Approximation of Analytic Functions by Chebyshev Functions
We solve the inhomogeneous Chebyshev's differential equation and apply this result for approximating analytic functions by the Chebyshev functions.
Let be a normed space over a scalar field , and let be an open interval, where denotes either or . Assume that , and are given continuous functions and that is an times continuously differentiable function satisfying the inequality for all and for a given . If there exists an times continuously differentiable function satisfying and for any , where is an expression of with , 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 [1–7].
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 satisfies the inequality , where is an open subinterval of , then there exists a constant such that for any . 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 (see also [12, 13]).
Moreover, Miura et al.  investigated the Hyers-Ulam stability of th order linear differential equation with complex coefficients. They  also proved the Hyers-Ulam stability of linear differential equations of first order, , where is a continuous function.
Jung also proved the Hyers-Ulam stability of various linear differential equations of first order [16–19]. 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 for all . 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 [20–26], we investigate the general solution of the inhomogeneous Chebyshev's differential equation of the form where 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 and define, for all , where we refer to (1.4) for the 's and we follow the convention . We can easily check that 's satisfy the following relation: for any .
Theorem 2.1. Assume that is a positive integer and the radius of convergence of the power series is . Let . Then, every solution of the Chebyshev's differential equation (1.4) can be expressed by where is a Chebyshev function and the 's are given in (2.1).
Proof. It is not difficult to see that, if and , then
Hence, we have with . If , then it follows from (2.1) that
We now suppose . Then it holds true that , and we have
Hence, we conclude from the above two inequalities that
for all , where we set
On the other hand, if and , then Hence, we get with . If , then it follows from (2.1) that If , then we have , and it follows from (2.1) that since and hence . Furthermore, we have Thus, we may conclude from the last two inequalities that for any , where Let be an arbitrary positive number less than . Then it follows from (2.7) and (2.13) that for any .
Because of , we obtain for all . Thus, we have for all . Since is arbitrarily given with , inequality (2.17) holds true for all . Moreover, the power series is absolutely convergent on . Hence, we conclude that for all . That is, the power series is convergent for each .
We will now prove that satisfies the inhomogeneous Chebyshev's differential equation (1.4) for all . If we substitute for in (1.4), then it follows from (2.2) that for all . That is, is a particular solution of the inhomogeneous Chebyshev's differential equation (1.4), and hence every solution of (1.4) can be expressed by where is a Chebyshev function.
3. Approximate Chebyshev Differential Equation
In this section, let and be constants. We denote by the set of all functions with the following properties: (a) is expressible by a power series whose radius of convergence is at least ; (b) for any , where for all and set .
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 be a positive integer, and assume that a function satisfies the differential inequality for all and for some . Let . Then there exists a Chebyshev function such that for all , where the constant is defined in (2.8).
Proof. It follows from (a) and (b) that
for all (cf. (2.19)). Moreover, by using (b) and (3.2), we get
for any .
According to Theorem 2.1 and (3.4), can be written as for all , where is some Chebyshev function and 's are given in (2.1). It moreover follows from (2.17) and (3.5) that for all .
Remark 3.2. We give some values for , , , and in Table 1.
Corollary 3.3. Let be a positive integer, and assume that a function satisfies the differential inequality (3.2) for all and for some . Let . Then there exists a Chebyshev function such that as .
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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