Given a regular compact set E in the complex plane C, a unit measure µ supported by ∂E, a triangular point set β:={{βn,k}k=1n}n=1, β ⊂ ∂E and a function f, holomorphic on E, let πn,mβ,f be the associated multipoint β− Padé approximant of order (n, m). We show that if the sequence πn,mβ,f, n ∈ Λ, m− fixed, converges exact maximally to f relatively to the measure µ, then the points βn,k are uniformly distributed on ∂E with respect to µ as n ∈ Λ. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence Λ is provided, under the condition that Λ is ”dense enough.”

1.
H. P.
Blatt
,
R. K.
Kovacheva
,
Distribution of interpolation points of maximally convergent multipoint Pade’ approximants
,
Journal of approximation theory
, Volume
191
, (
2015
), pp.
46
57
. DOI:
2.
L.
Carleson
,
Mergelyan’s theorem on uniform polynomial approximation
,
Math. Scand.
, (
1974
),
15
,
34
,
167
175
.
3.
A. A.
Gonchar
,
On a theorem of Saff
,
Mat. Sbornik
,
94
(
136
) (
1974
),
152
157
.
A. A.
Gonchar
, English translation in
Math. USSR Sbornik
,
23
, No.
1
(
1974
),
149
154
.
4.
A. A.
Gonchar
,
On the convergence of generalized Padé approximants of meromorphic functions
,
Mat. Sbornik
,
98
(
140
),
1975
,
564
577
A. A.
Gonchar
, English translation in
Math. USSR Sbornik
,
27
,
1975
, No.
4
,
503
514
.
5.
R.
Grothmann
,
Distribution of interpolation points, Arkiv för matematik
, Volume
34
,
1996
,
103
117
.
6.
M. Bello
Hernándes
,
De la Calli
Ysern
,
Meromorphic continuation of functions and arbitrary distribution of interpolation points
,
J. of mathematical analysis and applications
, Volume
403
,
2013
, pp.
107
119
.
7.
N.
Ikonomov
,
Generalized Pade approximants for plane condenser
,
Mathematica Slovaca, -Springer
, accepted,
2014
.
8.
N.
Ikonomov
,
Multipoint Padé approximants and uniform distribution of points
,
C.R.Acad. Bulg.
, Volume
66
, Issue
8
,
2013
, pp.
1097
1105
.
9.
N.
Ikonomov
,
R. K.
Kovacheva
,
Distribution of points of interpolation of multipoint Padéapproximants
,
AIP Conference Proceedings, AMEE2014
,
1631
,
292
,
2014
,
405
412
.
10.
R. K.
Kovacheva
,
Normal families of meromorphic functions
,
Comp. R. de l’Académie bulg. des Sciences
, Tome
63
, N.
6
,
2010
,
807
814
.
11.
R. K.
Kovacheva
, Generalized Padé approximants of Kakehashi’s type and meromorphic continuation of functions,
Deformation of Mathematical Structures
,
1989
,
Kluwer Academic Publishers
,
151
159
.
12.
O.
Perron
,
Die Lehre von den Kettenbrüchen
,
Teubner, Leipzig
,
1929
.
13.
E. B.
Saff
,
V.
Totik
, Logarithmic potentials with external fields, Springer,
Grundlehren der mathematischen Wissenschaften
,
316
,
New York/Berlin
,
1997
.
14.
E. B.
Saff
,
An extension of Montessus de Ballore theorem on the convergence of interpolation rational functions
,
J. Approx. Theory
, Volume
6
,
1972
, pp.
63
67
.
15.
M.
Tsuji
,
Potential Theory in Modern Function Theory
,
Maruzen, Tokyo
,
1959
.
16.
J. L.
Walsh
,
Overconvergence, degree of convergence, and zeros of sequences of analytic functions
,
Duke Math. J.
,
13
,
1946
,
195
234
.
18.
J. L.
Walsh
,
The analogue for maximally convergent polynomials of Jentzsch’s theorem
,
Duke Math. J.
,
26
,
1959
,
605
616
.
18.
J. L.
Walsh
,
Interpolation and Approximation by Rational Functions in the complex domain
,
Amer. Math. Soc. Colloq. Pub.
, New York, Vol.
20
,
1969
.
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