Approximation of Analytic Functions by Chebyshev Functions

Abstract

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⊂ℝ be an open interval, where 𝕂 denotes either ℝ or ℂ. Assume that 𝑎₀,𝑎₁,…,𝑎_𝑛∶𝐼→𝕂, and 𝑔∶𝐼→𝑋 are given continuous functions and that 𝑦∶𝐼→𝑋 is an 𝑛 times continuously differentiable function satisfying the inequality 𝑎_𝑛(𝑡)𝑦^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝑦^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)𝑦^′(𝑡)+𝑎₀(𝑡)𝑦(𝑡)+𝑔(𝑡)≤𝜀 (1.1) for all 𝑡∈𝐼 and for a given 𝜀>0. If there exists an 𝑛 times continuously differentiable function 𝑦₀∶𝐼→𝑋 satisfying 𝑎_𝑛(𝑡)𝑦₀^(𝑛)(𝑡)+𝑎_𝑛−1(𝑡)𝑦₀^(𝑛−1)(𝑡)+⋯+𝑎₁(𝑡)𝑦^′₀(𝑡)+𝑎₀(𝑡)𝑦₀(𝑡)+𝑔(𝑡)=0 (1.2) and ‖𝑦(𝑡)−𝑦₀(𝑡)‖≤𝐾(𝜀) for any 𝑡∈𝐼, where 𝐾(𝜀) is an expression of 𝜀 with lim_𝜀→0𝐾(𝜀)=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 [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 [10]. They proved that if a differentiable function 𝑓∶𝐼→ℝ satisfies the inequality |𝑦′(𝑡)−𝑦(𝑡)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a constant 𝑐 such that |𝑓(𝑡)−𝑐𝑒^𝑡|≤3𝜀 for any 𝑡∈𝐼. Their result was generalized by Takahasi et al. Indeed, it was proved in [11] 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. [14] investigated the Hyers-Ulam stability of 𝑛th order linear differential equation with complex coefficients. They [15] 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 [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 equation1−𝑥²𝑦^′′(𝑥)−𝑥𝑦^′(𝑥)+𝑛²𝑦(𝑥)=0 (1.3) 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 [22]).

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 form1−𝑥²𝑦^′′(𝑥)−𝑥𝑦^′(𝑥)+𝑛²𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚, (1.4) 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∈ℕ,𝑐_2𝑚=12𝑚^𝑚−1_𝑖=0𝑎_2𝑖2𝑖+1^𝑚−1_𝑗=𝑖+12𝑗²−𝑛²,𝑐2𝑗2𝑗+1_2𝑚+1=12𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+12𝑖+2^𝑚−1_𝑗=𝑖+12𝑗+1²−𝑛²,2𝑗+12𝑗+2 (2.1) where we refer to (1.4) for the 𝑎_𝑚's and we follow the convention ∏^𝑚−1_𝑗=𝑚[⋯]=1. We can easily check that 𝑐_𝑚's satisfy the following relation:(𝑚+2)(𝑚+1)𝑐_𝑚+2−𝑚²−𝑛²𝑐_𝑚=𝑎_𝑚 (2.2) 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=0𝑎_𝑚𝑥^𝑚 is 𝜌>0. Let 𝜌₀=min{1,𝜌}. Then, every solution 𝑦∶(−𝜌₀,𝜌₀)→ℂ of the Chebyshev's differential equation (1.4) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_m=2𝑐_𝑚𝑥^𝑚, (2.3) where 𝑦_ℎ(𝑥) is a Chebyshev function and the 𝑐_𝑚's are given in (2.1).

Proof. It is not difficult to see that, if 𝑗∈ℕ and |(2𝑗)²−𝑛²|>2𝑗(2𝑗+1), then √𝑗<−1+1+8𝑛²8<√8𝑛²8for2𝑗<𝑛. (2.4) Hence, we have 1≤𝑗≤𝑛_𝑒 with 𝑛_𝑒=[𝑛/√8]. If 𝑚>𝑛_𝑒, then it follows from (2.1) that 𝑐_2𝑚≤12𝑚^𝑛𝑒−1_𝑖=0𝑎_2𝑖2𝑖+1^𝑛𝑒_𝑗=𝑖+12𝑗²−𝑛²2𝑗2𝑗+1^𝑚−1_{𝑗=𝑛𝑒+1}2𝑗²−𝑛²+12𝑗2𝑗+12𝑚^𝑚−1_{𝑖=𝑛𝑒}𝑎_2𝑖2𝑖+1^𝑚−1_𝑗=𝑖+12𝑗²−𝑛²≤12𝑗2𝑗+12𝑚^𝑛𝑒−1_𝑖=0𝑎_2𝑖2𝑖+1^𝑛𝑒_𝑗=𝑖+1𝑛²−4+12𝑗2𝑗+12𝑚^𝑚−1_{𝑖=𝑛𝑒}𝑎_2𝑖≤12𝑖+12𝑚^𝑛𝑒−1_𝑖=0(2𝑖)!𝑛²−4^{𝑛𝑒−𝑖}2𝑛_𝑒!𝑎+1^2𝑖+12𝑚^𝑚−1_{𝑖=𝑛𝑒}𝑎_2𝑖2𝑛_𝑒≤+1max_{0≤𝑖≤𝑛𝑒}(2𝑖)!/2𝑛_𝑒+1!𝑛²−4^{𝑛𝑒−𝑖}2𝑚^𝑚−1_𝑖=0𝑎_2𝑖. (2.5) We now suppose 1≤𝑚≤𝑛_𝑒. Then it holds true that 𝑛≥3, and we have 𝑐_2𝑚≤12𝑚^𝑚−1_𝑖=0𝑎_2𝑖2𝑖+1^𝑚−1_𝑗=𝑖+12𝑗²−𝑛²≤12𝑗2𝑗+12𝑚^𝑚−1_𝑖=0𝑎_2𝑖2𝑖+1^𝑚−1_𝑗=𝑖+1𝑛²−4=12𝑗2𝑗+12𝑚^𝑚−1_𝑖=0(2𝑖)!𝑛²−4^{𝑚−1−𝑖}()!𝑎2𝑚−1^2𝑖≤max_{0≤𝑖≤𝑚−1}((2𝑖)!/(2𝑚−1)!)𝑛²−4^{𝑚−1−𝑖}2𝑚^𝑚−1_𝑖=0𝑎_2𝑖. (2.6) Hence, we conclude from the above two inequalities that 𝑐_2𝑚≤𝑀_𝑒2𝑚^𝑚−1_𝑖=0𝑎_2𝑖 (2.7) for all 𝑚∈ℕ, where we set 𝑀_𝑒=max_{0≤𝑖≤ℓ≤𝑛𝑒}(2𝑖)!()!𝑛2ℓ+1²−4^ℓ−𝑖. (2.8)
On the other hand, if 𝑗∈ℕ and |(2𝑗+1)²−𝑛²|>(2𝑗+1)(2𝑗+2), then √𝑗<8𝑛²+1−58<√8𝑛²−48<𝑛2−12for2𝑗+1<𝑛. (2.9) Hence, we get 1≤𝑗≤𝑛_𝑜 with 𝑛_𝑜=[n/√8−1/2]. If 𝑚>𝑛_𝑜, then it follows from (2.1) that 𝑐_2𝑚+1≤12𝑚+1^𝑛𝑜−1_𝑖=0𝑎_2𝑖+12𝑖+2^𝑛𝑜_𝑗=𝑖+12𝑗+1²−𝑛²2𝑗+12𝑗+2^𝑚−1_{𝑗=𝑛𝑜+1}2𝑗+1²−𝑛²+12𝑗+12𝑗+22𝑚+1^𝑚−1_{𝑖=𝑛𝑜}𝑎_2𝑖+12𝑖+2^𝑚−1_𝑗=𝑖+12𝑗+1²−𝑛²≤12𝑗+12𝑗+22𝑚+1^𝑛𝑜−1_𝑖=0𝑎_2𝑖+12𝑖+2^𝑛𝑜_𝑗=𝑖+1𝑛²−9+12𝑗+12𝑗+22𝑚+1^𝑚−1_{𝑖=𝑛𝑜}𝑎_2𝑖+1≤12𝑖+22𝑚+1^𝑛𝑜−1_𝑖=0(2𝑖+1)!𝑛²−9^{𝑛𝑜−𝑖}2𝑛_𝑜!𝑎+2^2𝑖+1+12𝑚+1^𝑚−1_{𝑖=𝑛𝑜}𝑎_2𝑖+12𝑛_𝑜≤+2max_{0≤𝑖≤𝑛𝑜}(2𝑖+1)!/2𝑛_𝑜+2!𝑛²−9^{𝑛𝑜−𝑖}2𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+1. (2.10) If 1≤𝑚≤𝑛_𝑜, then we have 𝑛≥5, and it follows from (2.1) that 𝑐_2𝑚+1≤12𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+12𝑖+2^𝑚−1_𝑗=𝑖+12𝑗+1²−𝑛²≤12𝑗+12𝑗+22𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+12𝑖+2^𝑚−1_𝑗=𝑖+1𝑛²−92𝑗+12𝑗+2 (2.11) since 𝑗<𝑛_𝑜 and hence 2𝑗+1<2𝑛/√8<𝑛. Furthermore, we have 𝑐_2𝑚+1≤12𝑚+1^𝑚−1_𝑖=0(2𝑖+1)!𝑛²−9^{𝑚−1−𝑖}()!𝑎2𝑚^2𝑖+1≤max_{0≤𝑖≤𝑚−1}((2𝑖+1)!/(2𝑚)!)𝑛²−9^{𝑚−1−𝑖}2𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+1. (2.12) Thus, we may conclude from the last two inequalities that 𝑐_2𝑚+1≤𝑀_𝑜2𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+1 (2.13) for any 𝑚∈ℕ, where 𝑀_𝑜=max_{0≤𝑖≤ℓ≤𝑛𝑜}(2𝑖+1)!()!𝑛2ℓ+2²−9^ℓ−𝑖. (2.14) Let 𝜌₁ be an arbitrary positive number less than 𝜌₀. Then it follows from (2.7) and (2.13) that ^∞_𝑚=2𝑐_𝑚𝑥^𝑚≤^∞_𝑚=1𝑐_2𝑚|𝑥|^2𝑚+^∞_𝑚=1𝑐_2𝑚+1|𝑥|^2𝑚+1≤𝑀_𝑒^∞_𝑚=1|𝑥|^2𝑚2𝑚^𝑚−1_𝑖=0𝑎_2𝑖+𝑀_𝑜^∞_𝑚=1|𝑥|^2𝑚+12𝑚+1^𝑚−1_𝑖=0𝑎_2𝑖+1=𝑀_𝑒𝑎₀|𝑥|²2+|𝑥|⁴4+|𝑥|⁶6+|𝑥|⁸8+|𝑥|¹⁰10+⋯+𝑀_𝑒𝑎₂|𝑥|²|𝑥|²4+|𝑥|⁴6+|𝑥|⁶8+|𝑥|⁸+|𝑥|1012¹⁰+⋯+𝑀_𝑒𝑎₄|𝑥|⁴|𝑥|²6+|𝑥|⁴8+|𝑥|⁶+|𝑥|10⁸+|𝑥|1214¹⁰+⋯+⋯+𝑀_𝑜𝑎₁𝑥||𝑥||²3+|𝑥|⁴5+|𝑥|⁶7+|𝑥|⁸9+|𝑥|¹⁰11+⋯+𝑀_𝑜𝑎₃|𝑥|³|𝑥|²5+|𝑥|⁴7+|𝑥|⁶9+|𝑥|⁸+|𝑥|1113¹⁰+⋯+𝑀_𝑜𝑎₅|𝑥|⁵|𝑥|²7+|𝑥|⁴9+|𝑥|⁶+|𝑥|11⁸+|𝑥|1315¹⁰+⋯+⋯=𝑀_𝑒^∞_𝑚=0𝑎_2𝑚|𝑥|^∞2𝑚_𝑖=1|𝑥|^2𝑖2(𝑚+𝑖)+𝑀_𝑜^∞_𝑚=0𝑎_2𝑚+1|𝑥|^∞2𝑚+1_𝑖=1|𝑥|^2𝑖2(𝑚+𝑖)+1 (2.15) for any 𝑥∈[−𝜌₁,𝜌₁].
Because of 0<𝜌₁<𝜌₀≤1, we obtain ^∞_𝑖=1|𝑥|^2𝑖2()≤1𝑚+𝑖|𝑥|2𝑚+21−|𝑥²|²,^∞_𝑖=1|𝑥|^2𝑖2()≤1𝑚+𝑖+1|𝑥|2𝑚+31−|𝑥²|² (2.16) for all 𝑥∈[−𝜌₁,𝜌₁]. Thus, we have ^∞_𝑚=2𝑐_𝑚𝑥^𝑚≤𝑀_𝑒^∞_𝑚=0𝑎_2𝑚𝑥^2𝑚|𝑥|2𝑚+21−|𝑥²|²+𝑀_𝑜^∞_𝑚=0𝑎_2𝑚+1𝑥^2𝑚+1|𝑥|2𝑚+31−|𝑥²|²≤𝑀_𝑒|𝑥|²1−|𝑥|^2∞_𝑚=0𝑎_𝑚𝑥^𝑚𝑚+2 (2.17) 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=0𝑎_𝑚𝑥^𝑚 is absolutely convergent on (−𝜌,𝜌). Hence, we conclude that ^∞_𝑚=2𝑐_𝑚𝑥^𝑚<∞ (2.18) for all 𝑥∈(−𝜌₀,𝜌₀). That is, the power series ∑^∞_𝑚=2𝑐_𝑚𝑥^𝑚 is convergent for each 𝑥∈(−𝜌₀,𝜌₀).
We will now prove that ∑^∞_𝑚=2𝑐_𝑚𝑥^𝑚 satisfies the inhomogeneous Chebyshev's differential equation (1.4) for all 𝑥∈(−𝜌₀,𝜌₀). If we substitute ∑^∞_𝑚=2𝑐_𝑚𝑥^𝑚=∑^∞_𝑚=1𝑐_2𝑚𝑥^2𝑚+∑^∞_𝑚=1𝑐_2𝑚+1𝑥^2𝑚+1 for 𝑦(𝑥) in (1.4), then it follows from (2.2) that 1−𝑥²𝑦^′′(𝑥)−𝑥𝑦^′(𝑥)+𝑛²𝑦(𝑥)=^∞_𝑚=0(2𝑚+2)(2𝑚+1)𝑐_2𝑚+2𝑥^2𝑚+^∞_𝑚=0(2𝑚+3)(2𝑚+2)𝑐_2𝑚+3𝑥^2𝑚+1−^∞_𝑚=12𝑚(2𝑚−1)𝑐_2𝑚𝑥^2𝑚−^∞_𝑚=1(2𝑚+1)(2𝑚)𝑐_2𝑚+1𝑥^2𝑚+1−^∞_𝑚=12𝑚𝑐_2𝑚𝑥^2𝑚−^∞_𝑚=1(2𝑚+1)𝑐_2𝑚+1𝑥^2𝑚+1+^∞_𝑚=1𝑛²𝑐_2𝑚𝑥^2𝑚+^∞_𝑚=1𝑛²𝑐_2𝑚+1𝑥^2𝑚+1=2𝑐₂+6𝑐₃𝑥+^∞_𝑚=1(2𝑚+2)(2𝑚+1)𝑐_2𝑚+2+𝑛²−(2𝑚)²𝑐_2𝑚𝑥^2𝑚+^∞_𝑚=1(2𝑚+3)(2𝑚+2)𝑐_2𝑚+3+𝑛²−(2𝑚+1)²𝑐_2𝑚+1𝑥^2𝑚+1=2𝑐₂+6𝑐₃𝑥+^∞_𝑚=1𝑎_2𝑚𝑥^2𝑚+^∞_𝑚=1𝑎_2𝑚+1𝑥^2𝑚+1=^∞_𝑚=0𝑎_𝑚𝑥^𝑚 (2.19) for all 𝑥∈(−𝜌₀,𝜌₀). That is, ∑^∞_𝑚=2𝑐_𝑚𝑥^𝑚 is a particular solution of the inhomogeneous Chebyshev's differential equation (1.4), and hence every solution 𝑦∶(−𝜌₀,𝜌₀)→ℂ of (1.4) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_𝑚=2𝑐_𝑚𝑥^𝑚, (2.20) where 𝑦_ℎ(𝑥) is a Chebyshev function.

3. Approximate Chebyshev Differential Equation

In this section, let 𝐾≥0 and 𝜌>0 be constants. We denote by 𝒞_𝐾 the set of all functions 𝑦∶(−𝜌,𝜌)→ℂ with the following properties: (a)𝑦(𝑥) is expressible by a power series ∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚 whose radius of convergence is at least 𝜌; (b)∑^∞_𝑚=0|𝑎_𝑚𝑥^𝑚|≤𝐾|∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚| for any 𝑥∈(−𝜌,𝜌), where 𝑎_𝑚=(𝑚+2)(𝑚+1)𝑏_𝑚+2−𝑚²−𝑛²𝑏_𝑚 (3.1) for all 𝑚∈ℕ₀ 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 𝑦∈𝒞_𝐾 satisfies the differential inequality 1−𝑥²𝑦^′′(𝑥)−𝑥𝑦^′(𝑥)+𝑛²𝑦(𝑥)≤𝜀 (3.2) for all 𝑥∈(−𝜌,𝜌) and for some 𝜀>0. Let 𝜌₀=min{1,𝜌}. Then there exists a Chebyshev function 𝑦_ℎ∶(−𝜌₀,𝜌₀)→ℂ such that 𝑦(𝑥)−𝑦_ℎ(𝑥)≤𝐾𝑀_𝑒𝜀2𝑥²1−𝑥² (3.3) for all 𝑥∈(−𝜌₀,𝜌₀), where the constant 𝑀_𝑒 is defined in (2.8).

Proof. It follows from (a) and (b) that 1−𝑥²𝑦^′′(𝑥)−𝑥𝑦^′(𝑥)+𝑛²𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚 (3.4) for all 𝑥∈(−𝜌,𝜌) (cf. (2.19)). Moreover, by using (b) and (3.2), we get ^∞_𝑚=0𝑎_𝑚𝑥^𝑚≤𝐾^∞_𝑚=0𝑎_𝑚𝑥^𝑚≤𝐾𝜀 (3.5) for any 𝑥∈(−𝜌,𝜌).
According to Theorem 2.1 and (3.4), 𝑦(𝑥) can be written as 𝑦_ℎ(𝑥)+∑^∞_𝑚=2𝑐_𝑚𝑥^𝑚 for all 𝑥∈(−𝜌₀,𝜌₀), where 𝑦_ℎ is some Chebyshev function and 𝑐_𝑚's are given in (2.1). It moreover follows from (2.17) and (3.5) that 𝑦(𝑥)−𝑦_ℎ(𝑥)=^∞_𝑚=2𝑐_𝑚𝑥^𝑚≤𝑀_𝑒𝑥²1−𝑥²𝐾2𝜀 (3.6) 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 𝑛_𝑒, 𝑛_𝑜, 𝑀_𝑒, and 𝑀_𝑜 in Table 1.

Table 1 

Corollary 3.3. Let 𝑛 be a positive integer, and assume that a function 𝑦∈𝒞_𝐾 satisfies the differential inequality (3.2) for all 𝑥∈(−𝜌,𝜌) and for some 𝜀>0. Let 𝜌₀=min{1,𝜌}. Then there exists a Chebyshev function 𝑦_ℎ∶(−𝜌₀,𝜌₀)→ℂ such that 𝑦(𝑥)−𝑦_ℎ(𝑥)=𝑂𝑥² (3.7) as 𝑥→0.

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).