Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 641-652
641
ON CONVERGENCE OF (I,v)-SEQUENCES OF
UNISOLVENT RATIONAL APPROXIMANTS
TO MEROMORPHIC FUNCTIONS IN C
C. H. LUTTERODT
Department of Mathematics
Howard University
Washington, D.C. 20059
(Received June Ii, 1985)
ABSTRACT.
A generalization of the classical Montessus de Ballore theorem in
some related theorems are discussed.
Non-Montessus type of convergence in
cn and
2n
dimensional Lebesgue measure is presented linking it to a result of Gon6ar.
KEY WORDS AND PHRASES.
Unlsolvent rational approximants, meromorphic functions,
Montsus-type convergence, non-Montessus type convergence.
1980 AS SUBJECT CLASSIFICATION CODE. 41A20, 32A20.
O.
INTRODUCTION.
Let f be in the class of meromorphic functions analytic at the origin, having
fixed number of codimension i polar sets over some polydisk domain in Cn. We investigate the modes of convergence of
(,)-sequences of unisolvent rational approximants
(URA) to f and we divide them into two main types; those associated with "horizontal
rows" of (,)-sequences of URA (for which v is fixed and
is free to grow infinitely)
and those linked to the "slanted rows" (for which
and v are related but both grow to
infinity) including "diagonal
rows" (cases where
definition 2.1, one requirement
;i
vi’
i
1,2
v).
As will become clear from
,n creates a "Pad Table" of upper
triangular (,)-sequences.
Some study of "horizontal row" of (,v)-sequences have
been carried out by Karlsson and Wallin [3] using explanations expressed in terms of
homogeneous polynomials in
2.
Our investigation of the convergence behavior of the
"horizontal rows" of (,) sequences constructed from non-homogeneous polynomials,
show that the convergence to f is uniform on compact subsets of the domain of mero-
morphy except on the analytic set {z
sets of
n:
I/f
O} or on the remaining limit polar
(,v)-sequences unattracted by f.
These limit polar sets have measure zero.
We refer to this type of convergence for (,)-sequences of URA as the Montessus
type.
The case of the "slanted
rows"
of (,v) sequences in
Cn
has been studied by
Gon6ar [2] using diagonal sequences constructed from homogeneous polynomials.
Although in one variable, it is known that for certain limited classes of functions
there can be locally uniform convergence for "almost diagonal
rows" of (,v)
sequences, it is generally recognized that most meaningful ways to handle convergence
of "slanted
refers to
rows"
of (,)-sequences is in measure or capacity.
n-Lebesgue
Here the measure
measure and capacity refers to any reasonably defined capacity
642
in
C.H. LUTTERODT
n adaptable
We call this type of convergence in
to this kind of problem.
rows"
measure (capacity) for "horizontal
rows" of (,v)-sequences, the
or "slanted
non-Montessus type.
Either type of convergence for (,) sequences gives rise to a certain rapidity
For the Montessus-type over-convergence means
and over-convergence (see Walsh [9]).
the domain of convergence of the (,v)-sequences of URA, includes in its interior the
domain of convergence of the local power series representative of f at the origin in
E
n.
For the non-Montessus-type, according to Gontar [2] over-convergence in measure
(capacity) means that convergence in measure (capacity) in any finite domain implies
convergence in measure (capacity) in
En
for f
The paper is organized to reflect the Montessus-type convergence in 3 and the
non-Montessus-type in 4.
In I we introduce the notations used in the paper and in
2 we introduce the definition of the URA’s and discuss the sense in which they are
unique.
The main theorems of the paper are theorems 3.1, 3.3, 3.4 and 4.1 (which examine
the different cases associated with the horizontal rows) and theorem 4.2 (which
rows).
examines a case of the slanted
tion of theorem 3.1.
There is also theorem 3.6 which is an applica-
The theorem yields a result about global analytic behaviour of
f if all the horizontal rows of (,v)-sequences are constrained in a certain way.
Some extension of this idea is discussed in Lutterodt [5].
(z)’s over-converge
insight into the way in which
Theorem 3.5 provides some
in the case of Montessus-type
and theorem 4.4, that of the non-Montessus type.
i.
NOTAT ION.
(z
Let z:
:
n
(I
{z.
and A
e
z
1.
n)
:
be an n-tuple in
(v
and v:
Izjl
I
o}, i
<
Vn
j
n and :
(z
qn
be n-tuples in
<
n, then A
n
A
e n-l;
Zn_l)
I
n-i
e lq
with
x...x
A
n-times
Let o
n E:
disk centered at origin. We denote by E the following subset of
n
n
where
is a partial ordering in IN
e. IN
0 )
},
given by
’-’
0
-
<:->
0
< J <
.,
1
<
<
0
is a poly-
n:
{%
n.
We adopt the following short notation as well
i+...+ n
Il
3z
l
EE
Let
be a domain in
tions holomorphic in
dz
zll
Z
a
dz ...dz
1
n
znn
Z
,(
I
n
n such that
Z
ae n
=0
0 e
,
al,
then
;7()
and continuous on
holomorphic at the origin, but meromorphic in
nomial
Pi(z)
in
Kn
of multiple
degree
or simply
be written as
P(z)
where
g gy1...y n
and z 7
l
Zl
E
n
n
gyz
a =0
()
n
is the ring of all func-
.
is the set of all functions
with finite pole sets in
’degree’
at most
(%1
A poly-
%n
can
CONVERGENCE OF (,)-SEQUENCES
Let
j
be the class of rational functions of the form R
ively; moreover (P
for some
P(z)/Qv(z)
(z)
and v respect-
are polynomials of ’degree’ at most
Pl(z), Q(z)
(z) Q,(z))
# 0 and
Qv(0)
where
643
i on A
n
It’
except on a subvariety of codimension /> 2
0 and for all (,).
I)
RATIONAL APPROXIMANTS.
2.
n
Let U be an open neighborhood of the origin in
R,(z)
e
is said to be a ,af.otctJ
qn,
E
(i)
E
(iii)
(iv)
(v)
0 if
to f at z
(2.1)
0
P (z))
z=O
an index interpolation set with the following properties:
E
0
(ii)
app:wma.t
(Qv(z)f(x)
z%
for
(U) with f(O) # 0, a rational function
Suppose f
DEFINITION 2.1:
v
e E
,
0
,- A
E
E
Each projected variable has the Pad indexing set.
v,
For each pair (,v),
[ElV <
(vi)
n
n
I where
j=l
=i
E
(v. + I)
(. + i) +
H
IEI
is the cardinality of
v
REMARK i.
Pad
index set is the index set that is used in the one variable case
to define Pad Approximants.
O, then
If the function f used in definition 2.1 is such that f(O)
depending on the order of regularity and in which variable, zj say, Weierstrass
REMARK 2.
Preparation Theorem can be
polynomial
W(,zj)
employed
with W(0)
then apply to g in f
W
to write f
W
g where W is a Weierstrass
0 and g is a unit with g(O) # 0.
Definition 2.1 may
g.
The rational approximants defined above are not in general unique.
of uniqueness is firmly tied to the cardinality of E
employed.
The question
Uniqueness as known
v is maximal in the following sense.
in Pad case seems only possible if E
DEFINITION 2.2.
n
(. + I) +
The interpolation set E
n
v
is said to be
n
+l)-i
j=l(J
j=l
It should be pointed out that there are many maximal
defining unique approximants.
PROPOSITION 2.1.
(U)
w.r.t,
.
Let the pair
maximal E
satisfies definition 2.1
w.r.t,
Since f
its "Pad table" equivalent.
<Pi(z), Qv(z)>
Suppose
the same E
P (z)
PROOF:
P*(z)
define a (,)-rational approximant
<P(z), Qv*(z)
’. Then
is another pair that
v
Q(z Q*(z) w.r.t. E
4 (U) and P q are polynomials qf
z
Taylor development in U.
can be chosen for
We note that when E v is maximal, there is
one-to-one correspondence between E v and
to f e
v that
if
This is a feature peculiar to several variables,
unknown in the Pad case in one variable.
a
mxa
Now by definition 2.1 and with E
P
(U) and has a
maximal we get
644
C.H. LUTTERODT
Qv(z)f(z)
where
gu\%
P (z)
\EVg z
Z
are the coefficients of the expansion.
(2 2)
<P,Q>
Similarly from the pair
we
again have
Multiplying (2.2) by
*
z
Q(z)f(z) P*(z) en\E,0gz
Q(z) and (2.3) by Qu(z) and subtracting
(2.3)
the former from the
latter we get
Q(z)P(z)
Q(z)Pu(z)
>"
r
(2 4)
iz
.
rv
where
is the coefficient of the expansion after taking the desired difference.
Now the R.H.S. of (2.4) is a power series regular with at least order
+ 1 in each
3
z.-variable. Therefore the L.H.S. of (2 4) must vanish w r t the maximal E v chosen,
j
since the L.H.S.
is a polynomial of
’degree’
Q (z)P* (z)
Recalling that maximal E
Q*(z) P
n)
Qu(,Zn
and
Q(0,z_ n )’ Q*(0,z_ n
Q(,O) # O. (2.6)
+ v.
z)
E
w r t
Thus we obtain
(2.5)
by construction, has maximal Pad4 indexing in each variable
and thus on projecting (2.5) on z -axis
n
yields
P (0, z
where
at most
the uniqueness of Pad approximants then
P(O,z n
(2.6)
Qv*(O,z_ n
.
do not vanish in some neighborhood U
The desired result then follows w.r.t. E
Q(O,O)_
U since
0
but also for each fixed
holds not only for
# 0
n-I
U.
From equation (2.1), we separate out the following linear equations"
z
z
(Qv(z)f(z)
P (z))
(Qv(z)f(z))Iz=0
z=O
O; %
0;
e E
(2 la)
. ED\E
(2.1b)
whose solutions for coefficients of
Qv(z) and then those of P (z) give rise to a
(u,) rational approximant. Now (,9)-sequences of rational approximants will be
called ani5oueut if the underlying E
is maximal and a certain determinatal or rank
condition is satisfied for the system of linear equations that arises from (2.1b).
(see also Lutterodt [4].).
For the rest of this paper we shall assume that we have (,v)-sequences of unisolvent rational approximants (URA).
with respect to some chosen E
Quv(z)
The latter is denoted by
u maximal.
We then normalize
n(z)
Qv(z),
Pv(z)/Quv(z
Puv(z)
dividing
u(z) by the modulus of the largest coefficient of Qua(z).
tion leaves
nu(z) unchanged but since P(z), Q(z) change we adopt the
denotation of ,Dv(z) with 9(z) normalized,
and
in
This opera-
following
(2.7)
3.
MONTESSUS-TYPE CONVERGENCE.
This section gives the generalized Montessus de Ballore theorem, some related
theorems covering the Montessus-type convergence and an application of Montessus
theorem to the generalized Pad Table.
CONVERGENCE OF (p,v)-SEQUENCES
THEOREM 3.1 (MONTESSUS).
](A n)
Let p
0 and v
with finite polar set defined by
(Vl,
Gv:
645
v
{z e
n)
n:
be fixed.
Suppose
qg(z) O} and
f e
n) where qo(z) is a polynomial of exact minimal ’degree’
v and A n n
0
Suppose
(z)
is an (P,)-unisolvent rational approximant
;J
to f(z) with its polar set
-i
-i
satisfying for
sufficiently large,
#
Then as
C(
q
Q(O)
.
n
Q(O)
(u.)
rain
G
l<j<n
(i)
(ii)
-i
An0 Qp(0)
An D G
f(z)
uniformly
rp(z)
on compact subsets of
n\G
REMARK:
The degree of convergence in (ii) of theorem
3.1 is geometric and it
rain (.).
The ’degree* of
in
(z) has to be exactly
l<j<n
The following lemma is used in the proof of
the theorem.
depends on p’
pv(z)
LEMMA 3.2.
For 0
p’ <
and
’
p
l.qn\ E
(.)
rain
l<j<n
’+I
(i
The proofs of Theorem 3.1 and Lemma 3.2 were given in Lutterodt
[6]; that of Lemma
3.2 being in the appendix.
The following two theorems are closely related to theorem 3.1 not only in statement but also in proof.
We shall therefore state them together and then briefly in-
(see, [6]).
dicate where in their proofs they differ from theorem 3.1.
Let
THEOREM 3.3.
n
(i
a polynomial of minimal
QI(O)_
and
(i)
(ii)
A
n
p
n
Kn
is the polar set of
Q-I
(0)
p
p(z)
THEOREM 3.4.
Then as
in
qm(z)
o A
and G
v(z).
(ii)
A
n
P
n
qf
e
C(
e (An))
Then as
with
qw(z)
where
is
fixed
u’
n
tends to a subset of A n G
p
f(z) unifoly on compact subsets of
A_k.p
Suppose the hypothesis of theorem 3.3 is satisfied with
u’
(i)
.
O} and
(z) is a (U,u)-unisolvent rational approximant to f(z) with
Suppose n
but
’degree’
Suppose f
0 be fixed.
and p
{z c
a finite polar set defined by G
-i
np
Q,(O)
(z)
v
G
s
tends to a set containing
as a proper subset.
n
f(z) uniformly on compact subsets of A except on G u Z
p
Lebesgue measure zero where Z
q
q
of
n_
contains the remaining limiting polar set of
REMARK ON THE PROOFS OF THE THEOREMS 3.3 & 3.4.
The proofs of the two theorems 3.3 and 3.4 as already indicated are essentially
the same as that of theorem 3.1 in our paper [6] except for minor changes in the
second half of their (i)-parts.
the exceptional set has
For theorem 3.4 we show further that the set G u Z
w
q
Kn-Lebesgue
measure zero.
The proofs, for second half of the (i)-parts for theorems 3.3 and 3.4, which
n
-I
n A and -i (0) n An
focus on
lim Qp(O)
P
P
P
p ,-=o
their corresponding versions of equation (3.8) in our paper [6]. i.e.
determine the relations between G
Q(z)q(z)f(z)
Q
qa,(z)P(z).
An
(3.1)
C.H. LUTTERODT
646
q(a)(a)
codim > 2),
that R.H.S. of (3.1) must give
(except for some subvariety of
Q
n
Dealing with theorem 3.3 first, we find that a
0
O.
(0) implies
((z),
Since
(z))
we must have that
q(a)
(a)
0 so
i for z
A
0.
n
Here
u’
and hence as
-I
n
n A
Q()
Q
p
i (0)
Anp
G
A
n"
(3.2)
O
The validity of equation (3.1) remains in tact on passing from subsequences to the
full sequence as discussed in the proof of theorem 3.1.
in A
A
n"
n
Thus the polar set of
(z)
I
gets completely attracted by the polar set of f(z) with similar multiplicity on
0
In the case of theorem 3.4, the equation (3.1) yields the following: Suppose
A
O. But since q(z)f(z) # O, except on a sub-varlety of cothen q(a)
n0,
G
a E
dimension
>
%(a)
2 therefore
0.
I
An
p
-i
Q(O)
hav
we must
With
An
p
(0)
G
as
n An
o
p
’
(3.3)
This remains true on passing to the full sequence from subsequences as done in
the proof of theorem 3.1 of [6].
Next we proceed to the final part of theorem 3.4
which begins with a modified version of the inequality (3.7) from our paper
[6].
The
inequality in question is
If(z)
Let K be any compact
A
A and K
A, Y np
z
get
np,.
(z)l < 1o<--)1 lq(z)l (xEZqn\
EB
n
and choose p’ > 0 such that
subset of A
p,
p’
< Cl( p
If(z)
i(0)
imply that
n A
_(G
P
d
first note that
in
A
np
is
n A
n
+
Ap,
n+n
el
01 (0)
n
(3.4)
n
An
(G
u
Zq)
n A
np
n
q(z)
0 on A which we know is not the case. So
connected
n
A
in A p" Therefore we can choose a countable sequence {
mm
Anp, m 1,2
centered at
where
sponding polydisks U
m
of
Anp
and each
Um n QI(o)
where d is a IR
2n
from (3.3).
#
m
1,2,...
Invoking
Q
{Um }m
Em
,
An_\,Q,I(o)
Jensen’s
vanishes in
Um cannot
n
P
we
is
must be dense
forms a covering
inequality we have
(3.5)
-Lebesgue volume measure for U
d
We claim
(0) with corre-
iumlgIFd(Z)[d >
that the set on which F
have measure zero.
(z)q(z)
For if not then since A
must have empty interior.
P
Fd(Z)
and define
IR2n-Lebesgue measure zero. This consequently will
of IR2n-Lebesgue measure zero. To prove the result
is a set of
Zq) n
Q-.I(0)
u
(i
n
Fd(Z)
then the zero set of
that
B’+I
,(z)l ]-O,(z)’qo(z)
(d I
n
_
0 < p’ < p implies
Then using Lemma 3.2 we can tighten the above inequality to
K
Now if we let d
)"
m
The inequality (3.5) tells us
have a positive measure, i.e. it must
n
the claim
that cover
Since there is a countable number of
Um
Ap,
is proved and hence the desired result.
Now for z
> 6 and
IB(z)[
C2
C
2(,C I)
K\(G
u Z
lq(z)l
q
>
> 0 such that
and
e.
’
sufficiently large we can find
> 0 such that
Thus on passing to sup-norm in (3.3) we can find
CONVERGENCE OF (,u)-SEQUENCES
(z)II < C2( P’)P’+I
lf(z)
647
p’
0 <- < 1
(3 6)
From (3.6) the desired result on degree of convergence
follows, showing the uniform
convergence.
-
The next theorem compares the rate of convergence of (,) sequences of URA
v(z)
to f
’(A)
[o(Z)
polynomials
is fixed, with the rate of convergence of the Taylor
where
The following definition shows that ’degree’ of
to f.
same as that of
DEFINITION 3.1.
A rational function
if in each z.-variable,
polynomials in z. las degree given by
COROLLARY 3.1a.
j*
is said to have
RIj
Ru(z)
follows from property (v) of E
is the
’degree’
expressed as a quotient of two pseudo-
(j
max
The ’degree’ of (,)
D
URA in this paper is always N.
This
in definition 2.1 where each pair satisfies
D,
making the "Pad@ Table" upper-triangular.
Suppose the hypothesis of theorem 3.1 is satisfied.
THEOREM 3.5.
A\G
compact subset of
Let K be any
then in terms of sup-norm on K, for sufficiently large
get
If(z)
where
0(z)
.
is the Taylor polynomial
PROOF.
1
where
F(z)
0
e
0
’degree’
(U) and let the
V, V n G
in U where
;
to f at the origin.
nr
Anr (so
that A
o(Z)
be the partial sum of f(z) of
n
and
being
A:)
be a neighborhood of
V being open and connected in A
p
Let
the closure of U.
Then
o
we
(z)ll K
z
Let 0 < r < p and let U
the origin such that f
’degree’
1K (llf(z)
(z)
’
FN(z) p(z)nvo(Z) 9(z).
and F9(z)
(U) and therefore by Cauchy’s
F(t)
1
F (z)
n
dtl...dt n
U
(2i)n
"
0
z.)
(t
J
C()
6
Integral formula
;
j=l
U is the distinguished boundary of U.
Using arguments similar to those
employed in the proof of theorem 3.1 of our paper [6] we can write
F (z)
E
z
If
where
fl))t
(2gi)n
f Ft)t+l
(t)
i
0U
dtl dtn
we claim that
Fv(z)
where
f%
I
)t
z
El+) \ E
(t) 0 (t)
U
tX+l
I
(2i)n 0
1
)f x))
(3.6a)
dt
I
.dt
n
(3.6b)
In order to substantiate this claim, we write
f(z)
+ g(z), where
is the
UO
f(z) and
is the remainder of the power series expansion of
f in
U. This makes the function
analytic but regular in each variable z. with order
partial sum of
g(z)
g(z)
o(Z)
C.H. LUTTERODT
648
.
+
Now observe from equations (2.1a) and (2.1b) that
i.
Dz
[N)(z) no(Z) PN(zlllz=0
x
:)za (blj,,(z)gl](z))I Z=0;
EI
X c
(3.7a)
and
However, the regularity conditions on
;iz
gN(Z)Iz=O
%
O,
%
. EIj
glj(z)
in each variable imply that
and hence invoking Leibnitz rule in the R.H.S. of (3.7a)
we get the R.H.S. to vanish, yielding the claim.
. E+\EN
We remark that even though for %
tive formula for (3.6b):
c E
form for
IJ+V
Recall that
equation (3.7b) provides an alterna-
there is no real advantage gained in adopting the second
",E
Qu)(t)
MQ
We then let
l(t)],
max
0U
t
independent of
;o(t)
being the partial sum of an absolute convergent development of
satisfy
l[uo(t)l
M, V t e
0U
G
since V
and U
Thus from (3.6a) we get using Cauchy inequality on
F ()
Q
FI v (z)
IQ
Thus
R.H.S. of (3.8) tends to zero.
Ng
>
.
u(z) i
<
,
G
v
(3.8)
Thus given
for z E U.
n
Thus on any compact subset K
Now since K
UR
0
]
and U
(0) n
R
G
[
An
v
G
UR,
0
and clearly as
,
l’
such that
R
where V
U
#, and
R
we must have for
D’
>
NI
el.
,
o
By Theorem 3 1 Q
K
0
large, we get by way of Huitz theorem in
z
NI
> O, N
i
n
This result holds verywhere in the complete
[o (z)n[ O(z) i(z)[ Ko <
2
that
l
1
Reinhardt domain of convergence of f(z) denoted by U
UR
on
z
The R.H.S. of (3.8) is the tal of a geometric series
;’ >
tlus
U must
0
Here M is independent U
V
f
.
v,
’degree’
is a normalized polynomial of fixed multiple
it is unifoly bounded on U.
H 6
....
0 such that
0
An
as
n
that
;
%(z)l
Thus for
lv(z) 0
>
J’
0
for all
sufficiently
for all z c K
0
and
furthermore
I0() (z) lEo < 60
Using triangular inequality for sup-nos on K O, we get
()lK
llf()
and for
’
> N
llf()
0
K
0
K0
0
we obtain
I
llf()
()iK
lf(z)
o(Z) IK0 +
since g > 0 is arbitrary, we get on K 0
If(z)
(z)ll
K0
llf(z)
uO
(z)l K
(3.9)
0
CONVERGENCE OF (,v)-SEQUENCES
’[’his
inequality is not violated if K
K
U
0
, n, "(;,.
K
R
&n\G
possible on K
is extended to any compact subset K where
0
Theorem 3.1 guarantees the L.H.S. of (3.9) to remain as small as
whereas R.H.S. of (3.9) cannot be made arbitrarily small in
’I’n-\U R for sufficiently large U’.
Ng
’
nD(z)llK < llf(z) -rDo(Z)IIK.
IIf(z)
>
Thus
implies
Suppose f(z) is analytic at the origin and is at most meromorphic
THEOREM 3.6.
Cn
with a finite polar set in
Suppose for each fixed
(z)
of each (,) unisolvent rational approximant
1’
649
(i’
to
the polar set
n
f(z) tends
to infinity as
n.
Then f(z) must be entire in
We need tlm following lemma in order to prove the above theorem.
to infinity as
U’
n be fixed.
and only if given any
(’I
Let
LEbMA 3.7.
if
-i
Quv(O)
for
I’
i
A
-i
n
(3.10)
For fixed
,
Qua(O)
suppose the polar set
.
tends to infinity as D
Then it is immediate that for any given 0 > 1 and a polydisk
n
-1
1
I:’ > N
QI,(O)_
bn,,
a
’
N
we have
A
n
n
0
Now on ^n
{ (z)}
_
ttowever, fix
lim
am
a
Q:.x)(a)
.
K
I
0.
A
n
0
#
.
I
An
{U<v(z)}
for which
Then
Now let
> 1 for
m m
|Jv’V
(z)
’degree’
(z)
as
so that
00
and suppose {a
K 1,
’0
is a normalized sequence of polynomials with fixed
and therefore we can choose a subsequence
uniformly on a compact subset K
a
-i
Qlv (0)
’.
such that
NI)
To prove the converse, we assume its opposite, i.e. we can choose
’0
tends
sufficiently large.
PROOF.
which
U (z)
Q(O)
of
The polar set
n
0 > 0 and a polydisk A
is a sequence of points in
Qlv(am) 0 for ali
, QI v(a) Q,(a)
Q:X
(0) for which
m and therefore by continuity we get
on K
I.
ConsequentIy
..,(a)
O.
Since
it follows that
A.n
’0
Thus the converse holds and this concludes the proof.
The proof of theorem 3.6 was given in Lutterodt
[4], so we merely provide an
outline for the present paper: Suppose
is analytic at the origin and meromorphic
in
with at most a finite polar set.
By theorems 3.1 or 3.4, the polar set of f(z)
attracts the whole polar set
or a subset of it as the case may be. But for
each fixed
-i
we know by hypothesis that the polar set of
tends to
infinity as
Thus the polar set of f must be drawn to infinity, making f
entire in
n
l(z)
n.’
,.
(z), Qua(O),
650
_
C.H. LUTTERODT
NON-MONTESSUS TYPE CONVERGENCE.
4.
In this section, we consider convergence of (I],)-sequences of unisolvent
not necessarily fixed as in 3. The convergence is
rational approximants with
’
Bisho
weakly expressed in terms of measure for the cases examined, using a lemma of
is replaced by
) where in
(
[i]. it should be noted that
n
(l
t;e latter v is no longer an n-tuple but a mere natural number.
0) with
(
I
6
Let
THEOREM 4.1.
e
qf
’
C(on)
g
IN but N remains an n-tuple.
g
where
z
(i
I
n)
<
) are such that 0 <
(
and
n
e
’degree’
is a polynomial of minimal
q0(z)
IN be fixed.
> 0 and 0 < I < l, and let 0,
n
and its finite polar set over A is G
()
Similarly
v <
q0(z)
(0
rain
|’
Suppose
O} and
). Suppose
(.).
l<j<n
(z) is a (,)-unisolvent rational approximant of f. Then for any
>
>
< 1 and |0
IN, positive such that
Hc
O, 0 <
A
compact subset K
Suppose
for all z
_
n,
K\ZII where ZDvl
where m is
En-Lebesgue
If(z)-
;](z)ll/’
{z
l]v(z)qo(z)
.
K:
’0
<
D)(z)l I/D’
If(z)
m[{z e K"
’
<
[;+o}
/>E}]
and
<
measure.
The difference between the above theorem 4.1 and the next one theorem 4.2 lies
not being fixed in Theorem 4.2.
in
X
’
K
v) is not fixed, with 0 <
),
<
rain
(j)
(,)
o(D’),
(z) is a (,)-unisolvent rational approximant to f(z).
but
n
<
00
l<j<.n
Suppose
A
,,
Suppose the hypothesis of theorem 4.1 is satisfied, except for
THEOREM 4.2.
(
c
compact,
< 1 and
> O, 0 <
If(z)
for all z e K\Z I- where m(Z
l/i)
l]
0
such that
,(z)l
< 6
’
’
>
0
-
as
Then for any
< e and m is a E n -Lebesgue measure
d
LEMMA 4.3 (Bishop). Let
Fd(Z be a normalized polynomial of degree
(d
d) in tli
> -and 0 < r < 1 be given. Then there
Let
exists a constant
c
c(n,o) such that the
n.
set
Z
has
n-Lebesgue
{z e
n ]Fd(Z)[
< N
d
measure satisfying
m(Z) < cn 2/n.
REMARK.
in Narasimhan
The proof of this Lemma uses an induction argument
and it is presented
[8].
PROOF OF THE THEOREM.
q(z)
same was as
qf_
e
.
(A)
i
f(z)
has a polar set
Without loss of generality we assume that
qc0(z) is
0}.
.(z).
n
(p)
and
tion given in the
Pl)(z) and u(z) are both
therefore qf
q ().
first part o
Since
te proof
G00
{z e
Kn
no--nalized in
polynomials in En and
the
6
Following the same presentaof theorem 3.1 of our paper [6], we extract
651
CONVERGENCE OF (p,)-SEQUENCES
the inequality (3.7) and transplant it to this paper as
IH.(z)l
<
)
M( >_,
n
%f. IN \E
From the above inequality we get,
where M is as before.
(z)l
If(z)
,
<-IQIz(z) qa(z)l
A,
INn\E
(4.1)
).
H
n
0’ < 0 ->
Let K be any compact subset of L and let 0’ > 0 be chosen so that 0
n
,n
.n
Then appealing to Lemma 3 2 for z e K, (4 i) becomes
A and K
r,
"
f,
_
bW-
P’
(p,n)
.z(z)l < Zqlo(z)
If(z)
(4.2)
qe(z)[ ()
where r(,,n) is a constant depending on p and n.
Let
z
2*.
(’iven
’--
z
n
0
< d}
lEd(Z)
e K-
d) is normalized in
(d
X+
< 1, let d
n.
+ 0)
( + z
.
+
with d
and
The polynomial
Fd(Z)
Fd(Z)
Thus by Lena 4.3, we must have the
)(z)q03(z).
of degree
En-Lebesgue
measure satisfying
m( -I"
Lrl--)
cr
2/n
R.H.S. being independent of the degree of F
have
d
IFd(z)l
,,,
KZ
Hu.
onsequentt
{z
t
}(z)
0 and cl
However, for
("i
we must therefore
K/Z
If(z)
K"
(p’)’+l
(4.4)
we can find
If(z)
for z
(p,n)
H(z)
p’
< 1 and
P
For any z
d.
O, so that
lf(z
Since 0
(4 3)
<
(SH’
2/n < g; the
set
t} is included in the set Z u and
(z)l I/H’
r
(4.5)
(.3).
[rom
The proof of Theorem 4.2 is identical to that of Theorem 4.1.
recognize that even though
because
But one has to
o(’), this does
not pre-
(0,1) for which the inequality (4.5) holds.
clude us finding 6
and some generalization for the case of meromorphic maps has
4.2
The Theorem
’
as
,
been discussed in Lutterodt [7].
The next theorem establishes over-convergence for the non-Montessus type.
(D) where D is a domain
Let 0 > 1 be fixed and let f
Suppose the conditions of either Theorem 4.1 or Theorem 4.2
THEORI 4.4 (Gondar).
in
n such
that A
n
P
D.
are satisfied and for any
m
Then
(z)
f(z) on any
(z)
n
-Lebesgue measure m as
RE.
f(z) in
n
K
compact
compact in D
The statement
’
m
(z)
f(z) on K as
H’
means that
as
f(z) as
’
and
l(z)
converges to
o(’)
C.H. LUTTERODT
652
We shall assume without loss of generality that 0
PROOF OF THEOREM 4.4.
En
of f in D
The pole set G
has
n-Lebesgue
measure zero (cf. final part of
Theorem 3.4 proof); since D is an open connected set, (D
in D.
Thus we can select distinct countable
disk neighborhoods A
n
centered at a
Oj
choose overlapping polydisks
&nP’ n
points,,
A
such that
J
n
,Aj.
,.j
0
n
so
al,a2,..,
D\G00
in
Since
must be dense
D\(;
and poly-
is compact, we
j=l
whose respective centers 0
a
aj i
are linked by a polygonal path which does not intersect the codimension 1 polar set
G
of f such that
c
A
0
n
The polygonal path then becomes a path along which f
n
with
0
sufficiently large we must have, for g > O.
cnn be analytically continued in D.
from theorem 4.1, for
N’
m{z e]( n
ANP
If(z)
Here we have identified A
n
P
(z)[ I/D >
For each of the overlapping polydisk neighborhoods
to
(4 6) in fact holds
Since
=n
u
n
Ojo
so that
} <
An
pj 1
(4 6)
n
,Ap.
a result identical
we obtain the following:
for p’
=00j
sufficiently large
m(z e X:
If(z)
(z)l I/’
>
} <
which gives the desired result.
ACKNOWLEDGEMENT.
The author is grateful to the referee for his comments.
The
research was partly supported by HURD grant #529053.
REFERENCES
i.
BISHOP, E., Holomorphic Completions, Analytic continuations and interpolation of
semi-norms, Ann. Math. 78 No. 3 (1963), 468-500.
2.
GONCAR, A.A., A local condition for single-valuedness of analytic functions of
several variables, Math. USSR Sb. 22 No. 2 (1974), 305-322.
3.
KARLSSON, J., WALLIN, H., Rational approximation by an Interpolation procedure
in several variables, Pad and Rational Approximation, Eds:
S&V. A.P. (1977),
83-100.
4.
LUTTERODT, C.H., On a theorem of Montessus de Ballore for (,)-type R.A. in
Approximation Theory III (1980), 603-609.
5.
LUTTERODT, C.H., On Uniform Convergence for (,v)-type rational approximants in
En II Internat. J. Math. & Math. Sci. 4 No. 4 (1981), 655-660.
6.
LUTTERODT, C.H., On a Partial Converse of Montessus de Ballore theorem in
Approx. Theory 40 (1984), 216-225.
7.
LUTTERODT, C.H., Meromorphic functions, maps and their Rational Approximants in
ed. Singh
Approximation Theory and Spline Functions Reidel (1984a)
et. al., 379-396.
8.
NAkSIMHAN, R., Several Complex Variables, Univ. of Chicago Press (1971).
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WALSH, J.L., Interpolation and Approximation by Rational functions in Complex
Domain, AMS Col. Pub. 20 (1969).
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En
n,
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