AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation43296110.1155/2011/432961432961Research ArticleApproximation of Analytic Functions by Chebyshev FunctionsJungSoon-Mo1RassiasThemistocles M.2ZhouYong1Mathematics SectionCollege of Science and TechnologyHongik University Jochiwon 339-701Republic of Koreahongik.ac.kr2Department of MathematicsNational Technical University of AthensZografou Campus15780 AthensGreecentua.gr201108092011201107052011110720112011Copyright © 2011 Soon-Mo Jung and Themistocles M. Rassias.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.

We solve the inhomogeneous Chebyshev's differential equation and apply this result for approximating analytic functions by the Chebyshev functions.

1. Introduction

Let X be a normed space over a scalar field 𝕂, and let I be an open interval, where 𝕂 denotes either or . Assume that a0,a1,,an:I𝕂, and g:IX are given continuous functions and that y:IX is an n times continuously differentiable function satisfying the inequality an(t)y(n)(t)+an-1(t)y(n-1)(t)++a1(t)y(t)+a0(t)y(t)+g(t)ɛ for all tI and for a given ɛ>0. If there exists an n times continuously differentiable function y0:IX satisfying an(t)y0(n)(t)+an-1(t)y0(n-1)(t)++a1(t)y0(t)+a0(t)y0(t)+g(t)=0 and y(t)-y0(t)K(ɛ) for any tI, where K(ɛ) is an expression of ɛ with limɛ0K(ɛ)=0, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to .

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [8, 9]). Here, we will introduce a result of Alsina and Ger . They proved that if a differentiable function f:I satisfies the inequality |y(t)-y(t)|ɛ, where I is an open subinterval of , then there exists a constant c such that |f(t)-cet|3ɛ for any tI. Their result was generalized by Takahasi et al. Indeed, it was proved in  that the Hyers-Ulam stability holds true for the Banach space valued differential equation y(t)=λy(t) (see also [12, 13]).

Moreover, Miura et al.  investigated the Hyers-Ulam stability of nth order linear differential equation with complex coefficients. They  also proved the Hyers-Ulam stability of linear differential equations of first order, y(t)+g(t)y(t)=0, where g(t) is a continuous function.

Jung also proved the Hyers-Ulam stability of various linear differential equations of first order . Moreover, he applied the power series method to the study of the Hyers-Ulam stability of Legendre's differential equation (see [20, 21]). Recently, Jung and Kim tried to prove the Hyers-Ulam stability of the Chebyshev's differential equation(1-x2)y′′(x)-xy(x)+n2y(x)=0 for all x(-1,1). However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of the Chebyshev's differential equation in a strict sense (see ).

In Section 2 of this paper, by using the ideas from , we investigate the general solution of the inhomogeneous Chebyshev's differential equation of the form(1-x2)y′′(x)-xy(x)+n2y(x)=m=0amxm, where n is a given positive integer. Section 3 will be devoted to the investigation of the Hyers-Ulam stability and an approximation property of the Chebyshev functions.

2. Inhomogeneous Chebyshev’s Equation

Every solution of the Chebyshev's differential equation (1.3) is called a Chebyshev function. The Chebyshev's differential equation has regular singular points at −1, 1, and , and it plays a great role in physics and engineering. In particular, this equation is most useful for treating the boundary value problems exhibiting certain symmetries.

In this section, we set c0=c1=0 and define, for all m,c2m=12mi=0m-1a2i2i+1j=i+1m-1(2j)2-n22j(2j+1),c2m+1=12m+1i=0m-1a2i+12i+2j=i+1m-1(2j+1)2-n2(2j+1)(2j+2), where we refer to (1.4) for the am's and we follow the convention j=mm-1[]=1. We can easily check that cm's satisfy the following relation:(m+2)(m+1)cm+2-(m2-n2)cm=am for any m{0,1,2,}.

Theorem 2.1.

Assume that n is a positive integer and the radius of convergence of the power series m=0amxm is ρ>0. Let ρ0=min{1,ρ}. Then, every solution y:(-ρ0,ρ0) of the Chebyshev's differential equation (1.4) can be expressed by y(x)=yh(x)+m=2cmxm, where yh(x) is a Chebyshev function and the cm's are given in (2.1).

Proof.

It is not difficult to see that, if j and |(2j)2-n2|>2j(2j+1), then j<-1+1+8n28<8n28(for  2j<n). Hence, we have 1jne with ne=[n/8]. If m>ne, then it follows from (2.1) that |c2m|12mi=0ne-1|a2i|2i+1(j=i+1ne|(2j)2-n2|2j(2j+1))(j=ne+1m-1|(2j)2-n2|2j(2j+1))+12mi=nem-1|a2i|2i+1j=i+1m-1|(2j)2-n2|2j(2j+1)12mi=0ne-1|a2i|2i+1j=i+1nen2-42j(2j+1)+12mi=nem-1|a2i|2i+112mi=0ne-1(2i)!(n2-4)ne-i(2ne+1)!|a2i|+12mi=nem-1|a2i|2ne+1max0ine((2i)!/(2ne+1)!)(n2-4)ne-i2mi=0m-1|a2i|. We now suppose 1mne. Then it holds true that n3, and we have |c2m|12mi=0m-1|a2i|2i+1j=i+1m-1|(2j)2-n2|2j(2j+1)12mi=0m-1|a2i|2i+1j=i+1m-1n2-42j(2j+1)=12mi=0m-1(2i)!(n2-4)m-1-i(2m-1)!|a2i|max0im-1((2i)!/(2m-1)!)(n2-4)m-1-i2mi=0m-1|a2i|. Hence, we conclude from the above two inequalities that |c2m|Me2mi=0m-1|a2i| for all m, where we set Me=max0ilne(2i)!(2l+1)!(n2-4)l-i.

On the other hand, if j and |(2j+1)2-n2|>(2j+1)(2j+2), then j<8n2+1-58<8n2-48<n2-12(for  2j+1<n). Hence, we get 1jno with no=[n/8-1/2]. If m>no, then it follows from (2.1) that |c2m+1|12m+1i=0no-1|a2i+1|2i+2(j=i+1no|(2j+1)2-n2|(2j+1)(2j+2))(j=no+1m-1|(2j+1)2-n2|(2j+1)(2j+2))+12m+1i=nom-1|a2i+1|2i+2j=i+1m-1|(2j+1)2-n2|(2j+1)(2j+2)12m+1i=0no-1|a2i+1|2i+2j=i+1non2-9(2j+1)(2j+2)+12m+1i=nom-1|a2i+1|2i+212m+1i=0no-1(2i+1)!(n2-9)no-i(2no+2)!|a2i+1|+12m+1i=nom-1|a2i+1|2no+2max0ino((2i+1)!/(2no+2)!)(n2-9)no-i2m+1i=0m-1|a2i+1|. If 1mno, then we have n5, and it follows from (2.1) that |c2m+1|12m+1i=0m-1|a2i+1|2i+2j=i+1m-1|(2j+1)2-n2|(2j+1)(2j+2)12m+1i=0m-1|a2i+1|2i+2j=i+1m-1n2-9(2j+1)(2j+2) since j<no and hence 2j+1<2n/8<n. Furthermore, we have |c2m+1|12m+1i=0m-1(2i+1)!(n2-9)m-1-i(2m)!|a2i+1|max0im-1((2i+1)!/(2m)!)(n2-9)m-1-i2m+1i=0m-1|a2i+1|. Thus, we may conclude from the last two inequalities that |c2m+1|Mo2m+1i=0m-1|a2i+1| for any m, where Mo=max0ilno(2i+1)!(2l+2)!(n2-9)l-i. Let ρ1 be an arbitrary positive number less than ρ0. Then it follows from (2.7) and (2.13) that |m=2cmxm|m=1|c2m||x|2m+m=1|c2m+1||x|2m+1Mem=1|x|2m2mi=0m-1|a2i|+Mom=1|x|2m+12m+1i=0m-1|a2i+1|=Me|a0|(|x|22+|x|44+|x|66+|x|88+|x|1010+)+Me|a2||x|2(|x|24+|x|46+|x|68+|x|810+|x|1012+)+Me|a4||x|4(|x|26+|x|48+|x|610+|x|812+|x|1014+)++Mo|a1||x|(|x|23+|x|45+|x|67+|x|89+|x|1011+)+Mo|a3||x|3(|x|25+|x|47+|x|69+|x|811+|x|1013+)+Mo|a5||x|5(|x|27+|x|49+|x|611+|x|813+|x|1015+)+=Mem=0|a2m||x|2mi=1|x|2i2(m+i)+Mom=0|a2m+1||x|2m+1i=1|x|2i2(m+i)+1 for any x[-ρ1,ρ1].

Because of 0<ρ1<ρ01, we obtain i=1|x|2i2(m+i)12m+2|x|21-|x|2,i=1|x|2i2(m+i)+112m+3|x|21-|x|2 for all x[-ρ1,ρ1]. Thus, we have |m=2cmxm|Mem=0|a2mx2m|2m+2|x|21-|x|2+Mom=0|a2m+1x2m+1|2m+3|x|21-|x|2Me|x|21-|x|2m=0|amxm|m+2 for all x[-ρ1,ρ1]. Since ρ1 is arbitrarily given with 0<ρ1<ρ0, inequality (2.17) holds true for all x(-ρ0,ρ0). Moreover, the power series m=0amxm is absolutely convergent on (-ρ,ρ). Hence, we conclude that |m=2cmxm|< for all x(-ρ0,ρ0). That is, the power series m=2cmxm is convergent for each x(-ρ0,ρ0).

We will now prove that m=2cmxm satisfies the inhomogeneous Chebyshev's differential equation (1.4) for all x(-ρ0,ρ0). If we substitute m=2cmxm=m=1c2mx2m+m=1c2m+1x2m+1 for y(x) in (1.4), then it follows from (2.2) that (1-x2)y′′(x)-xy(x)+n2y(x)=m=0(2m+2)(2m+1)c2m+2x2m+m=0(2m+3)(2m+2)c2m+3x2m+1-m=12m(2m-1)c2mx2m-m=1(2m+1)(2m)c2m+1x2m+1-m=12mc2mx2m-m=1(2m+1)c2m+1x2m+1+m=1n2c2mx2m+m=1n2c2m+1x2m+1=2c2+6c3x+m=1[(2m+2)(2m+1)c2m+2+(n2-(2m)2)c2m]x2m+m=1[(2m+3)(2m+2)c2m+3+(n2-(2m+1)2)c2m+1]x2m+1=2c2+6c3x+m=1a2mx2m+m=1a2m+1x2m+1=m=0amxm for all x(-ρ0,ρ0). That is, m=2cmxm is a particular solution of the inhomogeneous Chebyshev's differential equation (1.4), and hence every solution y:(-ρ0,ρ0) of (1.4) can be expressed by y(x)=yh(x)+m=2cmxm, where yh(x) is a Chebyshev function.

3. Approximate Chebyshev Differential Equation

In this section, let K0 and ρ>0 be constants. We denote by 𝒞K the set of all functions y:(-ρ,ρ) with the following properties:

y(x) is expressible by a power series m=0bmxm whose radius of convergence is at least ρ;

m=0|amxm|K|m=0amxm| for any x(-ρ,ρ), where am=(m+2)(m+1)bm+2-(m2-n2)bm for all m0 and set b0=b1=0.

We now investigate the (local) Hyers-Ulam stability problem of the Chebyshev differential equation. More precisely, we try to answer the question, whether there exists a Chebyshev function near any approximate Chebyshev function.

Theorem 3.1.

Let n be a positive integer, and assume that a function y𝒞K satisfies the differential inequality |(1-x2)y′′(x)-xy(x)+n2y(x)|ɛ for all x(-ρ,ρ) and for some ɛ>0. Let ρ0=min{1,ρ}. Then there exists a Chebyshev function yh:(-ρ0,ρ0) such that |y(x)-yh(x)|KMeɛ2x21-x2 for all x(-ρ0,ρ0), where the constant Me is defined in (2.8).

Proof.

It follows from (a) and (b) that (1-x2)y′′(x)-xy(x)+n2y(x)=m=0amxm for all x(-ρ,ρ) (cf. (2.19)). Moreover, by using (b) and (3.2), we get m=0|amxm|K|m=0amxm|Kɛ for any x(-ρ,ρ).

According to Theorem 2.1 and (3.4), y(x) can be written as yh(x)+m=2cmxm for all x(-ρ0,ρ0), where yh is some Chebyshev function and cm's are given in (2.1). It moreover follows from (2.17) and (3.5) that |y(x)-yh(x)|=|m=2cmxm|Mex21-x2K2ɛ for all x(-ρ0,ρ0).

If ρ is assumed to be less than 1, then ρ0=ρ<1 and Theorem 3.1 implies the Hyers-Ulam stability of the Chebyshev's differential equation (1.3).

Remark 3.2.

We give some values for ne, no, Me, and Mo in Table 1.

nnenoMeMo
1 0 −1 1 -
2 0 0 1 1/2
3 1 0 1 1/2
4 1 0 2 1/2
5 1 1 7/22/3
6 2 1 128/159/8
Corollary 3.3.

Let n be a positive integer, and assume that a function y𝒞K satisfies the differential inequality (3.2) for all x(-ρ,ρ) and for some ɛ>0. Let ρ0=min{1,ρ}. Then there exists a Chebyshev function yh:(-ρ0,ρ0) such that |y(x)-yh(x)|=O(x2) as x0.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).

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