles in B (poles are counted with their multiplicities). We say that a function f is holomorphic on the compactum E and write f ∈ A(E), if it is holomorphic in some open neighborhood of E.
Let β be an infinite triangular table of points, β := {{β n,k } n k=1 } n=1,2,... , β n,k ∈ E, with no limit points outside E. Set Let f ∈ A(E) and (n, m) be a fixed pair of nonnegative integers. The rational function π β,f n,m := p/q, where the polynomials p ∈ Π n and q ∈ Π m are such that is called a β-multipoint Padé approximant of f of order (n, m) . As is well known, the function π β,f n,m always exists and is unique ( [14], [3]). In the particular case when β ≡ 0, the multipoint Padé approximant π β,f n,m coincides with the classical Padé approximant π f n,m of order (n, m) ( [12]). Set π β,f n,m := where the polynomials P β,f n,m and Q β,f n,m do not have common divisors. The zeros of Q β,f n,m are called free zeros of π β,f n,m ; deg Q n,m ≤ m. We say that the points β n,k are uniformly distributed relatively to the measure µ, if µ ωn −→ µ, n → ∞.
For Ω ⊂ C, we set where the infimum is taken over all coverings {V ν } of Ω by disks and |V ν | is the radius of the disk V ν .
Let D be a domain in C and ϕ a function defined in D with values in C.
A sequence of functions {ϕ n }, meromorphic in D, is said to converge to a function ϕ m 1 -almost uniformly inside D if for any compact subset K ⊂ D and every ε > 0 there exists a set K ε ⊂ K such that m 1 (K \ K ε ) < ε and the sequence {ϕ n } converges uniformly to ϕ on K ε . (U µ is superharmonic on E; hence it attains its minimum (on E)). As is known ( [15], [13]), Set, for r > ρ min , Because of the upper semicontinuity of the function χ(z) Let f ∈ A(E) and m ∈ N be fixed. Let R m,µ (f ) = R m,µ and D m,µ (f ) = D m,µ := E µ (R m,µ ) denote, respectively, the radius and domain of m− meromorphy with respect to µ; that is Furthermore, we introduce the notion of a µ−maximal convergence to f with respect to the m−meromorphy of a sequence of rational functions {r n,ν } (a µ-maximal convergence): that is, for any ε > 0 and each compact set Hernandez and Calle Ysern proved the following: Theorem A, [6] : Let E, µ, β and ω n , n = 1, 2, ..., be defined as above.
Suppose that µ ωn −→ µ as n → ∞ and f ∈ A(E). Then, for each fixed m ∈ N, the sequence π β,f n,m converges to f µ−maximally with respect to the m− meromorphy.
We now utilize the normalization of the polynomials Q n,m (z) with respect to a given open set D m,µ ; that is, where α ′ n,k , α ′′ n,k are the zeros lying inside, resp. outside D m,µ . Under this normalization, for every compact set K and n large enough there holds In the sequel, we denote by C i positive constant, independent on n and different at different occurances.
For points z ∈ Ω(ε), we have where k n stands for the number of the zeros of Q β,f n,m in D m,µ ; k n ≤ m. Let Q be the monic polynomial, the zeros of which coincide with the poles of f in D m,µ ; deg Q ≤ m. It was proved in [6] [6] With E, µ, m, ω n and f as in Theorem A, assume that K is a regular compact set for which e −U µ K is not attained at a point on E. Suppose that the function f is defined on K and satisfies lim sup , centered at a point z 0 of radius r > 0 and such that f is analytic on V. Fix r 1 , 0 < r 1 < r and set A := {z, r 1 ≤ |z − z 0 | ≤ r}.Fix a number ε < (r − r 1 )/4. Introduce, as before, the set Ω(ε). Recall that It is clear that the set A \ Ω(ε) contains a concentric circle Γ (otherwise we would obtain a contradiction with m 1 (Ω(ε)) < (r − r 1 )/4.) We note that the function f and the rational functions π β,f n,m are well defined on Γ. Viewing using Theorem B, we arrive at R m,µ + σ < R m,µ . The contradiction yields where V Γ is the disk bounded by Γ.
Then the function −U µ − ln R m,µ is an exact harmonic majorant of the family {|f QQ β,f n,m − QP β,f n,m | 1/n } in D m,µ (see (3)). Therefore, there exists a subsequence Λ such that for every compact subset (see [16], [17]) for a discussion of exact harmonic majorant)).We will refer to this sequences as to an exact maximally convergent sequence.

We prove
Theorem 1: Under the same conditions on E, assume that µ ∈ B(∂E) and that β ⊂ ∂E is a triangular set of points. Let m ∈ N be fixed, f ∈ A(E) and ̺ max < R m,µ < ∞. Suppose that D m,µ is connected. If for a subsequence Λ of the multipoint Padé approximants π β,f n,m condition (4) holds, then µ ωn −→ µ as n → ∞, n ∈ Λ.
The problem of the distribution of the points of interpolation of multipoint Padé approximants was investigated, so far, only for the case when the measure µ coincides with the equilibrium measure µ E of the compact set E. It was first rased by J. L. Walsh ([18], Chp. 3) while considering maximally convergent polynomials with respect to the equilibrium measure. He showed that the sequence µ ωn converges weakly to µ E through the entire set N (respectivey their associated measures onto the boundary of E) iff the interpolating polynomials of every function f t (z) of the form f t (z) := (t − z), t ∈ E, t− fixed, converge maximally to f t . Walsh's result was extended to multipoint Padé approximants with a fixed number of the free poles by N. Ikonomov in [8], as well as to generalized Padé generalized approximants, associated with a regular condenser ( [7]). The case of polynomial interpolation of an arbitrary function f holomorphic in E was considered by R. Grothmann ([5]); he established the existence of an appropriate sequence Λ such that µ ωn −→ µ E , n → ∞, n ∈ Λ, respectively the balayage measures onto ∂E. Grothmann's result was generalized in relation to multipoint Padé approximants π β,f n,m with a fixed number of the free poles (see [9]). Finally, in [1] was considered the case when the degrees of the denominators tend slowly to infinity, namely m n = o(n/ ln n).

As a consequence of Theorem 1, we derive
Theorem 2: Under the conditions of Theorem 1, suppose that the exact maximally convergent sequence Λ := {n k } ∞ k=1 satisfies the condition to be "dense enough"; that is lim sup n k+1 n k < ∞.
Then there is at least one point z 0 ∈ ∂D m,µ (f ) such that for every disk V z 0 (r) centered at z 0 of radius r lim sup n→∞, n∈Λ Proof of Theorem 1: Set Q β,f n,m := Q n , P β,f n,m := P n and F := f Q. Fix numbers R, τ, r such that ̺ max < R < τ < r < R m,µ and E µ (R) is connected. Then, by the conditions of the theorem, for every compactum K ⊂ D m,µ (comp. (4)) Select a positive number η such that R + η < τ < τ + η < r < R m,µ . Let Γ be an analytic curve in E µ (r) \ E µ (τ + η) such that Γ winds around every point in E µ (τ ) exactly once. In an analogous way, we select a curve γ ⊂ E µ (R + η) \ E µ (R). Additionally, we require that U µ is constant on Γ and γ. Set F n (z) := 1 n ln |F Q n − P n Q|(z) + U µ (z) + ln R m,µ , n ∈ Λ.
On the other hand, by (5), for any compact set K ⊂ E r \ E R and n large enough there is a point z n,K ∈ K such that − min K U µ (z n,K )−ln R m,µ −σ ≤ 1 n ln |F Q n (z n,K )−QP n (z n,K )|, n ≥ n 3 (K), n ∈ Λ.
Hence, φ ≡ 0. Then the definition of φ yields The function U µ (z) − U ω (z) is harmonic in the unbounded complement G of γ, and by the maximum principle, consequently, On the other hand, (U µ − U ω )(∞) = 0, which yields U µ ≡ U ω in E c . By Carleson's Lemma, µ = ω. On this, Theorem 1 is proved.
The proof of Theorem 2 will be preceded by an auxiliary lemma Lemma 1, [10] : Given a domain U , a regular compact subset S and a sequence ϑ := {n k } of positive integers, n k < n k+1 , k = 1, 2, · · · , such that lim sup n k+1 n k < ∞.
Suppose that {φ n k } is a sequence of rational functions, φ n k ∈ R n k ,n k , k − 1, 2, · · · , φ n k = φ ′ n k /φ" n k having no more that m poles in U and converging uniformly of ∂S to a function φ ≡ 0 such that lim sup Assume, in addition, that on each compact subset of U Then the function φ admits a continuation into U as a meromorphic function with no more that m poles.
Proof of Theorem 2. We preserve the notations in the proof of Theorem 1.
The proof of Theorem 2 follows from Lemma 1 and Theorem 1. Indeed, under the conditions of the theorem the sequence {π n } n∈Λ converges maximally to f with respect to the measure µ and the domain D m,µ . Hence, inside D m,µ condition (11) if fulfilled. From the proof of Theorem 1, we see that there is a regular compact subset S of D m,µ such that lim sup n∈∆ f − π n 1/n S < 1.
Suppose now that the statement of Theorem 2 is not true. Then there is an open strip W containing ∂D m,µ such that on each compact subset of W condition (11) holds. Applying Lemma 1 with respect to the sequence π n and the domain D m,µ W, we conclude that f ∈ M m (D m,µ ). This contradicts the definition of D m,µ .
On this, the proof of Theorem 2 is completed. Q.E.D.
Using again Lemma 1 and applying Theorem A, we obtain a result related to the zero distribution of the sequence {π β,f n,m }.
Theorem 3: Let E be a regular compactum in C with a connected complement, let µ ∈ B(E) and β ∈ E be a triangular point set. Let the polynomials ω n , n = 1, 2, ..., be defined as above. Suppose that µ ωn −→ µ as n → ∞ and f ∈ A(E). Let m ∈ N be fixed, and suppose that R m,µ < ∞. Then, there is at least one point z 0 ∈ ∂D m,µ such that lim sup n→∞ µ π β,f n,m (V z 0 (r)) > 0 for every positive r.