Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 905635, 11 pages
doi:10.1155/2008/905635
Research Article
Matrix Transformations and Disk of Convergence
in Interpolation Processes
Chikkanna R. Selvaraj and Suguna Selvaraj
Department of Mathematics, Pennsylvania State University, Shenango Campus, 147 Shenango Avenue,
Sharon, PA 16146, USA
Correspondence should be addressed to Chikkanna R. Selvaraj, ulf@psu.edu
Received 23 May 2008; Accepted 22 July 2008
Recommended by Huseyin Bor
Let Aρ denote the set of functions analytic in |z| < ρ but not on |z| ρ 1 < ρ < ∞. Walsh proved
that the difference of the Lagrange polynomial interpolant of fz ∈ Aρ and the partial sum of
the Taylor polynomial of f converges to zero on a larger set than the domain of definition of f. In
1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation,
and Hermite-Birkhoff interpolation processes in a similar manner. In this paper, we apply a certain
matrix transformation on the sequences of operators given in the above-mentioned interpolation
processes to prove the convergence in larger disks.
Copyright q 2008 C. R. Selvaraj and S. Selvaraj. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
If x xk is a complex number sequence and A ank is an infinite matrix, then Ax is the
sequence whose nth term is given by
Axn ∞
ank xk .
1.1
k0
A matrix A is called X − Y matrix if Ax is in the set Y whenever x is in X. For 0 ≤ α < ∞, let
Gα x : lim sup |xk |1/k ≤ α .
1.2
For various values of α, this sequence space has been studied extensively by several authors
1–5. In particular, Jacob Jr. 2, page 186 proved the following result.
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International Journal of Mathematics and Mathematical Sciences
Theorem 1.1. An infinite matrix A is a Gu − Gv matrix if and only if for each number w such that
0 < w < 1/v, there exist numbers B and s such that 0 < s < 1/u and
|ank |wn ≤ Bsk
1.3
for all n and k.
Let Aρ denote the collection of functions analytic in the disk Dρ {z ∈ C : |z| < ρ}
for some 1 < ρ < ∞ and having a singularity on the circle |z| ρ. In Section 2, we state the
results proved by Cavaretta Jr. et al. 6 on the Lagrange interpolation, Hermite interpolation,
and Hermite-Birkhoff interpolation of fz ∈ Aρ in the nth roots of unity, which will be
required. Main results are given in Section 3 and deal with the application of a certain matrix
to the various polynomial interpolants in the above results. Interestingly, we are able to show
that under the matrix transformation the difference of the interpolant polynomials and the
corresponding Taylor polynomials converges to zero in a larger region.
2. Preliminaries
Throughout the paper, we assume that fz ∈ Aρ with 1 < ρ < ∞ and fz has the Taylor
series expansion
fz ∞
2.1
ak zk .
k0
For each integer n ≥ 0, let Ln z; f denote the unique Lagrange interpolation polynomial of
degree n which interpolates fz in the n 1st roots of unity, that is,
Ln ω; f fω,
2.2
where ω is any n 1st root of unity. Setting Sn z; f equiconvergence theorem 7 states that
lim Ln z; f − Sn z; f 0,
n→∞
n1
k
k0 ak z ,
the well-known Walsh
∀|z| < ρ2 ,
2.3
with the convergence being uniform and geometric on any closed subset of |z| < ρ2 . This
theorem has been extended in various ways by several authors 6, 8–10. In all that follows,
we state some of the results of 6 which are needed for our main results.
Under Lagrange interpolation, letting
Sn,j z; f n
akjn1 zk ,
j 0, 1, . . . ,
k0
the authors 6, Theorem 1, page 156 have proved the following result.
2.4
C. R. Selvaraj and S. Selvaraj
3
Theorem 2.1. For each positive integer l,
lim
n→∞
l−1
Ln z; f − Sn,j z; f
0,
∀|z| < ρl1
2.5
j0
and this result is best possible.
In the proof of Theorem 2.1, it has been shown by the authors that
Ln z; f −
l−1
Sn,j z; f j0
1
2πi
fttn1 − zn1 dt,
n1 − 1tln1
Γ t − zt
2.6
where Γ is any circle |t| R with 1 < R < ρ.
In 6 the authors have studied the Hermite interpolation also in a similar way. For a
fixed positive integer r ≥ 2, let hrn1−1 z; f be the unique Hermite polynomial interpolant
to f, f , . . . , f r−1 in the n 1st roots of unity, that is,
j
hrn1−1 ω; f f j ω,
j 0, 1, . . . , r − 1,
2.7
where ω is any n 1st root of unity.
Setting
Hrn1−1,0 z; f rn1−1
ak zk ,
k0
n
Hrn1−1,j z; f βj z akn1rj−1 · zk ,
2.8
j 1, 2, . . . ,
k0
where
βj z r−1
k
rj −1
zn1 − 1 ,
k
k0
j 1, 2, . . . ,
2.9
in 6, Theorem 3, page 162 the authors proved the following result.
Theorem 2.2. For each positive integer l,
lim
n→∞
hrn1−1 z; f −
and this result is best possible.
l−1
j0
Hrn1−1,j z; f
0,
∀|z| < ρ1l/r
2.10
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International Journal of Mathematics and Mathematical Sciences
In the proof of Theorem 2.2, it was shown by the authors in 6, page 165 that
hrn1−1 z; f −
l−1
Hrn1−1,j z; f j0
1
2πi
ft
Kt, zdt,
Γ t − z
2.11
where Γ is any circle |t| R with 1 < R < ρ and
|Kt, z| ≤ M
Rn1 |z|n1 |z| 1r−1
Rrln1
2.12
.
Under Hermite-Birkhoff interpolation, the authors in 6 established several results for
different cases. We consider here only the 0, m case. Let n and m be integers with n ≥ m ≥ 0.
0,m
Let b2n1 z; f be the unique Hermite-Birkhoff polynomial of degree 2n1 which interpolates
f in the n 1st roots of unity and whose mth derivative interpolates f m in the n 1st
roots of unity, that is,
0,m
b2n1 ω; f fω,
0,m
b2n1 ω; f
m
f m ω,
2.13
where ω is any n 1st root of unity.
Setting
0,m
B2n1,0 z; f 2n1
ak zk ,
k0
n
0,m
ajν1n1 · zj qj,ν z,
B2n1,ν z; f 2.14
ν 1, 2 . . . ,
j0
where qj,ν z is a polynomial of degree n 1 given by
qj,ν z zn1 ν 1n 1 jm − n 1 jm n1
z − 1,
n 1 jm − jm
j 0, 1, . . . , n,
2.15
with the following conventional notation
jm ⎧
⎨jj − 1 · · · j − m 1,
⎩
0,
if m ≤ j,
if m > j,
in 6, Theorem 4, page 170 the authors proved the following result.
2.16
C. R. Selvaraj and S. Selvaraj
5
Theorem 2.3. For each positive integer l,
lim
n→∞
0,m
b2n1 z; f
−
l−1
0,m
B2n1,j z; f
0,
for |z| < ρ1l/2
2.17
j0
and this result is best possible.
In the proof, it was shown in 6, Theorem 4, page 171 that
0,m
b2n1 z; f −
l−1
0,m
B2n1,j z; f j0
1
2πi
Γ
ftKn,l t, zdt,
2.18
where Γ is any circle |t| R with 1 < R < ρ and |Kn,l t, z| is bounded on the circle |t| R < |z|
by
|Kn,l t, z| ≤
|z|n1 |t|n1 |z|n1
|t − z||t|l1n1 |tn1 − 1|
|z|n1 1 |z|n1 Rn1
|z| − RRl2n1
.
2.19
3. Main results
Our aim in this section is to apply a Gu − Gv matrix to the polynomial sequences of operators
in each of the above three theorems given in Section 2 and prove that the difference of
transformed sequences in each case converges to zero in a lager disk Dρ . To simplify, let
us denote Ln z; f and l−1
j0 Sn,j z; f in Theorem 2.1 by Ln and Sn,l , respectively.
Theorem 3.1. Let fz ∈ Aρ and let Γ be any circle |t| R with 1 < R < ρ. For any ρ > ρ, choose
u > ρ/R
and 0 < v < 1. Let A be a Gu − Gv matrix and define
λn z ∞
ank Lk ,
σn,l z k0
∞
3.1
ank Sk,l
k0
for n 0, 1, . . . . Then, for each l,
lim λn z − σn,l z 0,
n→∞
∀z ∈ Dρ.
3.2
Proof. Using the integral representation given in 2.6, for each l 1, 2, . . . , we have
λn z − σn,l z ∞
ank Lk − Sk,l k0
∞
1
ank
2πi
k0
∞
1
ank
2πi
k0
fttk1 − zk1 dt
k1 − 1tlk1
Γ t − zt
ft
1−
lk1
Γ t − zt
z
t
k1 3.3
tk1
dt.
tk1 − 1
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International Journal of Mathematics and Mathematical Sciences
Since |t| R > 1, we have
∞
ft
1
λn z − σn,l z ank
1−
2πi Γ t − ztlk1
k0
1
2πi
z
t
∞
∞
ft 1 1−
a
nk
lq
Γ t − z q0 t
k0
k1 ∞
z
t
dt
tk1
q0
k1 q
1
1
tlq
3.4
k
dt.
The interchange of the integral and the summations is justified by showing that the series
∞
ank
k0
k
1
,
tlq
∞
ank
k0
z
t
k
1
k
tlq
3.5
converge absolutely for each q as follows. From 1.3, for any 1 < w < 1/v we have that
|ank |wn ≤ Bsk , where s < 1/u < R/ρ < 1. Thus, for each q, l, and n, we have
∞
|ank |
k0
1
|t|lq
k
≤
∞
B s
n
w k0 Rlq
k
Rlq
B
,
n
lq
w R − s
3.6
because s/Rlq < R/ρR
lq < 1 and similarly for |z| < ρ
k1
z
1
|ank | t
|t|lq
k0
∞
k
≤
∞
sρ
B ρ n
lq1
w R k0 R
k
Bρ
Rlq1
,
wn R Rlq1 − sρ
3.7
because sρ/R
lq1 < R/Rlq1 < 1. Therefore, from 3.6 and 3.7, identities 3.4 become
B
|λn z − σn,l z| ≤
2πwn
B
2πwn
∞
ρR
lq
|ft| 1
Rlq
lq1
dt
lq Rlq − s
R
− sρ
Γ |t − z| q0 R
∞ |ft| ρ
1
lq1
dt.
lq − s
R
− sρ
Γ |t − z| q0 R
3.8
It can be easily proved that the two series on the right side of the above expression converge
by using the ratio test. Assuming that ft is bounded on Γ, w > 1 implies that
lim λn z − σn,l z 0,
n→∞
∀|z| < ρ.
3.9
Next, we prove a similar result of convergence on a larger disk in the case of the
Hermite interpolation. To simplify, let us denote
hrn1−1 z; f,
l−1
j0
in Theorem 2.2 by hn and Hn,l , respectively.
Hrn1−1,j z; f
3.10
C. R. Selvaraj and S. Selvaraj
7
Theorem 3.2. Let fz ∈ Aρ and let Γ be any circle |t| R with 1 < R < ρ. For any ρ > ρ, choose
u > ρ/R
and 0 < v < 1. Let A be a Gu − Gv matrix and define
λn z ∞
σn,l z ank hk ,
k0
∞
ank Hk,l
3.11
k0
for n 0, 1, . . . . Then, for each l,
lim λn z − σn,l z 0,
n→∞
∀z ∈ Dρ.
3.12
Proof. Using the integral representation given in 2.11, for each l 1, 2, . . ., we have
∞
ft
1
Kt, zdt.
ank hk − Hk,l ank
λn z − σn,l z 2πi
t
Γ −z
k0
k0
∞
3.13
From 2.12 we obtain that
∞
1
|λn z − σn,l z| ≤
|ank |
2π
k0
|ft| M Rk1 |z|k1 |z| 1
Rrlk1
Γ |t − z|
M|z| 1r1
≤
2π
r−1
dt
∞
Rk1 |z|k1
|ft| |ank |
dt.
Rrlk1
Γ |t − z| k0
3.14
The interchange of the integral and the summation is justified by showing that the series
∞
|ank |
,
rl−1k1
k0 R
∞
|z|
Rrl
|ank |
k0
k1
3.15
converge as follows. From 1.3 we have that for any 1 < w < 1/v, |ank |wn ≤ Bsk , where
s < 1/u < R/ρ < 1.
Thus for any fixed positive integer r ≥ 2 and for each l and n
∞
|ank |
k0
k1
1
Rrl−1
≤
∞
B s
n
rl
rl−1
w R k0 R
k
B
wn Rrl−1
− s
,
3.16
because s/Rrl−1 < 1/ρR
rl−2 < 1, and similarly for |z| < ρ,
∞
k0
|ank |
|z|
Rrl
kl
because s|z|/Rrl < 1/Rrl−1 < 1.
≤
∞
B|z| s|z|
n
rl
w R k0 Rrl
k
1
B|z|
,
n
rl
w R − s|z|
3.17
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International Journal of Mathematics and Mathematical Sciences
Therefore, using 3.16 and 3.17 in 3.14 we get that
|λn z − σn,l z| ≤
MB|z| 1r1
2πwn
|ft|
1
|z|
dt.
rl−1 − s
Rrl − s|z|
Γ |t − z| R
3.18
Assuming that ft is bounded on Γ, w > 1 implies that
lim λn z − σn,l z 0,
n→∞
for |z| < ρ.
3.19
Thus in the Lagrange case, the Gu −Gv matrix transformation of the sequence operators
produces new sequences such that the difference between the transformed sequences of
polynomials Ln and Sn,l converges to zero in an arbitrarily large disk Dρ by choosing u > ρ/R.
Similarly, in the Hermite case also, the domain of convergence to zero is arbitrarily large
But as we see in the following
under Gu − Gv matrix transformation by choosing u > ρ/R.
theorem, in the case of Hermite-Birkhoff interpolation, the domain of convergence to zero of
the difference of the transformed polynomials of Theorem 2.3 is arbitrarily large only if we
choose u > ρ2 /R.
0,m
0,m
To simplify, let us denote b2n1 z; f and l−1
j0 B2n1,j z; f of Theorem 2.3 by bn and
Bn,l , respectively.
Theorem 3.3. Let fz ∈ Aρ and let Γ be any circle |t| R with 1 < R < ρ. For any ρ > ρ, choose
u > ρ2 /R and 0 < v < 1. Let A be a Gu − Gv matrix and define
λn z ∞
ank bk ,
σn,l z k0
∞
ank Bk,l
3.20
k0
for n 0, 1, . . . . Then, for each l,
lim λn z − σn,l z 0,
n→∞
∀z ∈ Dρ.
3.21
Proof. Using the integral representation given in 2.18, for each l 1, 2, . . . , we have
λn z − σn,l z ∞
1
ank
ftKk,l t, zdt.
2πi Γ
k0
3.22
From 2.19 we obtain that
|λn z − σn,l z| ≤
∞
1
|ank |
|ft||Kk,l t, z|dt
2π Γ
k0
1
≤
2π
∞ |a ||z|k1 |t|k1 |z|k1
|ft| nk
dt
|t
−
z|
|t|l1k1 |tk1 − 1|
Γ
k0
1
2π
∞ |a | |z|k1 1 |z|k1 Rk1
|ft| nk
dt.
Rl2k1
Γ |z| − R k0
3.23
C. R. Selvaraj and S. Selvaraj
9
The interchange of the integral and the summations is justified by showing that the two series
in 3.23 converge as follows. From 1.3 we have that for any 1 < w < 1/v, |ank |wn ≤ Bsk ,
using the same method used in 3.4 we
where s < 1/u < R/ρ 2 < 1. Since |t| R and |z| < ρ,
write the first summation as
∞
|ank ||z|k1 |t|k1 |z|k1
k0 |t|
l1k1
Rk1 − 1
≤
∞
Bsk |z|k1
1
n l1k1
k0 w R
|z|
R
∞
∞
ρ
sρ
B ≤ n
w q0 Rlq1 k0 Rlq1
k1 ∞
q0
k
1
q
Rk1
∞
sρ
k0
Rlq1
k
ρ
R
k1
∞
ρ
ρ
B Rlq2
Rlq1
,
wn q0 Rlq1 Rlq1 − sρ
R Rlq2 − sρ2 3.24
because sρ/R
lq1 < 1/ρR
lq < 1 and sρ2 /Rlq2 < 1/Rlq1 < 1 for each l and q. Now, for the
second sum in 3.23 we have
∞ |a | |z|k1 1 |z|k1 Rk1
nk
k0
Rl2k1
≤
∞
sk
B k1
2k2
k1
k1
ρ
ρR
R
ρ
wn k0 Rl2k1
ρ
ρ
ρ2
1
B
,
wn Rl2 − sρ2 Rl2 − sρ Rl1 − sρ Rl1 − s
3.25
because for each l,
sρ2
1
< l1 < 1,
l2
R
R
sρ
1
<
< 1,
l2
R
ρR
l1
sρ
1
<
< 1,
l1
R
ρR
l
s
1
<
< 1.
Rl1 ρ2 Rl
3.26
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International Journal of Mathematics and Mathematical Sciences
Using 3.24 and 3.25, the expression 3.23 becomes
B
|λn z − σn,l z| ≤ n
w 2π
∞ |ft| ρ2
ρ
dt
lq1 − sρ
Rlq2 − sρ2
Γ |t − z| q0 R
B
2πwn
ρ
|ft|
ρ
ρ2
1
dt.
l2 − sρ
2 Rl2 − sρ Rl1 − sρ Rl1 − s
Γ |z| − R R
3.27
It can be easily proved that the two series on the right side of the above expression converge
by using the ratio test. Assuming that ft is bounded on Γ, w > 1 implies that
lim λn z − σn,l z 0
3.28
n→∞
for each |z| < ρ.
In two of the above three theorems, for any R and ρ satisfying the stated conditions,
we have chosen u > ρ/R
> 1 and 0 < v < 1. In the case of the last theorem, we have chosen
u > ρ2 /R > 1 and 0 < v < 1. Now, to see if there exists such a Gu − Gv matrix, we give below
an obvious example.
Define a matrix A by
ank vn
,
pk
3.29
p > u.
Then for any w such that 0 < w < 1/v, we have that
|ank |wn vwn
<
pk
1
p
k
,
3.30
where 1/p < 1/u. Hence by Theorem 1.1, A is a Gu − Gv matrix.
Acknowledgment
The authors are thankful to the referee for his valuable remarks, in particular his suggestion
on improving the presentation of Theorem 3.3.
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