Distribution of Points of Interpolation and of Zeros of Exactly Maximally Convergent Multipoint Padé Approximants ()
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 Fn 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.