Matrix Transformations and Disk of Convergence in Interpolation Processes

Abstract

Let 𝐴_𝜌 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 𝑓(𝑧)∈𝐴_𝜌 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.

1. Introduction

If 𝑥=(𝑥_𝑘) is a complex number sequence and 𝐴=[𝑎_𝑛𝑘] is an infinite matrix, then 𝐴𝑥 is the sequence whose 𝑛th term is given by(𝐴𝑥)_𝑛=^∞_𝑘=0𝑎_𝑛𝑘𝑥_𝑘. (1.1)A matrix 𝐴 is called 𝑋−𝑌 matrix if 𝐴𝑥 is in the set 𝑌 whenever 𝑥 is in 𝑋. For 0≤𝛼<∞, let𝐺_𝛼={𝑥∶limsup|𝑥_𝑘|^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.

Theorem 1.1. An infinite matrix 𝐴 is a 𝐺_𝑢−𝐺_𝑣 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|𝑎_𝑛𝑘|𝑤^𝑛≤𝐵𝑠^𝑘 (1.3)for all 𝑛 and 𝑘.

Let 𝐴_𝜌 denote the collection of functions analytic in the disk 𝐷_𝜌={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 𝑓(𝑧)∈𝐴_𝜌 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 𝑓(𝑧)∈𝐴_𝜌 with 1<𝜌<∞ and 𝑓(𝑧) has the Taylor series expansion𝑓(𝑧)=^∞_𝑘=0𝑎_𝑘𝑧^𝑘. (2.1)For each integer 𝑛≥0, let 𝐿_𝑛(𝑧;𝑓) denote the unique Lagrange interpolation polynomial of degree 𝑛 which interpolates 𝑓(𝑧) in the (𝑛+1)st roots of unity, that is,𝐿_𝑛(𝜔;𝑓)=𝑓(𝜔), (2.2)where 𝜔 is any (𝑛+1)st root of unity. Setting 𝑆_𝑛(𝑧;𝑓)=∑^𝑛+1_𝑘=0𝑎_𝑘𝑧^𝑘, the well-known Walsh equiconvergence theorem [7] states thatlim_𝑛→∞[𝐿_𝑛(𝑧;𝑓)−𝑆_𝑛(𝑧;𝑓)]=0,∀|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, letting𝑆_𝑛,𝑗(𝑧;𝑓)=^𝑛_𝑘=0𝑎_{𝑘+𝑗(𝑛+1)}𝑧^𝑘,𝑗=0,1,…, (2.4)the authors [6, Theorem 1, page 156] have proved the following result.

Theorem 2.1. For each positive integer 𝑙,lim_𝑛→∞{𝐿_𝑛(𝑧;𝑓)−^𝑙−1_𝑗=0𝑆_𝑛,𝑗(𝑧;𝑓)}=0,∀|𝑧|<𝜌^𝑙+1 (2.5)and this result is best possible.

In the proof of Theorem 2.1, it has been shown by the authors that𝐿_𝑛(𝑧;𝑓)−^𝑙−1_𝑗=0𝑆_𝑛,𝑗1(𝑧;𝑓)=2𝜋𝑖_Γ𝑓(𝑡)(𝑡^𝑛+1−𝑧^𝑛+1)(𝑡−𝑧)(𝑡^𝑛+1−1)𝑡^𝑙(𝑛+1)𝑑𝑡, (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 ℎ_{𝑟(𝑛+1)−1}(𝑧;𝑓) be the unique Hermite polynomial interpolant to 𝑓,𝑓^′,…,𝑓^(𝑟−1) in the (𝑛+1)st roots of unity, that is,ℎ^(𝑗)_{𝑟(𝑛+1)−1}(𝜔;𝑓)=𝑓^(𝑗)(𝜔),𝑗=0,1,…,𝑟−1, (2.7)where 𝜔 is any (𝑛+1)st root of unity.

Setting𝐻_{𝑟(𝑛+1)−1,0}(𝑧;𝑓)=∑^{𝑟(𝑛+1)−1}_𝑘=0𝑎_𝑘𝑧^𝑘,𝐻_{𝑟(𝑛+1)−1,𝑗}(𝑧;𝑓)=𝛽_𝑗(𝑧)∑^𝑛_𝑘=0𝑎_{𝑘+(𝑛+1)(𝑟+𝑗−1)}⋅𝑧^𝑘,𝑗=1,2,…, (2.8)where𝛽_𝑗(𝑧)=^𝑟−1_𝑘=0𝑘𝑟+𝑗−1(𝑧^𝑛+1−1)^𝑘,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→∞{ℎ_{𝑟(𝑛+1)−1}(𝑧;𝑓)−^𝑙−1_𝑗=0𝐻_{𝑟(𝑛+1)−1,𝑗}(𝑧;𝑓)}=0,∀|𝑧|<𝜌^1+𝑙/𝑟 (2.10)and this result is best possible.

In the proof of Theorem 2.2, it was shown by the authors in [6, page 165] thatℎ_{𝑟(𝑛+1)−1}(𝑧;𝑓)−^𝑙−1_𝑗=0𝐻_{𝑟(𝑛+1)−1,𝑗}1(𝑧;𝑓)=2𝜋𝑖_Γ𝑓(𝑡)(𝑡−𝑧)𝐾(𝑡,𝑧)𝑑𝑡, (2.11)where Γ is any circle |𝑡|=𝑅 with 1<𝑅<𝜌 and|𝐾(𝑡,𝑧)|≤𝑀(𝑅^𝑛+1+|𝑧|^𝑛+1)(|𝑧|+1)^𝑟−1𝑅^{(𝑟+𝑙)(𝑛+1)}. (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. Let 𝑏^(0,𝑚)_2𝑛+1(𝑧;𝑓) be the unique Hermite-Birkhoff polynomial of degree 2𝑛+1 which interpolates 𝑓 in the (𝑛+1)st roots of unity and whose 𝑚th derivative interpolates 𝑓^(𝑚) in the (𝑛+1)st roots of unity, that is,𝑏^(0,𝑚)_2𝑛+1(𝜔;𝑓)=𝑓(𝜔),(𝑏^(0,𝑚)_2𝑛+1(𝜔;𝑓))^(𝑚)=𝑓^(𝑚)(𝜔), (2.13)where 𝜔 is any (𝑛+1)st root of unity.

Setting𝐵^(0,𝑚)_2𝑛+1,0(𝑧;𝑓)=∑^2𝑛+1_𝑘=0𝑎_𝑘𝑧^𝑘,𝐵^(0,𝑚)_2𝑛+1,𝜈(𝑧;𝑓)=∑^𝑛_𝑗=0𝑎_{𝑗+(𝜈+1)(𝑛+1)}⋅𝑧^𝑗𝑞_𝑗,𝜈(𝑧),𝜈=1,2…, (2.14)where 𝑞_𝑗,𝜈(𝑧) is a polynomial of degree (𝑛+1) given by𝑞_𝑗,𝜈(𝑧)=𝑧^𝑛+1+((𝜈+1)(𝑛+1)+𝑗)_𝑚−(𝑛+1+𝑗)_𝑚(𝑛+1+𝑗)_𝑚−(𝑗)_𝑚(𝑧^𝑛+1−1),𝑗=0,1,…,𝑛, (2.15)with the following conventional notation(𝑗)_𝑚=𝑗(𝑗−1)⋯(𝑗−𝑚+1),if𝑚≤𝑗,0,if𝑚>𝑗, (2.16)in [6, Theorem 4, page 170] the authors proved the following result.

Theorem 2.3. For each positive integer 𝑙, lim_𝑛→∞{𝑏^(0,𝑚)_2𝑛+1(𝑧;𝑓)−^𝑙−1_𝑗=0𝐵^(0,𝑚)_2𝑛+1,𝑗(𝑧;𝑓)}=0,for|𝑧|<𝜌^1+𝑙/2 (2.17)and this result is best possible.

In the proof, it was shown in [6, Theorem 4, page 171] that𝑏^(0,𝑚)_2𝑛+1(𝑧;𝑓)−^𝑙−1_𝑗=0𝐵^(0,𝑚)_2𝑛+1,𝑗1(𝑧;𝑓)=2𝜋𝑖_Γ𝑓(𝑡)𝐾_𝑛,𝑙(𝑡,𝑧)𝑑𝑡, (2.18)where Γ is any circle |𝑡|=𝑅 with 1<𝑅<𝜌 and |𝐾_𝑛,𝑙(𝑡,𝑧)| is bounded on the circle |𝑡|=𝑅<|𝑧| by|𝐾_𝑛,𝑙(𝑡,𝑧)|≤|𝑧|^𝑛+1(|𝑡|^𝑛+1+|𝑧|^𝑛+1)|𝑡−𝑧||𝑡|^{(𝑙+1)(𝑛+1)}|𝑡^𝑛+1+−1|(|𝑧|^𝑛+1+1)(|𝑧|^𝑛+1+𝑅^𝑛+1)(|𝑧|−𝑅)𝑅^{(𝑙+2)(𝑛+1)}. (2.19)

3. Main Results

Our aim in this section is to apply a 𝐺_𝑢−𝐺_𝑣 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 𝐿_𝑛(𝑧;𝑓) and ∑^𝑙−1_𝑗=0𝑆_𝑛,𝑗(𝑧;𝑓) in Theorem 2.1 by 𝐿_𝑛 and 𝑆_𝑛,𝑙, respectively.

Theorem 3.1. Let 𝑓(𝑧)∈𝐴_𝜌 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 𝐺_𝑢−𝐺_𝑣 matrix and define𝜆_𝑛(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝐿_𝑘,𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝑆_𝑘,𝑙 (3.1)for 𝑛=0,1,…. Then, for each 𝑙,lim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=0,∀𝑧∈𝐷𝜌. (3.2)

Proof. Using the integral representation given in (2.6), for each 𝑙=1,2,…, we have𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘(𝐿_𝑘−𝑆_𝑘,𝑙)=^∞_𝑘=0𝑎_𝑛𝑘12𝜋𝑖_Γ𝑓(𝑡)(𝑡^𝑘+1−𝑧^𝑘+1)(𝑡−𝑧)(𝑡^𝑘+1−1)𝑡^𝑙(𝑘+1)=𝑑𝑡_∞𝑘=0𝑎_𝑛𝑘12𝜋𝑖_Γ𝑓(𝑡)(𝑡−𝑧)𝑡^𝑙(𝑘+1)𝑧[1−(𝑡)^𝑘+1]𝑡^𝑘+1𝑡^𝑘+1−1𝑑𝑡. (3.3)Since |𝑡|=𝑅>1, we have𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘12𝜋𝑖_Γ𝑓(𝑡)(𝑡−𝑧)𝑡^𝑙(𝑘+1)𝑧[1−(𝑡)^𝑘+1]^∞_𝑞=0(1𝑡^𝑘+1)^𝑞=1𝑑𝑡2𝜋𝑖_Γ𝑓(𝑡)(𝑡−𝑧)^∞_𝑞=01𝑡^∞𝑙+𝑞_𝑘=0𝑎_𝑛𝑘𝑧[1−(𝑡)^𝑘+11](𝑡^𝑙+𝑞)^𝑘𝑑𝑡. (3.4)The interchange of the integral and the summations is justified by showing that the series^∞_𝑘=0𝑎_𝑛𝑘(1𝑡^𝑙+𝑞)^𝑘,^∞_𝑘=0𝑎_𝑛𝑘(𝑧𝑡)^𝑘(1𝑡^𝑙+𝑞)^𝑘 (3.5)converge absolutely for each 𝑞 as follows. From (1.3), for any 1<𝑤<1/𝑣 we have that |𝑎_𝑛𝑘|𝑤^𝑛≤𝐵𝑠^𝑘, where 𝑠<1/𝑢<𝑅/𝜌<1. Thus, for each 𝑞,𝑙, and 𝑛, we have^∞_𝑘=0|𝑎_𝑛𝑘1|(|𝑡|^𝑙+𝑞)^𝑘≤𝐵𝑤^𝑛∞_𝑘=0(𝑠𝑅^𝑙+𝑞)^𝑘=𝐵𝑤^𝑛𝑅^𝑙+𝑞(𝑅^𝑙+𝑞,−𝑠) (3.6)because 𝑠/𝑅^𝑙+𝑞<𝑅/𝜌𝑅^𝑙+𝑞<1 and similarly for |𝑧|<𝜌^∞_𝑘=0|𝑎_𝑛𝑘𝑧||𝑡|^𝑘+1(1|𝑡|^𝑙+𝑞)^𝑘≤𝐵𝜌𝑤^𝑛𝑅^∞_𝑘=0(𝑠𝜌𝑅^𝑙+𝑞+1)^𝑘=𝐵𝜌𝑤^𝑛𝑅𝑅^𝑙+𝑞+1𝑅^𝑙+𝑞+1𝜌,−𝑠 (3.7)because 𝑠𝜌/𝑅^𝑙+𝑞+1<𝑅/𝑅^𝑙+𝑞+1<1. Therefore, from (3.6) and (3.7), identities (3.4) become|𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙𝐵(𝑧)|≤2𝜋𝑤^𝑛_Γ|𝑓(𝑡)||𝑡−𝑧|^∞_𝑞=01𝑅^𝑙+𝑞[𝑅^𝑙+𝑞𝑅^𝑙+𝑞+−𝑠𝜌𝑅^𝑙+𝑞𝑅^𝑙+𝑞+1𝜌=𝐵−𝑠]𝑑𝑡2𝜋𝑤^𝑛_Γ|𝑓(𝑡)||𝑡−𝑧|^∞_𝑞=0[1𝑅^𝑙+𝑞+𝜌−𝑠𝑅^𝑙+𝑞+1−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 that 𝑓(𝑡) is bounded on Γ,𝑤>1 implies thatlim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=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 denoteℎ_{𝑟(𝑛+1)−1}(𝑧;𝑓),^𝑙−1_𝑗=0𝐻_{𝑟(𝑛+1)−1,𝑗}(𝑧;𝑓) (3.10)in Theorem 2.2 by ℎ_𝑛 and 𝐻_𝑛,𝑙, respectively.

Theorem 3.2. Let 𝑓(𝑧)∈𝐴_𝜌 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 𝐺_𝑢−𝐺_𝑣 matrix and define𝜆_𝑛(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘ℎ_𝑘,𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝐻_𝑘,𝑙 (3.11)for 𝑛=0,1,…. Then, for each 𝑙,lim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=0,∀𝑧∈𝐷𝜌. (3.12)

Proof. Using the integral representation given in (2.11), for each 𝑙=1,2,…, we have𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘(ℎ_𝑘−𝐻_𝑘,𝑙)=^∞_𝑘=0𝑎_𝑛𝑘12𝜋𝑖_Γ𝑓(𝑡)𝑡−𝑧𝐾(𝑡,𝑧)𝑑𝑡. (3.13)From (2.12) we obtain that|𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)|≤^∞_𝑘=0|𝑎_𝑛𝑘|12𝜋_Γ|𝑓(𝑡)||𝑡−𝑧|𝑀(𝑅^𝑘+1+|𝑧|^𝑘+1)(|𝑧|+1)^𝑟−1𝑅^{(𝑟+𝑙)(𝑘+1)}≤𝑑𝑡𝑀(|𝑧|+1)^𝑟+12𝜋_Γ|𝑓(𝑡)||𝑡−𝑧|^∞_𝑘=0|𝑎_𝑛𝑘|(𝑅^𝑘+1+|𝑧|^𝑘+1)𝑅^{(𝑟+𝑙)(𝑘+1)}𝑑𝑡. (3.14)The interchange of the integral and the summation is justified by showing that the series^∞_𝑘=0|𝑎_𝑛𝑘|𝑅^{(𝑟+𝑙−1)(𝑘+1)},^∞_𝑘=0|𝑎_𝑛𝑘|(|𝑧|𝑅^𝑟+𝑙)^𝑘+1 (3.15)converge as follows. From (1.3) we have that for any 1<𝑤<1/𝑣,|𝑎_𝑛𝑘|𝑤^𝑛≤𝐵𝑠^𝑘, where 𝑠<1/𝑢<𝑅/𝜌<1.
Thus for any fixed positive integer 𝑟≥2 and for each 𝑙 and 𝑛^∞_𝑘=0|𝑎_𝑛𝑘1|(𝑅^{𝑟+𝑙−1})^𝑘+1≤𝐵𝑤^𝑛𝑅^∞𝑟+𝑙_𝑘=0(𝑠𝑅^{𝑟+𝑙−1})^𝑘=𝐵𝑤^𝑛(𝑅^{𝑟+𝑙−1},−𝑠) (3.16)because 𝑠/𝑅^{𝑟+𝑙−1}<1/𝜌𝑅^{𝑟+𝑙−2}<1, and similarly for |𝑧|<𝜌,^∞_𝑘=0|𝑎_𝑛𝑘|(|𝑧|𝑅^𝑟+𝑙)^𝑘+𝑙≤𝐵|𝑧|𝑤^𝑛𝑅^∞𝑟+𝑙_𝑘=0(𝑠|𝑧|𝑅^𝑟+𝑙)^𝑘=𝐵|𝑧|𝑤^𝑛1(𝑅^𝑟+𝑙,−𝑠|𝑧|) (3.17)because 𝑠|𝑧|/𝑅^𝑟+𝑙<1/𝑅^{𝑟+𝑙−1}<1.
Therefore, using (3.16) and (3.17) in (3.14) we get that|𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)|≤𝑀𝐵(|𝑧|+1)^𝑟+12𝜋𝑤^𝑛_Γ|𝑓(𝑡)|[1|𝑡−𝑧|𝑅^{𝑟+𝑙−1}+−𝑠|𝑧|𝑅^𝑟+𝑙−𝑠|𝑧|]𝑑𝑡. (3.18)Assuming that 𝑓(𝑡) is bounded on Γ,𝑤>1 implies thatlim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=0,for|𝑧|<𝜌. (3.19)

Thus in the Lagrange case, the 𝐺_𝑢−𝐺_𝑣 matrix transformation of the sequence operators produces new sequences such that the difference between the transformed sequences of polynomials 𝐿_𝑛 and 𝑆_𝑛,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 under 𝐺_𝑢−𝐺_𝑣 matrix transformation by choosing 𝑢>𝜌/𝑅. 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 𝑢>𝜌²/𝑅.

To simplify, let us denote 𝑏^(0,𝑚)_2𝑛+1(𝑧;𝑓) and ∑^𝑙−1_𝑗=0𝐵^(0,𝑚)_2𝑛+1,𝑗(𝑧;𝑓) of Theorem 2.3 by 𝑏_𝑛 and 𝐵_𝑛,𝑙, respectively.

Theorem 3.3. Let 𝑓(𝑧)∈𝐴_𝜌 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 𝐺_𝑢−𝐺_𝑣 matrix and define𝜆_𝑛(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝑏_𝑘,𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘𝐵_𝑘,𝑙 (3.20)for 𝑛=0,1,…. Then, for each 𝑙,lim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=0,∀𝑧∈𝐷𝜌. (3.21)

Proof. Using the integral representation given in (2.18), for each 𝑙=1,2,…, we have𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)=^∞_𝑘=0𝑎_𝑛𝑘12𝜋𝑖_Γ𝑓(𝑡)𝐾_𝑘,𝑙(𝑡,𝑧)𝑑𝑡. (3.22)From (2.19) we obtain that|𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)|≤^∞_𝑘=0|𝑎_𝑛𝑘|12𝜋_Γ|𝑓(𝑡)||𝐾_𝑘,𝑙≤1(𝑡,𝑧)|𝑑𝑡2𝜋_Γ|𝑓(𝑡)||𝑡−𝑧|^∞_𝑘=0|𝑎_𝑛𝑘||𝑧|^𝑘+1(|𝑡|^𝑘+1+|𝑧|^𝑘+1)|𝑡|^{(𝑙+1)(𝑘+1)}|𝑡^𝑘+1+1−1|𝑑𝑡2𝜋_Γ|𝑓(𝑡)||𝑧|−𝑅^∞_𝑘=0|𝑎_𝑛𝑘|(|𝑧|^𝑘+1+1)(|𝑧|^𝑘+1+𝑅^𝑘+1)𝑅^{(𝑙+2)(𝑘+1)}𝑑𝑡. (3.23)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,|𝑎_𝑛𝑘|𝑤^𝑛≤𝐵𝑠^𝑘, where 𝑠<1/𝑢<𝑅/𝜌²<1. Since |𝑡|=𝑅 and |𝑧|<𝜌, using the same method used in (3.4) we write the first summation as^∞_𝑘=0|𝑎_𝑛𝑘||𝑧|^𝑘+1|𝑡|^{(𝑙+1)(𝑘+1)}(|𝑡|^𝑘+1+|𝑧|^𝑘+1)(𝑅^𝑘+1≤−1)^∞_𝑘=0𝐵𝑠^𝑘|𝑧|^𝑘+1𝑤^𝑛𝑅^{(𝑙+1)(𝑘+1)}[1+(|𝑧|𝑅)^𝑘+1]^∞_𝑞=0(1𝑅^𝑘+1)^𝑞≤𝐵𝑤^𝑛∞_𝑞=0𝜌𝑅^𝑙+𝑞+1[^∞_𝑘=0(𝑠𝜌𝑅^𝑙+𝑞+1)^𝑘+^∞_𝑘=0(𝑠𝜌𝑅^𝑙+𝑞+1)^𝑘(𝜌𝑅)^𝑘+1]=𝐵𝑤^𝑛∞_𝑞=0𝜌𝑅^𝑙+𝑞+1[𝑅^𝑙+𝑞+1(𝑅^𝑙+𝑞+1+𝜌−𝑠𝜌)𝑅(𝑅^𝑙+𝑞+2)(𝑅^𝑙+𝑞+2−𝑠𝜌²)], (3.24)because 𝑠𝜌/𝑅^𝑙+𝑞+1<1/𝜌𝑅^𝑙+𝑞<1 and 𝑠𝜌²/𝑅^𝑙+𝑞+2<1/𝑅^𝑙+𝑞+1<1 for each 𝑙 and 𝑞. Now, for the second sum in (3.23) we have^∞_𝑘=0|𝑎_𝑛𝑘|(|𝑧|^𝑘+1+1)(|𝑧|^𝑘+1+𝑅^𝑘+1)𝑅^{(𝑙+2)(𝑘+1)}≤𝐵𝑤^𝑛∞_𝑘=0𝑠^𝑘𝑅^{(𝑙+2)(𝑘+1)}[𝜌^2𝑘+2+𝜌^𝑘+1+(𝜌𝑅)^𝑘+1+𝑅^𝑘+1]=𝐵𝑤^𝑛[𝜌²𝑅^𝑙+2−𝑠𝜌²+𝜌𝑅^𝑙+2𝜌+𝜌−𝑠𝑅^𝑙+1𝜌+1−𝑠𝑅^𝑙+1−𝑠], (3.25)because for each 𝑙,𝑠𝜌²𝑅^𝑙+2<1𝑅^𝑙+1𝑠𝜌<1,𝑅^𝑙+2<1𝜌𝑅^𝑙+1𝑠𝜌<1,𝑅^𝑙+1<1𝜌𝑅^𝑙𝑠<1,𝑅^𝑙+1<1𝜌²𝑅^𝑙<1. (3.26)Using (3.24) and (3.25), the expression (3.23) becomes|𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙𝐵(𝑧)|≤𝑤^𝑛2𝜋_Γ|𝑓(𝑡)||𝑡−𝑧|^∞_𝑞=0[𝜌𝑅^𝑙+𝑞+1𝜌+𝜌−𝑠𝑅^2𝑙+𝑞+2−𝑠𝜌²+𝐵]𝑑𝑡2𝜋𝑤^𝑛_Γ|𝑓(𝑡)|[𝜌|𝑧|−𝑅𝑅^𝑙+2−²𝑠𝜌²+𝜌𝑅^𝑙+2𝜌+𝜌−𝑠𝑅^𝑙+1𝜌+1−𝑠𝑅^𝑙+1−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 that 𝑓(𝑡) is bounded on Γ,𝑤>1 implies thatlim_𝑛→∞[𝜆_𝑛(𝑧)−𝜎_𝑛,𝑙(𝑧)]=0 (3.28)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 𝐺_𝑢−𝐺_𝑣 matrix, we give below an obvious example.

Define a matrix 𝐴 by𝑎_𝑛𝑘=𝑣^𝑛𝑝^𝑘,𝑝>𝑢. (3.29)Then for any 𝑤 such that 0<𝑤<1/𝑣, we have that|𝑎_𝑛𝑘|𝑤^𝑛=(𝑣𝑤)^𝑛𝑝^𝑘1<(𝑝)^𝑘, (3.30)where 1/𝑝<1/𝑢. Hence by Theorem 1.1, 𝐴 is a 𝐺_𝑢−𝐺_𝑣 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.