Received 20 March 2015; accepted 30 April 2015; published 5 May 2015

ABSTRACT

Given a regular compact set in, a unit measure supported by, a triangular point set, and a function, holomorphic on, let be the associated multipoint -Padé approximant of order. We show that if the sequence, , , -fixed, converges exactly -maximally to with respect to the -meromorphy, then the points are uniformly distributed on with respect to as. Furthermore, a result about the behavior of the zeros of the exact maximally convergent sequence is provided, under the condition that is “dense enough”.

Keywords:

Multipoint Padé Approximants, Maximal Convergence, Domain of m-Meromorphy

1. Introduction

We first introduce some needed notations.

Let, be the class of the polynomials of degree ≤ n and.

Given a compact set, we say that is regular, if the unbounded component of the complement is solvable with respect to Dirichlet problem. We will assume throughout the paper that E possesses a connected complement. In what follows, we will be working with the max-norm on E, that is.

Let be the class of the unit measures supported on E, that is. We say that the infinite sequence of Borel measures converges in the weak topology to a measure and write, if

for every function continuous on. We associate with a measure, the logarithmic potential, that is,

.

Recall that ([1] ) is a function superharmonic in, subharmonic in, harmonic in and

.

We also note the following basic fact ( [2] ):

Carleson’s lemma: Given the measures supported by, suppose that for every. Then,.

Finally, we associate with a polynomial, the normalized counting measure of, that is

,

where F is a point set in.

Given a domain, a function g and a number, we say that g is m-meromorphic in B if g 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, if it is holomorphic in some open neighborhood of E.

Let be an infinite triangular table of points, , , with no limit points out-

side E (we write). Set

.

Let and be a fixed pair of nonnegative integers. The rational function where the polynomials and are such that

is called a β-multipoint Padé approximant of f of order. As is well known, the function always exists and is unique [3] [4] . In the particular case when, the multipoint Padé approximant co- incides with the classical Padé approximant of order ( [5] ).

Set

, (1)

where the polynomials and do not have common divisors. The zeros of are called free zeros of;.

We say that the points are uniformly distributed relatively to the measure, if

.

We recall the notion of -Hausdorff measure (cf. [6] ). For, we set

where the infimum is taken over all coverings of by disks and is the radius of the disk.

Let D be a domain in and a function defined in D with values in. A sequence of functions, meromorphic in D, is said to converge to a function -almost uniformly inside D if for any compact subset and every there exists a set such that and the sequence converges uniformly to on.

For, define

,

and

;

(is superharmonic on E; hence, it attains its minimum (on E)). As is known ( [1] [7] ),

,

Set, for,

.

Because of the upper semicontinuity of the function, the set is open; clearly if and if.

Let and be fixed. Let and denote, re- spectively, the radius and domain of m-meromorphy with respect to, that is

Furthermore, we introduce the notion of a -maximal convergence to f with respect to the m-meromorphy of a sequence of rational functions (a -maximal convergence), that is, for any and each compact set, there exists a set such that and

.

Hernandez and Calle Ysern proved the followings:

Theorem A [8] : Let and, be defined as above. Suppose that as and. Then, for each fixed, the sequence converges to f -maximally with respect to the -meromorphy.

Theorem A generalizes Saff’s theorem of Montessus de Ballore’s type about multipoint Padé approximants (see [3] ).

We now utilize the normalization of the polynomials with respect to a given open set, that is,

, (2)

where, are the zeros lying inside, resp. outside. Under this normalization, for every compact set and large enough there holds

,

where is a positive constant, depending on. In the sequel, we denote by positive constant, independent on and different at different occurrences.

In [8] , the set (look at the definition of a -maximal convergence) is explicitly written, namely, where

.

For we have

.

For points, we have

,

where stands for the number of the zeros of in;.

Let be the monic polynomial, the zeros of which coincide with the poles of in;. It was proved in [8] (Proof of Lemma 2.3) that for every compact subset of

. (3)

Hence, is a harmonic majorant in of the family.

Theorem B [8] : With and f as in Theorem A, assume that K is a regular compact set for which

is not attained at a point on. Suppose that the function f is defined on K and satisfies

.

Then.

Suppose that and is connected. Let V be a disk in, centered at a

point of radius and such that f is analytic on V. Fix, and set. Fix a number. Introduce, as before, the set. Recall that

.

It is clear that the set contains a concentric circle (otherwise we would obtain a contradiction with.) We note that the function f and the rational functions are well defined on. Viewing (3), we may write

,

Suppose that

.

or, what is the same,

.

for an appropriate. Then,

.

for all and large enough. This leads to

.

using Theorem B, we arrive at. The contradiction yields

,

where is the disk bounded by.

Then the function is an exact harmonic majorant of the family in

(see (3)). Therefore, there exists a subsequence such that for every compact subset

. (4)

(see [9] [10] ) for a discussion of exact harmonic majorant)). We will refer to this sequences as to an exact - maximal convergent sequence to f with respect to the m-meromorphy.

It is clear that for any and each compactum there exists a set such that and

.

2. Main Results and Proofs

The main result of the present paper is

Theorem 1: Under the same conditions on, assume that and that is a triangular set of points. Let be fixed, and. Suppose that is connected. If for a subsequence of the multipoint Padé approximants condition (4) holds, then as,.

The problem of the distribution of the points of interpolation of multipoint Padé approximants has been investigated, so far, only for the case when the measure coincides with the equilibrium measure of the compact set E. It was first raised by Walsh ([11] , Chp. 3) while considering maximally convergent polynomials with respect to the equilibrium measure. He showed that the sequence converged weakly to through the entire set (respectively their associated balayage measures onto the boundary of E) iff the interpolating polynomials at the points of β of every function of the form, -fixed, , converged -maximally to. Walsh’s result was extended to multipoint Padé approximants with a fixed number of the free poles by Ikonomov in [12] , as well as to generalized Padé approximants, associated with a regular condenser [13] . The case of polynomial interpolation of an arbitrary function was con- sidered by Grothmann [14] ; he established the existence of an appropriate sequence such that, , , respectively the balayage measures onto. Grothmann’s result was extended to multipoint Padé approximants with a fixed number of the free poles (see [15] ). Finally, in [16] the case was considered, when the degrees of the denominators tended slowly to infinity, namely, ,.

As a consequence of Theorem 1, we derive

Theorem 2: Under the conditions of Theorem 1, suppose that the -exact maximally convergent sequence satisfies the condition to be “dense enough”, that is

.

Then, there is at least one point such that

.

Proof of Theorem 1: Set, and. Fix numbers such that and is connected. Then, by the conditions of the theorem, for every compactum (comp. (4))

. (5)

Select a positive number such that. Let be an analytic curve in such that winds around every point in exactly once. In an analogous way, we select a curve. Additionally, we require that is constant on and. Set

. (6)

Let be arbitrary. The functions F_n are subharmonic in. By (5) and the choice of,

,

and, analogously,

.

Then, by the max-principle of subharmonic functions,

, (7)

where is the “annulus”, bounded by and.

On the other hand, by (5), there exists, for every compact set and large enough, a point such that

.

Therefore,

. (8)

Further, by the formula of Hermite-Lagrange, for we have

.

Hence, by (5),

where. To simplify the notations, we set (the correctness will be not lost, since is fixed). Involving into consideration the functions (see (6)), we get for

.

By Helly’s selection theorem [1] , there exists a subsequence of which we denote again by such that

,. Passing to the limit, we obtain

. (9)

Consider the function, harmonic in and

From (7) and (9), we arrive at

,

for in. Being harmonic, obeys the maximum and the minimum principles in this region. The de- finition yields

,

We will show that

, (10)

Suppose that (10) is not true. Let be a closed curve in the set, where stands for the interior of. Then there exists a number such that for every. This inequality con- tradicts (8), for close enough to the zero and sufficiently large.

Hence,. Then the definition of yields

.

The function is harmonic in the unbounded complement of, and by the maximum principle,

,

consequently,

.

On the other hand, , which yields in. By Carleson’s Lemma,. On this, Theorem 1 is proved. Q.E.D.

The proof of Theorem 2 will be preceded by an auxiliary lemma

Lemma 1 [17] : Given a domain, a regular compact subset and a sequence of positive integers, , , such that

,

Suppose that is a sequence of rational functions, , , having no more that poles in and converging uniformly of to a function such that

.

Assume, in addition, that on each compact subset of

. (11)

Then the function admits a continuation into U as a meromorphic function with no more than m poles.

Proof of Theorem 2: We preserve the notations from 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 converges maximally to f with respect to the measure and the domain. Hence, inside (on compact subsets) condition (11) if fulfilled. From the proof of Theorem 1, we see that there is a regular compact subset of such that.

Suppose now that the statement of Theorem 2 is not true. Then there is, for every a disk , with. We select a finite covering of disks such that

. Condition (11) holds inside. Applying Lemma 1 with respect to the sequence and

to the domain, we conclude that. This contradicts the definition of.

On this, the proof of Theorem 2 is completed. Q.E.D.

Using again Lemma 1 and applying Theorem A, we obtain a more general result about the zero distribution of the sequence.

Theorem 3: Let E be a regular compactum in with a connected complement, let and be a triangular point set. Let the polynomials, , be defined as above. Suppose that as and. Let be fixed, and suppose that. Then there is at least one point

such that for every positive.

Acknowledgements

The author is very thankful to Prof. E. B. Saff for the useful discussions.

References