I nt. J. Math. & Math. Sci.
Vol. 4 No. 4 (1981)655-660
655
ON UNIFORM CONVERGENCE FOR
TYPE RATIONAL APPROXIMANTS IN (n. ii
CLEMENT H. LUTTERODT
6935 Spinning Seed Road
Columbia, Maryland 21045 U.S.A.
(Received June 3, 1980)
ABSTRACT.
This paper shows that if f(z) is analytic in some neighborhood of the
origin, but meromorphic in
n
sections in
otherwise, with a denumerable non-accumulatlng pole
and if for each fixed
rational approxlmant w
n.
be entire in
n
e
the pole set of each ( )
’
(z) tends to infinity as
min(i)
i<_n
+
unlsolvent
oo, then f(z)
This paper also shows a monotonicity property for the
lf(z)
must
"error
n.
m on compact subsets m of
KEY WORDS AND PHRASES. uniform convergence, ee functions, approximations
expansions.
sequence"
H(z)
1980 4ATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
and
41.
INTRODUCTION.
Two earlier papers by Lutterodt [1,2] gave results on uniform convergence
under restricted assumptions made about the (,9)
the
Bl-type
rational approximants.
In [i],
(,v)-rational approximants, were assumed to be uniformly bounded on a
polydisk; whereas, in [2], the (,l)-rational approximants were under the assumption
that the coefficients of the denominator polynomial of degree
vanished as
+(,...,) except for
bo()...O # O.
In fact,
bO()...O
(i,I,...,i)
is normalized to
unity.
In this paper, we attempt to provide a general result about uniform conver-
gence of (,9)-rational approximants to entire functions in
n.
I
C. H. LUTTERODT
656
The main results of this paper are Theorems i and 2.
-
uniform convergence for (,) unlsolvent
pole sections that tend to infinity as
e
.
on any compact-subset of
(=o,
Let z:
(l,...,n)
(z
z
I
If(z)
p
n and
(z)
shows that e
K
is monotonic in
n)
be an n-tuple point in
P(z)
ively, with
and
%(z)
and
P]j (z) 1% (z),
O (0)
9:
# 0
and 9, respect-
i in some neighborhood of the origin.
Suppose f(z) is analytic at the origin and f(0) # 0.
An
is said to be a (p,)-type rational approxlmant to f(z) at z
(z) s
P(z))Iz=O
z % ((z)f(z)
p
X e E
(i)
(il)
(iii)
c
,a lattice interpolation set with the following properties:
p
0 e E
EPD T e EPg’ i < %1
EU:{% E n: 0 %i Ui’
X e
l,...,n
i
n}
1
i
c
up
E
n
n
(v)
Each projected variable has the
(vi)
Ea ch
vi
<
i
i
i’""
Ep] is the cardinality of E
81IX
p9
Pad6 index
n"
and
8
1
+
+n
Xn
XI
z
1
n
0 if
(2.1)
0
(v)
Here
(l"’’’n)
are polynomials of multiple degree of at most
(F(z), Qv(z))
DEFINITION i.
for
.
:
let
for sufficiently
be the class of all rational functions of the form
R) (z)
where
cn.,
be n-tuples of non-negatlve integers in
Let
n
,oo) on compact subsets of
NOTATION AND DEFINITIONS.
2.
R
rational approximants with infinite
"error sequence"
Theorem 2 introduces an
large values of
Theorem i establishes
set
657
UNIFORM CONVERGENCE FOR RATIONAL APPROXIMANTS
,
An
DEFINITION 2.
(I "’’’n
v
is said to have multiple degree
R(z)
,
in the z.-variable,
if
polynomials in
e
Rv(z
has degree given by
zj,
,
expressed as a quotient of two pseudo-
max(j,j),
j
*
I <
J
< n.
that the multiple degree of a (,)-type
It follows from property (vi) of E
rational approximant is always
We shall refer the reader to the definition of a unlsolvent (,9)-type rational
approxlmant to f(z) in
We then normalize the
Lutterodt
[3].
We shall denote this by
w].l)(z) P].Iv(z)/QI/x)( z)"
denominator polynomial Q(z),
dividing numerator and denomin-
ator by the modulus of largest coefficient of the denominator polynomial.
Thus,
we get
WV (z) P*(z) /Qv* (z)
where
*
Qv(z)
is a normalized polynomial.
CONVERGENCE.
3.
The uniform convergence for the (,v)-rational approximants to f(z) entire in
n
rests on the assumptions made about f(z) and the hypothesis that, for each fixed
multiple denominator degree
/
9
of
(z),
the pole set tends to infinity as
In Theorem i below, we assume that f(z) is possibly meromorphlc,
(oo,...,oo).
not with a finite pole set as in Theorem 2 of
[3], but with
a pole set having infin-
ite sections such that only a finite number of such pole sections overlap with any
given polydisk.
THEOREM I:
Thus, Theorem 1 of this paper extends the result in [3].
Suppose f(z) is analytic at the origin and is possibly meromorphic
with an infinite pole set in
n
without accumulation of pole sections such that
given O > i, the polydisk
n
(n
A
< 0,
E
j
i
n,
0 > i}
overlaps with only a finite number of these pole sections.
Suppose
wv(z)
is a unisolvent
for each fixed 9, the pole set of w
(i)
(ii)
f(z) must be entire in
w9(z)
/
(,9)-rational approximant to f(z) such that
(z) tends to infinity as
/
n
f(z) uniformly on every compact subset of
n.
(oo,... ,oo).
Then
658
C.H. LUTTERODT
THEOREM 2:
Suppose the conditions of Theorem i are satisfied.
compact subset of
n.
Let K be any
Let
e
(z) llK
lf(z)
sup
z e K
If(z)
v(z)
(3.1)
for each fixed
Then for sufficiently large 9, e
LEMMA I.
of
(z).
Let
e
<
e,+l
is monotonic in 9 and satisfies
with
(z)
3
* (z)
be fixed and let
The zero set of
_< 9.
9j
+ i,
I
_< j _< n
be a normalized denominator polynomial
tends to infinity as
/
to a constant.
PROOF.
Suppose the result is false; i.e., for fixed
,
ity, but
Q(z)
Suppose that
n
given 0 > 1 and a polydisk A
0’
Q* (z)
’ran) mi < i’
in less than
/
Q
,
Q-I9(0)
re(z)
n
1 < i < n and that
Now, taking
a contradiction.
0o
and
sufficiently large,
(3.2)
Q*m(Z)
Thus,
is a polynomial of multiple degree
Then since Q * (z) is non-constant, it
m
Q-im(0),
the zero set of
0".m(Z)
cannot
> i, we find that
An
#
(3.3)
Hence the above supposition must be false and the Lemma holds.
PROOF OF THEOREM i.
f(z) is analytic at z
with an infinite pole set
G=
and
(0) tends to infin-
An=0
in a partial ordered sense.
Q-Im (0)
where
tends
is not constant as
has a set of non zero coefficients.
be empty.
, Q-ll;
* (z)
Q
does not tend to a constant.
By Lemma 1 in [3]
(ml
(oo,...,oo)
U G
k=l
Ok
0 and is possibly meromorphic
UNIFORM CONVERGENCE FOR RATIONAL APPROXIMANTS
Gok
qo (z)
=’ e’
659
0}
qOk(Z)
is a polynomial of at most multiple degree,
k
(kl
k
kn
Given any real number 0 > i, and a polydisk An
p’ then
the zero set G
overlaps the polydisk A
n
then we must have on A as
o
k
ko
+
P
o
Anp n Q i(0)
.
k
k
o
o
(0) such that
Now, by Theorem i of [3], if we choose
(
+
But by hypothesis, the pole set of
for each fixed
n
A
,)
np
n
(z)
Gk
(3 4)
o
tends to infinity as
Therefore, for the given p > i above as
/
-
(,...,=)
(oo,...,), we must
have
np Q-I(0)
A
n
(3.5)
Thus by (3.4) and (3.5) we must have
AnnG
P
Since
k
/
o
ko ko (p)
and p is arbitrary, it follows that
be entire.
This completes
Gok
must tend to infinity as
o
must tend to infinity and
Hence, all the poles of f(z)
oo.
=
ako
f(z) must therefore
(1).
To prove (ii), we note that the result follows immediately from Theorem i of
[3] and the (i) part just proved above.
PROOF OF THEOREM 2.
p > i and a polydisk
Let K be any compact subset of
n such that K
c
A
n
Then, for
P
n.
Then we can find
sufficiently large and
z e K, we find by the hypothesis of Theorem i, that for each fixed
* (z)
QU
# 0
> 0
i.e.
such that
lQu(z)
>
Hence, under these conditions, we get
I,+l(z) (z) ll K <
o
2
K
*
(z)
*
,
C. H. LUTTERODT
660
By Lemma i, we know tat
fixed
.
Hence, given
E
(z)
> 0,
tends to a constant as
(i0
0
+
(,...,) for any
such that for
,n0
i0
<
lin
*
* (z) ll
(z)
Q
1Qu,+l
K
M
0
--lIP* (z)[IAn
liP*
>-
(z)
P
ll K
is dependent on p but independent of
for each fixed
and
i0
<
i’
.
< e
(3 7)
2M
by the maximum modulus principle, and M
P
Hence, by combining (3.6), (3.7) and (J.8)
i -< i < n, we obtain
,(z)l
K
<
(3 8)
To get the desired inequality, we note by triangular for sup-norms on K that
e ,+i e + lw
<
where we have used the definition
For
+i
of e
(z)
(z)
ll K
(3.9)
as in (3.1).
i < i < n, and for each fixed
i0 <i’
e, +i
<e
Since e > 0 is arbitrary, the results follows.
ACKNOWLEDGEMENT.
This paper was written while I was at the Mathematics Department,
University of South Florida, Tampa, Florida.
REFERENCES
i.
LUTTERODT, C.H.
J. Phys. A. Math. Gen. 7, 1027-1037, 1974.
2.
LUTTERODT, C.H.
Complex Analysis and Applications
3.
LUTTERODT, C.H.
"On
ll__!l 25-34, IAEA, 1976.
(,)-type Rational
Approximation Theory III, 603-609 A,P, 1980.
a Theorem of Montessus de Ballore for
Approximations in
n,,