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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008, Article IDΒ 905635, 11 pages
http://dx.doi.org/10.1155/2008/905635
Research Article

Matrix Transformations and Disk of Convergence in Interpolation Processes

Department of Mathematics, Pennsylvania State University, Shenango Campus, 147 Shenango Avenue, Sharon, PA 16146, USA

Received 23 May 2008; Accepted 22 July 2008

Academic Editor: HuseyinΒ Bor

Copyright Β© 2008 Chikkanna R. Selvaraj and Suguna 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.

Abstract

Let denote the set of functions analytic in but not on . Walsh proved that the difference of the Lagrange polynomial interpolant of and the partial sum of the Taylor polynomial of converges to zero on a larger set than the domain of definition of . 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.

1. Introduction

If is a complex number sequence and is an infinite matrix, then is the sequence whose th term is given byA matrix is called matrix if is in the set whenever is in . For , letFor various values of , this sequence space has been studied extensively by several authors [1–5]. In particular, Jacob Jr. [2, page 186] proved the following result.

Theorem 1.1. An infinite matrix is a matrix if and only if for each number such that , there exist numbers and such that andfor all and .

Let denote the collection of functions analytic in the disk for some and having a singularity on the circle . 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 in the th 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 with and has the Taylor series expansionFor each integer , let denote the unique Lagrange interpolation polynomial of degree which interpolates in the st roots of unity, that is,where is any st root of unity. Setting , the well-known Walsh equiconvergence theorem [7] states thatwith the convergence being uniform and geometric on any closed subset of 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, lettingthe authors [6, Theorem 1, page 156] have proved the following result.

Theorem 2.1. For each positive integer ,and this result is best possible.

In the proof of Theorem 2.1, it has been shown by the authors thatwhere is any circle with .

In [6] the authors have studied the Hermite interpolation also in a similar way. For a fixed positive integer , let be the unique Hermite polynomial interpolant to in the st roots of unity, that is,where is any st root of unity.

Settingwherein [6, Theorem 3, page 162] the authors proved the following result.

Theorem 2.2. For each positive integer ,and this result is best possible.

In the proof of Theorem 2.2, it was shown by the authors in [6, page 165] thatwhere is any circle with and

Under Hermite-Birkhoff interpolation, the authors in [6] established several results for different cases. We consider here only the case. Let and be integers with . Let be the unique Hermite-Birkhoff polynomial of degree which interpolates in the st roots of unity and whose th derivative interpolates in the st roots of unity, that is,where is any st root of unity.

Settingwhere is a polynomial of degree given bywith the following conventional notationin [6, Theorem 4, page 170] the authors proved the following result.

Theorem 2.3. For each positive integer , and this result is best possible.

In the proof, it was shown in [6, Theorem 4, page 171] thatwhere is any circle with and is bounded on the circle by

3. Main Results

Our aim in this section is to apply a 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 To simplify, let us denote and in Theorem 2.1 by and , respectively.

Theorem 3.1. Let and let be any circle with . For any , choose and . Let be a matrix and definefor Then, for each ,

Proof. Using the integral representation given in (2.6), for each we haveSince , we haveThe interchange of the integral and the summations is justified by showing that the seriesconverge absolutely for each as follows. From (1.3), for any we have that , where Thus, for each , and , we havebecause and similarly for because . Therefore, from (3.6) and (3.7), identities (3.4) becomeIt can be easily proved that the two series on the right side of the above expression converge by using the ratio test. Assuming that is bounded on implies that

Next, we prove a similar result of convergence on a larger disk in the case of the Hermite interpolation. To simplify, let us denotein Theorem 2.2 by and , respectively.

Theorem 3.2. Let and let be any circle with . For any choose and . Let be a matrix and definefor Then, for each ,

Proof. Using the integral representation given in (2.11), for each , we haveFrom (2.12) we obtain thatThe interchange of the integral and the summation is justified by showing that the seriesconverge as follows. From (1.3) we have that for any where .
Thus for any fixed positive integer and for each and because , and similarly for ,because
Therefore, using (3.16) and (3.17) in (3.14) we get thatAssuming that is bounded on implies that

Thus in the Lagrange case, the matrix transformation of the sequence operators produces new sequences such that the difference between the transformed sequences of polynomials and converges to zero in an arbitrarily large disk by choosing . Similarly, in the Hermite case also, the domain of convergence to zero is arbitrarily large under matrix transformation by choosing . But as we see in the following 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 .

To simplify, let us denote and of Theorem 2.3 by and , respectively.

Theorem 3.3. Let and let be any circle with . For any , choose and . Let be a matrix and definefor Then, for each ,

Proof. Using the integral representation given in (2.18), for each we haveFrom (2.19) we obtain thatThe 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 , where . Since and , using the same method used in (3.4) we write the first summation asbecause and for each and . Now, for the second sum in (3.23) we havebecause for each ,Using (3.24) and (3.25), the expression (3.23) becomesIt can be easily proved that the two series on the right side of the above expression converge by using the ratio test. Assuming that is bounded on implies thatfor each .

In two of the above three theorems, for any and satisfying the stated conditions, we have chosen and . In the case of the last theorem, we have chosen and . Now, to see if there exists such a matrix, we give below an obvious example.

Define a matrix byThen for any such that , we have thatwhere . Hence by Theorem 1.1, is a 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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