RATIONAL APPROXIMATION BY AN INTERPOLATION PROCEDURE IN SEVERAL VARIABLES J. Karlsson and H. Wallin We study the convergence properties of certain rational approximation schemes in several variables. These schemes are defined by interpolation conditions generalizing those that define Padé approximants in one variable. 0 Introduction If
f
define the function < V, of
is a (formal) power series in one complex variable we [ n , v ] Padé approximant to P/Q
of type
[n,v],
f
i.e. with
as the (unique) rational denominator
of degree
and numerator of degree <_ n, such that the Taylor expansion Qf-P
starts with terms of degree >_ n+V+1. Padë approximants
have been widely studied in recent years due to their usefulness in various applications (cf [l]). In many applications the need for approximants to functions of several variables arises naturally. This paper is concerned with generalizing the definition of Padé approximants to higher dimensions and studying properties of the approximants obtained. Seemingly, the first investigations in this line were carried out by Chisholm^s group in Canterbury (cf. Chisholm [3]). 1 Definitions and some properties Throughout this paper, we will restrict attention to two and
z?
for simplicity. We write
complex variables 2
ζη
(ζ^,ζ^) = ζ G C .
Most results carry over directly to any number
of variables and sometimes the appropriate changes will be indicated. A rational approximant to a power series f ( 2 ,l z 2) = J
kc jk z\ z\ 83
f
J. KARLSSONANDH.
84
WALL/Ν
will be of the form
P/Q
where
given classes such that
Qf-P
Taylor coefficients with
Ρ
and
Q
are polynomials from
satisfies given conditions on its
Q # 0.
Schemes of this type have
previously been described by Chisholm
et. a l . [ 2 ] , [ 5 ] , [10]
and Lutterodt [14]. These authors consider η
η
^
j= k=o
Jlc
1
and
1
0
V
ν
k
Jj
j=0 k=0
z
They also look at the more general setting when the sums range from 0
0
to
to
n-^, from
v -
0
We refer to
2
to
n^,
from
0
to
,
and from
[10] and [14] for details.
In Lutterodt the conditions on the coefficients of
I
Qf-P =
k
d.
k
z{ z
2
are d
jk
where
= 0
S, 2
(n+1)
(j,k)
the interpolation set, 2
+ (v+1)
_<(n,n)},
eS
- 1
is a subset of
Ν χ Ν
elements, containing the set
with
{ (j ,k) | (j ,k)<^
and satisfying the rectangle rule: (j ,k) € S =>
(£,m) E S
V(£,m) < (j,k)
under the usual partial order.
In Chisholm's case the interpolation set is smaller S = i(j,k)|(j,k) 1 (η,η) or j+k < v+n} and in addition he requires d
, +d ^ ,-, =0 1t v+n+1-ρ,ρ ρ,ν+η+1-p
for
p=l,
n+v.
The latter conditions may be changed to arbitrary linear conditions [ 5 ] . We propose rational approximants where P
z
J - . Σ jk i -2 a
j +k
r
THEORY
Q =
I
j+k
OF PADÉ
APPROXIMANTS
AND
85
GENERALIZATIONS
k
ZJ Z
β'jk l 2 '
Such approximants are said to be of type
[n,v]
or
approximants. We require that the interpolation set
[n,v] S
contains
{(j,k)Ij+k
Qf.
Ρ
is the appropriate partial sum of the
Furthermore the remaining conditions give
rise to linear equations in the coefficients of equating coefficients of
Qf
Q,
obtained by
or linear combinations of them to
zero. The number of equations is one less than the number of unknowns. By elementary linear algebra a nontrivial solution then always exists. It is clear that existence is independent of the actual choice of the interpolation set, only depending on the number of elements in it. Uniqueness of the approximants is a trickier matter, however. If we do not require that
Ρ
be the partial sum of
in trouble because then a valid approximant to e.g. z^/1 (1,0),
f = 0
Qf
we are would be
provided we do not ask for interpolation in the point
i.e. require that
THEOREM 1.1.
If_ P/Q
is a rational approximant of type
with an interpolation set
{(j»k)Ij+k
(1,0) € S.
S
[n,v]
containing, in addition to
as many points as possible from some given
enumeration of the points in
NxN, then
P/Q
is unique.
Remark. With the rectangle rule this gives in one variable an equivalent definition of Padé approximant (see [18], Prop. 1 ) . Proof. Suppose
Q^
and
are two linearly independent
polynomials giving the same (maximal) interpolation set. Then aQ^ + bQ^
gives at least the same interpolation set and we can
in fact choose
a
and
b
contradicting maximality.
to get an extra interpolation point
J. KARLSSON
86
AND
H,
WALLIN
Remark. If no finite maximal
S
methods in section 3 that
is rational provided
f
exists, it can be shown by the f
is analytic
at the origin. THEOREM 1,2.
Whenever an interpolation set satisfies the
rectangle rule and
P/Q
is an approximant to
f,
f(0,0) φ 0,
then
Q/P
is an approximant with the same interpolation set to
1/f.
(Lutterodt's, Chisholm's, or our definition)(c.f. [ 2 ] , [14])
Proof. By definition (j,k) € S. e.
= 0
Qf-P = ) d., z\
Multiplying by
for
(j,k) G S
zt
with
1/f: Q-P/f = \ e ^
d., = 0 z| z£
for
with
by the rectangle rule. For the proof in
Chisholm^s case we refer to [ 2 ] , Note that this works also with the approximants of Th. 1.1 because if we do not get maximal interpolation to
1/f
we would get more interpolation to
taking the reciprocal of the approximant to
f
by
1/f.
In the same way, under the same conditions, for all three definitions, but provided approximant to
f
and
η = V
we get: If_
ad - be φ 0
then
P/Q
is an
(a P/Q + b)/(c P/Q + d)
is an approximant with the same interpolation set to (af + b)/(cf +'d)
( c f . [ 2 ] , [14]).
Furthermore, provided
η = V,
approximants under all three
definitions are invariant under certain changes of variables,!.e. = if
w^
are functions of
f(w^,w^) as a function of P/Q
is the approximant to
z^., k (ζ^,ζ^)
1> 2, then the approximant to is
P(w^,w^)/Q(w^,w^)
where
f.
The relevant variable changes are: In Lutterodt's case: [14] — , k = 1, 2. wk = -1 —+^ B,^ z, ' k
k
In Chisholm'sA case: Z [2] k
w
k
=
, k = 1, 2. 1 + B, z, ' ' k
k
(By changing the second set of defining conditions different constants in the numerator can be allowed [5].)For our case the relevant variable changes are
THEORY
W
\
=
k
1 +
OF PADÉ
APPROXIMANTS
k
=Χ
+ Cz2'
'
AND
GENER
ALIZA
87
TIONS
2
'
This invariance is in all cases established by first noting that P(w^jW^)/Q(w^jW^)
will have correct kinds of numerator and
denumerator and then checking the interpolation. ChisholnTs and Lutterodt's schemes (with suitable interpolation sets) give the approximants the property that on setting
z^ or
= 0
we get Pade-approximants in the remaining
variable (projection property [2]). Furthermore a product of two z g^(z^) and § 2 ^ 2 ^
functions
have as a Chisholm
(or Lutterodt) type approximant the product of the P a d ap roximant i n
s gen ra
l
sati f e
I n
Lut erod
an
t
project
,
fo
o f
typ z ^
d
e =
ont
unde
l
hav
e
r
nic
. Ou an
y
[n,v]
.
r comple
se
t
s
e
n Th
e
e
wil
on
transformation
s l
e
s s
s
propertie
t
s
ap roximant
ap roximant
[n,v],
firs
ap roximant
s
lin
e
e
i s
b e
n +v}.
relationship
x
Th
thes
s
£
o f
certai
o f
contain
<_ j
type
.
approximants of type gives type
al
r
defin tion
) | 0
arguments
0) o
r
{(j,0 r
e d
ou
. Neithe
n
n +v}U <
[6]
interpolatio
invari nc s
d
b y
: h
=0
e
comparison
function
^
th
<_ k
establis
(z
an d
i f
) | 0
g ^
satisf e
d
{(0,k
t o
é
hav e ζ °^ χ
t o
o f
bot
n h
Chishol
function th
ca
o f m
s
o n
o r th
e
axe
s
+ZCe n i c = e ep r o p e r t Wet y o rt h a t n a ,a t
2 2
^
^
^
^-
while the Chisholm-Lutterodt approach
on the axes and type
[2n,2v]
on all other
lines. 2 Convergence of [η,η] approximants The fact that our approximants have the same type on all complex lines makes it natural to prove results by projecting onto complex lines and using standard techniques from onevariable Padé theory on every line. Our approximants will not be Padé approximants when projected but throughout this section our approximants are chosen so that
S
contains
{ ( j ,k) | j +k<_[n/2 ] } ,
the square brackets denoting the integer part. This will give
88
J. KARLSSON
AND
H.
WALLIN
sufficient interpolation to use the same techniques. This was first exploited by Goncar [8] who proved the following analog of a theorem of Nuttall's [15] for functions analytic at THEOREM 2.1. lf_
ζ = 0.
2 fCz^z^
rational approximants
is meromorphic in
^ n/ Q n of type
C
[η,η]
then the
(for a class of
definitions including ours) converge in measure to 2 subsets of C .
f
on compact
By the same method we get the following theorem, for the notation of which we refer to [12] and [8]. THEOREM 2.2. Suppose
1
1
f € R Q( A ( n ) ) .
Then on any complex line
11
capilf - Ρ /Q I > ε } < const. λ(η) η η —
/ε.
Note that the degree of convergence in capacity obtained here is not the same as in the one variableN case (Karlsson [8]). In general the exponent Jl-l
is
2"^ -1
where
Ν
is the complex
dimension of the space studied. The following generalizes a theorem of Wallin's [19]. THEOREM 2.3. Suppose that the power series
f = \
z^ z^
has coefficients that satisfy
ν ( I IV
ι
Λ
α/η
^ <
k |J 1 1 max I c., V [.n v/2"]-n v
α > 0
and some sequence
the Taylor polynomial of degree
k
complex line fr /r--. - Ρ /Q +0 [n v/2] nv nv
00 <
n^ -> °°. to_
Then if
f^
denotes
f, we have on every
0 as —
V ->
except on a set of q-dimensional Hausdorff measure zero. Remark 1 on proof. Projecting on a complex line we get a function, the n*"* Taylor coefficient :oefficient of which we can majorize by 1 ,2 + η - 1, J max c, · ( ) . (In Ν variabels the 2 is changed ι . tι ik η |j+k|=n
THEOR
to
N.)
Y OF PADÉ
APPROXIMANTS
AND
GENERALIZA
TIONS
89
Our approximants interpolate to this function of order
[n^/2]. If we carry out the same calculations as in Wallin [19] we get the desired result. The reason for the surprisingly neat formulation of theorem 2.1 is that once we have proved convergence in (2~dimensional Lebesgue) measure on every line it follows that we have convergence in (2N-dimensional Lebesgue) measure. In theorems 2.2 and 2.3 it is not easy to summarize the results in this way. We also want to note the following result (cf. Baker [1, p. 1 7 0 ] , Jones, Thron [11] for the one-variable case,* in several variables cf. Lutterodt
P
THEOREM 2.4. Let
[13]).
v/ Q v be rational approximants to f such that
the sum of the degrees of and such that
V^/Q^
P^
tends to infinity with
is uniformly (in v)
containing the origin, where converges uniformly to
and
f
f
ν
bounded in an open set,
is analytic. Then
V^/Q^
on compact subsets. This theorem is
true for any of the definitions of rational approximants. Proof. Take any subsequence of
P^/Q^.
By the Montel theorem in
several variables there exists a subsequence uniformly convergent on compact subsets. By the interpolation properties (use e.g. Cauchy^s estimates) we get convergence to
f
in a polydisc about
the origin, and by analytic continuation the convergence is to
f
everywhere. Thus every subsequence has a subsequence converging to
f.
Hence the whole sequence converges to 3 Convergence of [ n , v ]
f.
approximants, ν fixed
The purpose of this section is to analyze the possibility to generalize the Montessus de Ballore theorem (Theorem 3.2) from one to several variables. However, we also give a simple unified treatment in the one-variable case of some more or less wellknown results. The calculations are based on Cauchy^s estimates.
90
J. KARLSSON
AND H.
WALLIN
3.1 The one-variable case, ζ € C. THEOREM 3.1. Let variable in
f
be a meromorphic function of one complex
|ζ| < R
ξ , .. . , ξ - counted with their
with poles
χ
0 < Ιι ^ξ j. ιΙ < R.
multiplicities - where £
a rational approximant of type that (3.1)
n
[n,v],
Ρ η/Qη ,
0 <_ y <_ ν ,
Qη # 0 , ' be— V fixed, such
( )n j
I
(Q f - Ρ )(z) =
Let
z .
d
j>n+y
Then there exist V-y points the sequence
^
n^ n^
ζμ+^ > ···> ?
and a subsequence of
v
converging uniformly to
f,
and even with
geometric degree of convergence (see (3.5)), on compact subsets of_
{|z| < R} ^ {ξ^ |l <_ j <_ v } .
Proof. We put (3.2)
choose
Ρ η
and
R^ < R.
Q η
\
<
Since degree
J
max
e nt b g
Y (3.1)
j>n+y so that m a x { I Q ( ζ ) I : I ζ I = R} = 1. η
and thennormalization of ) \e)
ad
n) μ J J e| z .
I
(Q O f - Q Ρ )(z) =
Normalize
ζ
Q(z) = ( ζ - ξ ^ ... ( ~ ξ )
Q P^ <^ n+y,
We
Cauchy's estimates
give
2
I(Qf)(ζ)|R
UI=T?
1
,
for
j > n+y.
ι~ι
By summing a geometric series this and (3.2) give, with some constant
M(R^)
n y+
(3.3)
|(Q Q f - Q P ) ( z ) | < M ( R ) ( | z | / R )
Since
ÎQ -^ i-
n sa
n
n
1
|ζ| <_ R
By (3.3) the analogous subsequence of
and hence on
|z| < R.
(3.4)
Qqf - Qp = 0
in
|z| < R.
If
is a pole of
f,
then
q(z) = 0
which
•> q Φ 0, q a polynomial J
converges uniformly to a function
ζ
±
we can choose a subsequence
converging uniformly on compact sets,
|z| < R^
|z| < Κ .
sequence of polynomials of degree _< ν
are uniformly bounded in
of degree <_ v.
for
1
ρ
r
{^ *
η
on compact subsets of
Also
Q(z) = 0,
(Qf)(z) f 0,
by (3.4). Consequently, the zeros of
q
and hence
are completely
THEORY
OF PADÉ
APPROXIMANTS
determined by the zeros of ξ -, .... ξ μ+1 ν Let
Ε
where
Q
if
GENERALIZATIONS
μ = V.
If
be the rest of the zeros of
.
μ < V,
let
q.
{ | ζ | <_ R^} \ {ξ^ 11 <_ j £ ν}
be a compact subset of <
AND
From (3.3) we get
1 n/
lim sup max | (Q Q^f - Q P n) ( z ) | n-**> ζ EE
<_ R 2/ R .
If we combine this with the fact that Q Q
-> Qq
uniformly on
Qq φ 0
E,
we finally get what we want to prove: 1
E,
/ n J
i
, (3.5)
on
lim sup max |(f " P n / Q n )(z)| j-x» z€E j j
< R 2/ R
< 1.
Remark. With the same assumptions and proof we get that every subsequence of
^
n^ n^
contains a convergent subsequence of
the kind mentioned in the theorem. As a corollary we also get: THEOREM 3.2.
(Montessus de Ballore)
at the origin and meromorphic with the
[n,v]
Padé approximant to
Suppose that f is analytic V
f,
poles in
|z| < R.
P
n/ Q n>
i s,
when
η
Then is
sufficiently large, the unique rational function of type
[n,v]
which interpolates to
n+v+1
The poles of £
Ρ /Q ^n
f
at the origin of order at least
converge to the £poles of s n
converges uniformly to
f,
and
Ρ /Q η
with geometric degree of convergence
in those compact subsets of poles of
f
|z| < R
which do not contain any
f.
Proof. With the same notation as in the proof of Theorem 3.1 q
and
Q
have the same zeros. By multiplying
suitable constant we can make
q = Q.
with a
According to the above
remark this is true even if we start from a subsequence of {P^/Q^}.
Consequently
-> Q
uniformly on compact sets which
gives the required convergence. As lose, for large
n,
Q n^ )
Q(0) ^ 0>
any interpolation when dividing by
we do not in
the definition of the Padé approximant. This gives the desired interpolation and the uniqueness follows from the uniqueness of
92
J. KARLSSON
AND
H.
WALLIN
the Padé approximant. Remarks. a) The method with the normalization of
to give
a simple proof of Montessus de Ballore"s theorem is due to Harold Shapiro (personal communication). In fact, our proof is only a technical modification of Shapiro^s proof. Other treatments are in [17] and [8]. b) Theorem 3.1 has a connection to a conjecture by Baker and Graves-Morris (see [9] pp. 61-62) saying that if at the origin and meromorphic with
μ
then there exists a subsequence of
[n,v]
(v fixed) converging to of
f
in
poles,
f
is analytic
μ <^ v ,
in |z| < R,
Padé approximants
|z| < R
(except at the poles
f ) . Theorem 3.1 shows that the conjecture is true in the
weaker form allowing v - y
exceptional points where convergence
is not required. c) From (3.4) follows that if p ( z Q) = 0
then
iff
ZQ,
| Z ^ | < R,
q ( z Q) = 0
or
is not a pole f ( z Q) = 0,
i.e.
of
f,
ζ^
is, according to a standard theorem in complex analysis, an Z ^( Q) ~ 0
accumulation point of zeros of Padé approximants iff or
ζ^
is accumulation point of poles of Padé approximants.
3.2
The many-variable case, ζ = (ζ^,ζ^).
In this case the
transition from (3.1) to (3.3) does not work (see the discussion after Theorem 3.4) and the situation is more complicated. Among other things we shall give counterexamples to the natural generalization of the Montessus de Ballore theorem. However, if we in this theorem change the assertion that for all large there exist unique rational functions of type [ n , v ] to
f
of order
n+v+1
n+1
interpolating
to the assertion that there exist
rational functions of type at least
η
[n,v]
interpolating to
f
of order
at the origin, then we get a convergence theorem
also in several variables (for the one-variable case see [20]): THEOREM 3.3.
Let
f = F/Q
where
{ζ = (ζ^,ζ^): |z^| < R^, i = 1,2} degree
V,
Q(0) ^ 0.
F and
is analytic in the polydisc Q
is a polynomial of
Then there exist rational functions
R
THEORY
of type least as
OF PADÉ
APPROXIMANTS
AND
[n,v] interpolating to f
93
GENERALIZATIONS
at the origin of order at
n+1 (i.e. having the same Taylor polynomial of degree
f) and converging uniformly, as η
η
°°, _to f, with
geometric degree of convergence, in compact subsets of {z| | Z | i< R J ^ {z|Q(z) = 0 } . Proof. Let π η Choose
> 0
if necessary, analytic in
be the Taylor polynomial of degree
η
such that
By replacing,
R^ by smaller numbers, we may assume that |z^| _< R^.
for some constant max
(R - e - ^ / R ^ ( R 2^ 2) / R 2.
M(R^,R 2>
KfQ-π )(z)| <
max
l illV 53 z
£
e
7
s=n Ε
j
iR x/
/ ^ • " ^ N
If
F is
Consequently, Cauchy's estimates give,
l illV i z
of fQ.
1
£n +
e 7
i ~
z J 1 K R ) ' |z | | λ £_ , .jz, |.k< kR >R
M(R
nl 2 x
i \
1
is a compact subset of i|z^| <_ R^"^^} ^ {Q(z) = 0}
we conclude l i/n/ lim sup max |(f - π / Q ) ( z ) | <- V ^ < L n-*» zGE 1 This gives the desired convergence property for The interpolation follows from the choice of π The estimates in this proof work polynomial of degree
of degree <^ ν η
of O f η
and
if we normalize
R
= ^/QQ(0) φ 0.
since
if we replace
Q
by a
by the Taylor polynomial Q η
to have maximum 1
in a suitable polydisc. If we combine this with the method from section 3.1 of choosing a subsequence of ^ Q n ^ converging to q, we obtain, as in the proof of Theorem 3.1, the following generalization to several variables of Theorem 3.1 in the case μ = 0. THEOREM 3.4. Let type
{P /Q }, η η
Let f
be analytic in {|z i| < r £, i = 1,2}.
Q # 0 , be a sequence of rational functions of η '
[η,ν] , ν _> 0, ν fixed, such that
polynomial of degree
η
Ρ
η is the Taylor
oj[ ^ η^ · Then there exists a polynomial
94
J. KARLSSON
AND
H.
q, q Φ 0, of degree uniforml
y
compac
t Th
e
i n Le
th t
t
Q e
polydis P/
η
f a
<
give
a
}
a
rationa
o f
o f
{z|q(z
No
w e
an
d
functio
£
=
F n
i n o f
is the Taylor polynomial of degree
convergin
g
t
Q(0 th
onl tha
) e
^
y
[n,v Q nf *
of
i n
t
f
0 ,
h
Then
Q an lyti
^_z| suc
e /F
i s
c ]
th =
f
polydis
e
o n
.
e
typ η
0}
as um V ,
n^
,
resul
w e
n /Q
convergence
) e
degr
,
l
e
^
.
| < ^ r^
degr
convergenc
an lytic l
|z^
r_^
s
polynomia
b e
η
c |
i s
c Q
geometri
^z|{ m
e i s
h
o f
Theore
wher
wit
s
las c
wher
f ,
sub et e
polydis
t o
WALLIN
c | < ^
tha
t
. Ρ
η
we have an
expansion (3.6)
( Q nf - P n) ( z ) =
I d ^ z J j z ^ . j +k>n
In order to be able to use the fact that
z
e w
\ \} — ^i (3.7)
^ I t ^ p l y by
degree < n+V
; |e. | <
M(R .
The coefficients Q Ρ η
.
1
z.l < R.} = 1. Since ι — ι we get by Cauchy's estimates max{|Q(z)
is analytic in
and get
( Q Q nf - Q P n) ( z ) = _ I e
z}J , \ j +k>n
Suppose that
(3.8)
Q
F = Qf
Q Ρ
η
has
R ) for
j+k > n+v+1.
2 e^!^
with n+l < i+k < n+V depend also on . . ~ and can not be estimated in this way. There are
v+1 J (n+s) = vn + v(v+3)/2 s=2 such coefficients. We can impose by choosing the coefficients of v(v+3)/2
of the
(
v(v+3)/2 Q
so that, for instance, ^
\
d.^ -coefficients, Jk
conditions on them
+ l n< j+k < n+V —
become
—
zero. However, we cannot - as in the one-dimensional case- choose interpolation conditions which guarantee that all the coefficients e ,
n+l < j+k < n+V
become zero. This leads to counterexamples
to a general analogue of the Montessus de Ballore theorem (see section 4 ) . An estimate of
n+l < j+k <_ n+V,
or of the
THEORY
OF PADÉ
APPROXIMANTS
corresponding coefficients
d
i
AND
s
GENERALIZATIONS
95
given by the following lemma
where we use the same notation and assumptions as above. LEMMA 3.5.
For some constant
M
and
n+l <_ j +k <_ n+V,
j n ) j k k Jk n | d ^ | < M 1 max IQ ( Z ) - Q ( z ) | r 7 r ~ + MR7 R ~ . |z.|
The same inequality holds for e f . \ jk
Proof. Cauchy's integral formula gives
( 2) π ι d r,.^2,(n) jk
(£
1 ,/ 1
V
(Z)
j +i
J
k+i
|z.|-r.
,
z2
,
dz
j+1 k+1 z. =r. z^ z0 ι ι 1 2 f(Q -Q)(z) z d+
-,
=
,
Λ
"
U j+i
|z.|=R. JZ
s
z d
k+i z2
and this gives the desired estimate. Remark. Chisholm and Graves-Morris [4] have studied the problem of generalizing the Montessus de Ballore theorem to functions of several variables for the Chisholm type approximants. In their estimates they do not seem to have taken into consideration their problem corresponding to those coefficients in the right member of (3.7) which are influenced by
Q P^
(see the fourth line
after (4.57) in [4]). It is not clear how this gap in their proof can be filled. 4 Convergence of [n,l] approximants When
ν = 1
explicit calculations are easy. We give a
convergence result and two counterexamples. The results, which should be compared to the case
V = 1
in Montessus de Ballore"s
theorem,show that the several variable case is more complicated than the one variable case. THEOREM 4.1. f c
Let
is analytic in
f = F/Q
where
Q(z) = l - b ^ - b ^ ,
|z^| < r^, i = 1,2,
1
b
φ 0,
with Taylor coefficients
1y
in Iz.l < R. with Taylor coefficients a M , F is analytic jk' ι ι ^ jk F ( b 1 ,0) φ 0, jmd 0 < Γ χ<_ l/|b_L | < R , 0 < r2 <_ R 2. Let_
s
96
J. KARLSSON
AND
H.
WALLIN
P^/Q^
be a rational approximant of type
[n,l]
satisfying at
least the interpolation conditions determined by (4.D
n-v«
= . i
- < l
* $ 4 4 >
J
j+k>n Then
*
^ n/ Q n converges uniformly to
f,
= °-
*
with geometric degree of
convergence, in compact subsets S < s of ί|ζ^|<^, i=l,2}\{z|Q(z)=0} if_
l l l l l 0'0 b
l
Example 1.
R
r
= R^ =
2
<
l li i 2> b
R
r
s
<
2
V
gives convergence everywhere except where
Q = 0. Example 2. r^ - r^ - r,
R^ = R^ = R
polydisc
|z^| < |b^|Rr
which is larger than the polydisc
|z^|
where
< r
|b^|R.
f
gives convergence in the
is analytic - we have got an expansion factor
By choosing the interpolation conditions differently it
is possible to get other expansion factors, for instance Proof of Theorem 4.1. We first observe that is the one-dimensional Padé approximant to Q (0) -> Q(0) φ 0, η
that large
η
Q n(z)
put
The condition
Ρ η( ζ χ, 0 ) / Q n( z 1, 0 ) f(z^,0).
=
so that
zn1 d^ ] = 0 n+1,0 Λ
(4
.
n+1
2)
_ .
, ; i .
/ \
v
1 Note that
A
η
1
a n c
^
J )(b^+b^) (^(z),
η
| M ,
A n- | , . 0b - i .
η
•> F(b \ θ ) ^ 0 by the assumptions. Analogously, . ^ l d { = 0 gives after simplification with the same η, 1 as in (4.2)
η the condition
A
Q n( 0 )
gives, since
(fQ n)(z) = (F · 1/Q · Q n) ( z ) = I a jk zj z\ that
Because of
by Montessus de Ballore's theorem. For
we may therefore normalize n )
= 1- ej
|b 2|R.
THEORY
2 3„
(4.3)
OF PADÉ
APPROXIMANTS
()n
• (b, - 3 Î
Hence
f
.
) ^ r^
and
J
R^
Ί
by smaller numbers, we may
1z . l1 < r.
and
F
I z .1l < R.. ι — ι
in
ι — ι (4.2) gives
and
η)
β<
97
5-
bn A 1 η
is analytic in
a. = 0(R R ) jk 1 Ζ
(4.4)
n-1
)
Replacing, if necessary, assume that
GENERALIZATIONS
2
- b, = - ^ - + b? A 1 η n-1
n
AND
n
- b 1 = 0 ( l ) ( | b 1| R 1) " .
Analogously, a simple calculation and (4.3) give (4.5)
b
,(n) f$2 "
=
n
° ( n ) ( | b 1| R 1)
2
.
The last two formulas give an estimate of
- Q.
We can
consequently use Lemma 3.5. Combined with (3.6) - (3.8) this gives in the usual way afterS some calculation, if swe put S S
δ
= xm fal
ι
d
2\ - —J
a= m δnx a / l
2\ \ις ' îçj ·
2
n = 0 ( n ) ( | b1 max |(Q Q f - Q Ρ )(z)| .n| R . ) " |z.|
n
+1
(η+2)δ"
1
+
oo
+ 0(1)
y (j+DO^ . n+l -> Q this gives in the usual way the desired convergence R
Since if
<
-jJb-|J
and
6 ^
< 1
and the theorem follows.
Counterexample 1. We construct an example where b x φ 0,
f(z) = F(z)/(1 - b ^ ) ,
F(z) = I a jk z\
z\
,
2 and
F
is analytic in
an entire function in
C . z^
We choose
having
a^
so that
F(0,z^)
is
[n,l] Padé approximants with
poles clustering everywhere in the complex plane, which is possible according to an example by
a
choose
-^j
>
J
0>
o
s
that
F
Perron [16, p. 2 7 0 ] . We then 2 becomes analytic in C . Let
98
J. KARLSSONANDH.
WALLIN
P n/ Q n be a rational approximant of type [n,l] determined by
n )
(f Q
- Ρn) (ζ) =
y
J z\ ζ) ,
j+k>n
[n,l] Padé approximant of
cluster everywhere P in Hence, ζ = 0.
C
= na <
n+1,0
By these interpolation conditions the
> >
+1 = 0.
0,n+l
Ρ^(0,z^)/Q n(0,ζ^) ζ^) ί(09
= F ( 0 , z 2)
becomes and its poles
by the construction.
n/ Q n cannot converge to
f
in any polydisc around
A comparison between this counterexample and Theorem 4.1
(Example 1) shows the importance of the choice of the interpolation conditions. Furthermore, it is easy to see that Q^Cz)
does not converge to
Q(z) = 1 - b^z^,
i.e. the
singularities of the approximants do not even converge to the singularities of
f.
We now give a more complicated counter
example in a case where
-+ Q.
Counterexample 2. We use the same notation and assumptions as in Theorem 4.1 and determine f = F/Q by putting k oo j oo 1 |>1. F(z) = I z + Z I (pb ) ζ] ζ where 0
>z
(4.2) and (4.3) give with
Ώ) β{
(4.6)
Q n( z ) = 1 - &^
η) = b
x
β^
and
y
z± " β^' 2
n
- b 2= a^/b* = p
for
η > 1.
The coefficients d f ^ are given by (4.1) and we see that / \ n+1 . (η) ηγ Ν - ,n-j / d,(η) = V a b , n+l-j ab = I (0η j )2
0,n+l
h
I 0j 2
- (b_ - β_ ) J b^-J + 1. 0
> Since
ρ|b^|
1>
this gives by means of (4.6), with some
constant M > 0, Zn + 1 1 ιd , (n) l 0,n +l 2 I1
M P ni, (b in 1 Z} v l 2 l " l 2l _ 1 - 1
if d
|zJ > p
=
(n) r> ..ι 0,n+l
n e
|b |
(n) r> 0,n+l
9s
_ . ι
.
ι
n+1 -
00
If w e u s e (3.6) - (3.8) and n o t e
t
gives that
n
i
that
THEORY
« ,
£ „ n -
OF PADÉ
APPROXIMANTS
ο , ηχ ο . . 2> - 4
%
,
AND
GENERALIZATIONS
•
ς
' diverges for
ρ
|t>2 |
(f " P n/ Q n) ( 0 , z 2) -> -
for
·\
n+2
< | z 21 < 1 = R 2,
converges for these values of
·£>
99
z^.
since the last sum
1 But 1
p " | b 2| "
Q
and we get
< | z 2| < 1 = R 2.
This example shows that we can not get convergence in the z 2~plane in the whole disc
| z 2| < R 2 where
Q ( 0 , z 2) φ 0.
It is also easy to see that this counterexample shows that Theorem 4.1 is sharp. R ef
e
r se
n
c
e
1
Baker, G.A. Jr., Essentials of Padé Approximants, Academic Press, New York, 1975.
2
Chisholm, J.S.R., Rational approximants (defined from double power series, Math. Comp., 27 (1973), 841-848.
3
Chisholm, J.S.R., Rational polynomial approximants in Ν variables, Lecture Notes in Physics 47_ (1976), 33-54.
4
Chisholm, J.S.R. and P.R. Graves-Morris, Generalization of the theorem of de Montessus to two-variable approximants, Proc. Royal Society Ser. A, 342 (1975), 341-372.
5
Chisholm, J.S.R. and R. Hughes Jones, Relative scale covariance of N-variable approximants, U. of Kent preprint, Canterbury (1974).
6
Common, A.K. and P.R. Graves-Morris, Some properties of Chisholm approximants, J. Inst. Maths. Applies, L3 (1974), 229-232.
7
Gon?ar, A.A., A local condition for the single-valuedness of analytic functions of several variables, Math. USSR Sbornik, 22 (1974), 305-322, (Russian original Mat. Sb., 93_ (1974).)
8
Gonc^ar, A.A., On the convergence of generalized Padé approximants to meromorphic functions, Mat. Sb., 98_ (140) (1975), 564-577, (Russian).
9
Graves-Morris, P.R., Convergence of rows of the Padé table, Lecture Notes in Physics 47_ (1976), 55-68.
10
Hughes Jones, R., General rational approximants in Ν variables, J. Approximation Theory, jL6 (1976), 201-233.
11
Jones, W.B. and W.J. Thron, On convergence of Padé mants, SIAM J. Math. Anal., 6^ (1975), 9-16.
approxi
100
J. KARLSSON
AND
H.
WALLIN
12
Karlsson, J., Rational interpolation and best rational approximation, J. Math. Anal. Appl.,53_ (1976), 38-52.
13
Lutterodt, C H . , A two-dimensional analogue of Padé Approxi mant theory, U. of Birmingham preprint (1973).
14
Lutterodt, C H . , Rational approximants to holomorphic functions in η dimensions, J. Math. Anal. Appl., 53 (1976), 89-98.
15
Nuttall, J., The convergence of Padé approximants of meromorphic functions, J. Math. Anal. Appl., 31_ (1970), 147-153.
16
Perron, 0., Die Lehre von den Kettenbriichen, Band II, Teubner, Stuttgart, 1957.
17
Saff, E.B., An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions, J. Approximation Theory, 6^ (1972), 63-68.
18
Wallin, Η., On the convergence theory of Padé approximants, Linear Operators and Approximation proceedings of a conference in Oberwolfach 1971, Birkhâuser, Stuttgart, 1972, pp. 461-469.
19
Wallin, Η., The convergence of Padé approximants and the size of the power series coefficients, Applicable Analysis, j4 (1974), 235-252.
20
Walsh, J.L., The convergence of sequences of rational functions of best approximation, Math. Annalen, 155 (1964), 252-264.
J. Karlsson Department of Mathematics University of Umea S-901 87 Umea, Sweden
H. Wallin Department of Mathematics University of Umea S-901 87 Umea, Sweden