Volume 2008, Article ID 905635,11pages 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,[email protected]
Received 23 May 2008; Accepted 22 July 2008
Recommended by Huseyin Bor
LetAρdenote the set of functions analytic in|z|< ρbut not on|z|ρ1< ρ <∞. Walsh proved that the difference of the Lagrange polynomial interpolant offz ∈ Aρand the partial sum of the Taylor polynomial offconverges to zero on a larger set than the domain of definition off. In 1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation, and Hermite-Birkhoffinterpolation 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.
Copyrightq2008 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
Ifx xkis a complex number sequence andA ankis an infinite matrix, thenAxis the
sequence whosenth term is given by
Axn∞ k0
ankxk. 1.1
A matrixAis calledX−Y matrix ifAxis in the setYwheneverxis inX. For 0≤α <∞, let
Gα
x: lim sup|xk|1/k ≤α. 1.2
For various values ofα, this sequence space has been studied extensively by several authors
Theorem 1.1. An infinite matrixAis aGu−Gvmatrix if and only if for each numberwsuch that
0< w <1/v, there exist numbersBandssuch that 0< s <1/uand
|ank|wn≤Bsk 1.3
for allnandk.
LetAρdenote the collection of functions analytic in the diskDρ {z ∈ C :|z| < ρ}
for some 1 < ρ <∞and having a singularity on the circle|z| ρ. InSection 2, we state the results proved by Cavaretta Jr. et al.6on the Lagrange interpolation, Hermite interpolation, and Hermite-Birkhoffinterpolation of fz ∈ Aρ in the nth roots of unity, which will be
required. Main results are given inSection 3and 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 thatfz ∈Aρ with 1< ρ < ∞andfzhas the Taylor
series expansion
fz ∞ k0
akzk. 2.1
For each integern≥ 0, letLnz;fdenote the unique Lagrange interpolation polynomial of
degreenwhich interpolatesfzin then1st roots of unity, that is,
Lnω;f fω, 2.2
whereω is anyn1st root of unity. SettingSnz;f
n1
k0akzk, the well-known Walsh
equiconvergence theorem7states that
lim
n→ ∞Lnz;f−Snz;f 0, ∀|z|< ρ
2, 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 authors6,8–10. In all that follows, we state some of the results of6which are needed for our main results.
Under Lagrange interpolation, letting
Sn,jz;f n
k0
akjn1zk, j0,1, . . . , 2.4
Theorem 2.1. For each positive integerl,
lim
n→ ∞
Lnz;f− l−1
j0
Sn,jz;f
0, ∀|z|< ρl1 2.5
and this result is best possible.
In the proof ofTheorem 2.1, it has been shown by the authors that
Lnz;f− l−1
j0
Sn,jz;f
1 2πi
Γ
fttn1−zn1
t−ztn1−1tln1dt, 2.6
whereΓis any circle|t|Rwith 1< R < ρ.
In6the authors have studied the Hermite interpolation also in a similar way. For a fixed positive integerr ≥2, lethrn1−1z;fbe the unique Hermite polynomial interpolant
tof, f, . . . , fr−1in then1st roots of unity, that is,
hjrn1−1ω;f fjω, j0,1, . . . , r−1, 2.7
whereωis anyn1st root of unity. Setting
Hrn1−1,0z;f rn1−1
k0 akzk,
Hrn1−1,jz;f βjz n
k0
akn1rj−1·zk, j1,2, . . . ,
2.8
where
βjz r−1
k0
rj−1
k
zn1−1k
, j 1,2, . . . , 2.9
in6, Theorem 3, page 162the authors proved the following result.
Theorem 2.2. For each positive integerl,
lim
n→ ∞
hrn1−1z;f− l−1
j0
Hrn1−1,jz;f
0, ∀|z|< ρ1l/r 2.10
In the proof ofTheorem 2.2, it was shown by the authors in6, page 165that
hrn1−1z;f− l−1
j0
Hrn1−1,jz;f
1 2πi
Γ ft
t−zKt, zdt, 2.11
whereΓis any circle|t|Rwith 1< R < ρand
|Kt, z| ≤M
Rn1|z|n1|z|1r−1
Rrln1 . 2.12
Under Hermite-Birkhoffinterpolation, the authors in6established several results for different cases. We consider here only the0, mcase. Letnandmbe integers withn≥m≥0. Letb0,m2n1z;fbe the unique Hermite-Birkhoffpolynomial of degree 2n1 which interpolates
f in then1st roots of unity and whosemth derivative interpolatesfm in then1st roots of unity, that is,
b0,m2n1ω;f fω, b2n10,mω;fmfmω, 2.13
whereωis anyn1st root of unity. Setting
B0,m2n1,0z;f
2n1
k0 akzk,
B2n1,ν0,m z;f
n
j0
ajν1n1·zjqj,νz, ν1,2. . . ,
2.14
whereqj,νzis a polynomial of degreen1given by
qj,νz zn1
ν1n1 jm−n1jm
n1jm−jm z
n1−1, j0,1, . . . , n, 2.15
with the following conventional notation
jm
⎧ ⎨ ⎩
jj−1· · ·j−m1, ifm≤j,
0, ifm > j,
2.16
Theorem 2.3. For each positive integerl,
lim
n→ ∞
b2n10,mz;f−
l−1
j0
B2n1,j0,mz;f
0, for|z|< ρ1l/2 2.17
and this result is best possible.
In the proof, it was shown in6, Theorem 4, page 171that
b0,m2n1z;f−
l−1
j0
B0,m2n1,jz;f 1 2πi
ΓftKn,lt, zdt, 2.18
whereΓis any circle|t|Rwith 1< R < ρand|Kn,lt, z|is bounded on the circle|t|R <|z|
by
|Kn,lt, z| ≤
|z|n1|t|n1|z|n1 |t−z||t|l1n1|tn1−1|
|z|n11|z|n1Rn1
|z| −RRl2n1 . 2.19
3. Main results
Our aim in this section is to apply aGu−Gvmatrix 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 denoteLnz;fand
l−1
j0Sn,jz;finTheorem 2.1byLnandSn,l, respectively.
Theorem 3.1. Letfz∈Aρand letΓbe any circle|t|Rwith 1< R < ρ. For anyρ > ρ , choose u >ρ/R and 0< v <1. LetAbe aGu−Gvmatrix and define
λnz
∞
k0
ankLk, σn,lz
∞
k0
ankSk,l 3.1
forn0,1, . . . .Then, for eachl,
lim
n→ ∞λnz−σn,lz 0, ∀z∈Dρ. 3.2
Proof. Using the integral representation given in2.6, for eachl1,2, . . . ,we have
λnz−σn,lz
∞
k0
ankLk−Sk,l
∞ k0
ank 1
2πi
Γ
fttk1−zk1 t−ztk1−1tlk1dt
∞ k0
ank
1 2πi
Γ
ft t−ztlk1
1− z
t
k1
tk1 tk1−1dt.
Since|t|R >1, we have
λnz−σn,lz
∞ k0 ank 1 2πi Γ ft t−ztlk1
1− z
t
k1∞
q0 1 tk1 q dt 1 2πi Γ ft t−z
∞ q0 1 tlq ∞ k0 ank 1− z
t k1 1 tlq k dt. 3.4
The interchange of the integral and the summations is justified by showing that the series
∞ k0 ank 1 tlq k , ∞ k0 ank z
t
k 1
tlq
k
3.5
converge absolutely for eachqas follows. From 1.3, for any 1 < w < 1/v we have that
|ank|wn≤Bsk, wheres <1/u < R/ρ < 1.Thus, for eachq, l, andn, we have
∞
k0
|ank| 1 |t|lq
k ≤ B wn ∞ k0 s Rlq k B wn Rlq
Rlq−s, 3.6
becauses/Rlq< R/ρR lq <1 and similarly for|z|<ρ
∞
k0 |ank|z
t
k1 |t|1lq
k ≤ Bρ
wnR
∞ k0 sρ Rlq1 k Bρ
wnR
Rlq1
Rlq1−sρ , 3.7
becausesρ/R lq1< R/Rlq1<1. Therefore, from3.6and3.7, identities3.4become
|λnz−σn,lz| ≤ 2 B πwn
Γ |ft| |t−z|
∞ q0 1 Rlq Rlq Rlq−s
ρRlq Rlq1−sρ
dt
B
2πwn
Γ |ft| |t−z|
∞
q0
1
Rlq−s
ρ Rlq1−sρ
dt.
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 thatftis bounded onΓ, w >1 implies that
lim
n→ ∞λnz−σn,lz 0, ∀|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
hrn1−1z;f, l−1
j0
Hrn1−1,jz;f 3.10
Theorem 3.2. Letfz∈Aρand letΓbe any circle|t|Rwith 1< R < ρ. For anyρ > ρ, choose u >ρ/R and 0< v <1. LetAbe aGu−Gvmatrix and define
λnz
∞
k0
ankhk, σn,lz
∞
k0
ankHk,l 3.11
forn0,1, . . . .Then, for eachl,
lim
n→ ∞λnz−σn,lz 0, ∀z∈Dρ. 3.12
Proof. Using the integral representation given in2.11, for eachl1,2, . . ., we have
λnz−σn,lz
∞
k0
ankhk−Hk,l
∞
k0 ank 1
2πi
Γ ft
t−zKt, zdt. 3.13
From2.12we obtain that
|λnz−σn,lz| ≤
∞
k0 |ank| 1
2π
Γ |ft| |t−z|
MRk1|z|k1|z|1r−1
Rrlk1 dt
≤ M|z|1r1
2π
Γ |ft| |t−z|
∞
k0 |ank|
Rk1|z|k1 Rrlk1 dt.
3.14
The interchange of the integral and the summation is justified by showing that the series
∞
k0
|ank| Rrl−1k1,
∞
k0
|ank| |z| Rrl
k1
3.15
converge as follows. From1.3we have that for any 1 < w < 1/v, |ank|wn ≤ Bsk,where s <1/u < R/ρ < 1.
Thus for any fixed positive integerr≥2 and for eachlandn
∞
k0
|ank| 1 Rrl−1
k1 ≤ B
wnRrl
∞
k0 s Rrl−1
k
B
wnRrl−1−s, 3.16
becauses/Rrl−1<1/ρR rl−2<1, and similarly for|z|<ρ,
∞
k0
|ank| |z| Rrl
kl
≤ B|z| wnRrl
∞
k0 s|z| Rrl
k B|z|
wn
1
Rrl−s|z|, 3.17
Therefore, using3.16and3.17in3.14we get that
|λnz−σn,lz| ≤MB|z|
1r1 2πwn
Γ |ft| |t−z|
1
Rrl−1−s |z| Rrl−s|z|
dt. 3.18
Assuming thatftis bounded onΓ, w >1 implies that
lim
n→ ∞λnz−σn,lz 0, for|z|<ρ. 3.19
Thus in the Lagrange case, theGu−Gvmatrix transformation of the sequence operators
produces new sequences such that the difference between the transformed sequences of polynomialsLnandSn,lconverges to zero in an arbitrarily large diskDρby choosingu >ρ/R .
Similarly, in the Hermite case also, the domain of convergence to zero is arbitrarily large underGu−Gv matrix transformation by choosingu > ρ/R . But as we see in the following
theorem, in the case of Hermite-Birkhoffinterpolation, the domain of convergence to zero of the difference of the transformed polynomials ofTheorem 2.3is arbitrarily large only if we chooseu >ρ2/R.
To simplify, let us denoteb0,m2n1z;fand lj0−1B2n1,j0,mz;fofTheorem 2.3bybn and Bn,l, respectively.
Theorem 3.3. Letfz∈Aρand letΓbe any circle|t|Rwith 1< R < ρ. For anyρ > ρ , choose u >ρ2/Rand 0< v <1. LetAbe aG
u−Gvmatrix and define
λnz
∞
k0
ankbk, σn,lz
∞
k0
ankBk,l 3.20
forn0,1, . . . .Then, for eachl,
lim
n→ ∞λnz−σn,lz 0, ∀z∈Dρ. 3.21
Proof. Using the integral representation given in2.18, for eachl1,2, . . . ,we have
λnz−σn,lz
∞
k0 ank 1
2πi
ΓftKk,lt, zdt. 3.22
From2.19we obtain that
|λnz−σn,lz| ≤
∞
k0 |ank| 1
2π
Γ|ft||Kk,lt, z|dt
≤ 1
2π
Γ |ft| |t−z|
∞
k0
|ank||z|k1|t|k1|z|k1 |t|l1k1|tk1−1| dt
1
2π
Γ |ft| |z| −R
∞
k0
|ank||z|k11|z|k1Rk1 Rl2k1 dt.
The interchange of the integral and the summations is justified by showing that the two series in3.23converge as follows. From1.3we have that for any 1 < w <1/v, |ank|wn ≤Bsk,
wheres <1/u < R/ρ2<1. Since|t|Rand|z|<ρ, using the same method used in3.4we
write the first summation as
∞
k0
|ank||z|k1 |t|l1k1
|t|k1|z|k1 Rk1−1 ≤
∞
k0
Bsk|z|k1 wnRl1k1
1 |z|
R
k1∞
q0 1 Rk1 q ≤ B wn ∞ q0 ρ Rlq1 ∞ k0 sρ Rlq1 k ∞ k0 sρ Rlq1 k ρ R k1 B wn ∞ q0 ρ Rlq1 Rlq1 Rlq1−sρ
ρ R
Rlq2 Rlq2−sρ2
,
3.24
becausesρ/R lq1 <1/ρR lq <1 andsρ2/Rlq2<1/Rlq1<1 for eachlandq. Now, for the
second sum in3.23we have
∞
k0
|ank||z|k11|z|k1Rk1 Rl2k1 ≤
B wn ∞ k0 sk Rl2k1
ρ2k2ρk1 ρR k1Rk1
B wn
ρ2 Rl2−sρ2
ρ Rl2−sρ
ρ Rl1−sρ
1
Rl1−s
,
3.25
because for eachl,
sρ2 Rl2 <
1
Rl1 <1,
sρ Rl2 <
1
ρRl1 <1,
sρ Rl1 <
1
ρRl <1,
s Rl1 <
1
ρ2Rl <1.
Using3.24and3.25, the expression3.23becomes
|λnz−σn,lz| ≤ B wn2π
Γ |ft| |t−z|
∞
q0
ρ Rlq1−sρ
ρ2 Rlq2−sρ2
dt
B
2πwn
Γ |ft| |z| −R
ρ2 Rl2−sρ2
ρ Rl2−sρ
ρ Rl1−sρ
1
Rl1−s
dt. 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 thatftis bounded onΓ, w >1 implies that
lim
n→ ∞λnz−σn,lz 0 3.28
for each|z|<ρ.
In two of the above three theorems, for anyRandρsatisfying the stated conditions, we have chosenu >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 aGu−Gvmatrix, we give below
an obvious example. Define a matrixAby
ank v n
pk, p > u. 3.29
Then for anywsuch that 0< w <1/v, we have that
|ank|wn vw n
pk <
1
p
k
, 3.30
where 1/p <1/u. Hence byTheorem 1.1,Ais aGu−Gvmatrix.
Acknowledgment
The authors are thankful to the referee for his valuable remarks, in particular his suggestion on improving the presentation ofTheorem 3.3.
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