Not all limit points of poles of the Padé approximants are obstructions for poinwise convergence

0.09 MB·English

by  Victor M. Adukov

Not all limit points of poles of the Pad´e approximants
are obstructions for poinwise convergence *
Victor M. Adukov
5
0 Department of Differential Equations and Dynamical Systems,
0
Southern Ural State University, Lenin avenue 76, 454080 Chelyabinsk,
2
Russia
n
a E-mail address: [email protected]
J

9 Abstract
1
In the work it is shown that not all limit points of poles of the Pad´e approx-

] imants for the last intermediate row are obstructions for pointwise convergence
A
of the whole row to an approximable function. The corresponding examples are
N
constructed.
.
h
t The notion of intermediate rows was introduced in the work [2], where
a
m sufficient conditions for the convergence of the whole intermediate row were
[ obtained. We will use the explicit description of the set of limit points of

2 poles of the Pad´e approximants for the last intermediate row obtained in [1].
v The paper is the continuation of the works [1].
9
We will construct the examples in the class of rational functions. Let
1
1 r(z) = N(z) be a strictly proper rational fraction. Here N(z),D(z) are co-
1 D(z)
prime polynomials, degD(z) = λ, and all poles of r(z) lie in the disk |z| < R.
0
5 Let (V (z),U (z)) be the unique solution of the Bezout equation
k k
0
h/ N(z)Vk(z)+D(z)Uk(z) = zk, k ≥ 0,
t
a
where the polynomial V (z) satisfy the inequality degV (z) < λ. This solu-
m k k
tion is called the minimal solution of the Bezout equation. It is easily seen
:
v
that for n ≥ 0 the polynomials V (z),−U (z) are the denominator and
i n+λ n+λ
X numerator of the Pad´e approximants πr (z) of type (n,λ − 1) for r(z),
n,λ−1
r
respectively:
a
U (z)
πr (z) = − n+λ
n,λ−1 V (z)
n+λ
(see [1], Theorem 4.1). For V (z) there is the following explicit formula:
k
λ i−1
V (z) = d v zj
k i k+i−j−1
i=1j=0
XX
*This work was supported by Russian Foundation for Basic Research under Grant #
04-01-96006.
(see [1], formula (4.8)). Here D(z) = zλ +d zλ−1 +···+d and v is the
λ−1 0 k
coefficient of zλ−1 in the polynomial V (z).
k
Wewill assume that all roots z ,...,z of the polynomial D(z) are simple.
1 λ
Then v is found as follows
k
v = C zk +···+C zk
k 1 1 λ λ
(see [1], formula (4.10)), the numbers C are defined in this case by the
j
formula
1
C = ,
j D2(z )A
j j j
where D (z) = D(z) and A is a residue of r(z) at z (see [1], formula (4.14)).
j z−zj j j
Substituing v into V (z), we obtain
k k
λ
V (z) = C zkD (z). (0.1)
k j j j
j=1
X
We will also suppose that the dominant poles z ,...,z , 1 < ν < λ−1,
1 ν
are vertices of a regular σ-gon, σ ≥ ν, and ρ ≡ |z | = ... = |z | > |z | >
1 ν ν+1
|z | ≥ ... ≥ |z |.
ν+2 λ
Let us introduce the following family of polynomials:
ν
ω (z) = C zm∆ (z), m = 0,1,...,σ −1,
m j j k
j=1
X
where ∆(z) = (z −z )···(z −z ), ∆ (z) = ∆(z).
1 ν k z−z
k
Let n+λ ≡ m(modσ). If we divide the sum in (0.1) by two sums over
the dominant and nondominant poles, we get
λ C zn+λ
V (z) = (z −z )···(z −z )zn+λ−mω (z)+D(z) j j . (0.2)
n+λ ν+1 λ 1 m z −z
j=ν+1 j
X
Here we take into account the equality zn+λ−m = ··· = zn+λ−m for n+λ ≡
1 ν
m(modσ).
From the representation (0.2) we see that for n → ∞, n ∈ Λ ≡
m
n ∈ N n+λ ≡ m(modσ) there exists
n (cid:12) o
(cid:12)
(cid:12) limz−(n+λ−m)V (z) = (z −z )···(z −z )ω (z).
1 n+λ ν+1 λ m
2
This means that the set of limit points of poles of the Pad´e approximants
πr (z) consists of the poles z ,...,z and the additional limit
n,λ−1 n∈Λm ν+1 λ
npoints thaot are the roots of the polynomials ω (z), m = 0,1,...,σ − 1
m
(see also Theorem 2.7 in [1]). The additional limit points are different from
the dominant poles z ,...,z (see Proposition 2.1 in [1]), but they can be
1 ν
coincided with the poles z ,...,z . The corresponding example will be
ν+1 λ
given below. It should be noted that every additional limit point ζ generates
0
the sequence of defects of the Pad´e approximants πr (z), i.e. the sequence
n,λ−1
of their zeros and poles that converge to ζ .
0
Theorem 1. Let ζ be a zero of one of polynomials ω (z), m = 0,1,...,
0 m
σ −1, and ζ does not coincide with the poles z ,...,z .
0 ν+1 λ
If |ζ | < |z |, then at the point ζ the whole sequence πr (z) converges
0 ν+1 0 n,λ−1
to r(z).
If |ζ | ≥ |z |, then limπr (ζ ) does not exist and also for |ζ | > |z |
0 ν+1 n,λ−1 0 0 ν+1
lim πr (ζ ) = ∞, n ∈ Λ .
n→∞ n,λ−1 0 m
Proof. If ω (ζ ) 6= 0, then by Theorem 2.3 from [1] the subsequence
m 0
πr (z) uniformly converges to r(z) in a neighborhood of ζ . It
n,λ−1 n∈Λm 0
nremains tooconsider those sequences Λ for which ω (ζ ) = 0. In this case
m m 0
for n ∈ Λ we have by formula (0.2)
m
λ C zn+λ
j j
V (ζ ) = D(ζ ) .
n+λ 0 0
ζ −z
j=ν+1 0 j
X
Taking into account our assumption |z | > |z | ≥ ... ≥ |z |, we
ν+1 ν+2 λ
obtain
C
V (ζ ) = zn+λD(ζ ) ν+1 +o(1) . (0.3)
n+λ 0 ν+1 0 "ζ0 −zν+1 #
−(n+λ)
Hence, for all sufficiently large n ∈ Λ the polynomial z V (z) is
m ν+1 n+λ
bounded away from zero in a neighborhood of ζ . From the Bezout equation
0
it follows
zn+λ
r(z)−πr (z) = . (0.4)
n,λ−1 D(z)V (z)
n+λ
Thus at ζ we have
0
ζn+λ
r(ζ )−πr (ζ ) = 0
0 n,λ−1 0 D2(ζ )zn+λ Cν+1 +o(1)
0 ν+1 ζ0−zν+1
h i
3
Hence, if |ζ | < |z |, then at ζ the sequence πr (z) converges to
0 ν+1 0 n,λ−1 n∈Λm
the function r(z). Therefore the whole sequencne πr (zo) converges to r(z)
n,λ−1
at ζ .
0
If |ζ | ≥ |z |, then from (0.4) we obtain for n → ∞, n ∈ Λ ,
0 ν+1 m
n+λ
ζ 1
|r(ζ )−πr (ζ )| = 0 ,
i.e. |r(ζ )−πr0 (ζn),λ|−→1 ∞0 for(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)|zζν+|1>(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) |z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)D|2a(nζ)dh|ζr0C−(νζz+ν1)+1−+πro(1)i((cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ζ )| 6→ 0 for
0 n,λ−1 0 0 ν+1 0 n,λ−1 0
|ζ | = |z |. The theorem is proved.
0 ν+1
Now we construct the examples that illustrate Theorem 1..
Example 1. Let λ = 3, ν = 2, σ = 2, z = 1,z = −1,z = 1/2, A = 1,
1 2 3 1
A = 1/18, A = 1, i.e.
2 3
1 1 1
r(z) = + + .
z −1 18(z +1) z −1/2
Then C = 1,C = 2,C = 16/9 and zeros of the polynomials ω (z)
1 2 3 0,1
are ζ = 1/3,ζ = 3. The point ζ = 1/3 lies in the unit disk D and |ζ | <
0 1 0 0
|z |. Hence, ζ does not an obstruction for the poinwise convergence and
3 0
the sequence πr (z) uniformly converges to r(z) on compact subsets of the
n,2
domain UF = D\{1/3,1/2} and pointwise converges on D\{1/2}.
Example 2. Let A = A = A = 1,
1 2 3
1 1 1
r(z) = + + .
z −1 (z +1) z −1/2
Then C = 1,C = 1/9,C = 16/9 and ζ = −4/5, ζ = −5/4. Now the
1 2 3 0 1
point ζ is an obstruction for the poinwise convergence and
0
lim πr (−4/5) = ∞.
m→∞ 2m+1,2
In this case the sequence πr (z) uniformly converges to r(z) on compact
n,2
subsets of the domain U˜F = D\{1/2,−4/5}.
The following example show that an additional limit point can be coin-
cided with a nondominant pole.
4
Example 3. Let A = 2,A = 2/27,A = 1,
1 2 3
2 2 1
r(z) = + + .
z −1 27(z +1) z −1/2
Then C = 1/2,C = 3/2,C = 16/9 and ζ = 1/2,ζ = 2. The addi-
1 2 3 0 1
tional limit point ζ coincides with the pole z .
0 3
The following example show that there are a point ζ and a sequence
Λ ⊂ N such that there is lim π (ζ) 6= a(ζ),n ∈ Λ.
n→∞ n,λ−1
Example 4. Let A = 2/3,A = 2/9,A = 1,
1 2 3
2 2 1
r(z) = + + .
3(z −1) 9(z +1) z −1/2
Then C = 3/2,C = 1/2,C = 16/9 and ζ = −1/2,ζ = −2. By for-
1 2 3 0 1
mula (0.4) we obtain πr (−1/2) = 0. Thus πr (−1/2) is a stationary
2m+1,2 2m+1,2
sequence and πr (−1/2) 6= r(−1/2) = −1.
2m+1,2
References
[1] V.M.Adukov, Theuniformconvergenceofsubsequences ofthelastinter-
mediate row of the Pad´e table, J. Approx. Theory 122 (2003), 160–207.
[2] A. Sidi, Quantitative and constructive aspects of the generalized
Koenig’s and de Montessus’s theorems for Pad´e approximants, J. Com-
put. Appl. Math. 29 (1990), 257–291.
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