Mean Convergence Rate of Derivatives by Lagrange Interpolation on Chebyshev Grids

Abstract

We consider the rate of mean convergence of derivatives by Lagrange interpolation operators based on the Chebyshev nodes. Some estimates of error of the derivatives approximation in terms of the error of best approximation by polynomials are derived. Our results are sharp.

1. Introduction and Main Results

Mean convergence of Lagrange interpolation based on the zeros of orthogonal polynomials (and possibly some additional points) has been studied for at least 70 years. There is a vast literature on this topic. The authors of [1–3] considered the simultaneous approximation by the Hermite interpolation operators, and we will consider the simultaneous approximation by Lagrange interpolation operators based on the zeros of Chebyshev polynomials. The relevant results can be found in [4–6]. We introduce these results below.

Let 𝜔(𝑥)=^𝑁_𝑘=1𝑥−𝑦_𝑘^Γ𝑘|𝑥|≤1;−1=𝑦₁<𝑦₂<⋯<𝑦_𝑁=1;Γ_𝑘>−1;𝑘=1,…,𝑁 (1.1) be a so-called generalized Jacobi weight (𝜔∈𝐺𝐽), and let−1≤𝑥₁<𝑥₂<⋯<𝑥_𝑛≤1 (1.2) be the zeros of the 𝑛th orthogonal polynomial 𝑝_𝑛(𝜔) associated with the weight-function 𝜔∈𝐺𝐽. Let 𝐿_𝑛(𝜔,𝑓) denote the Lagrange interpolating polynomial which interpolates 𝑓 at the zeros of 𝑝_𝑛(𝜔). By using Markov-Bernstein type inequalities in 𝐿_𝑝 metric, J. Szabados and A. K. Varma [5] reduced the weighted mean convergence of derivatives 𝐿_𝑛^(𝑟)(𝜔,𝑓,𝑥) to the weighted mean convergence of 𝐿_𝑛(𝜔,𝑓,𝑥) and obtained the following. If 𝐿^𝑝 means functional space equipped with 𝐿_𝑝 norm and 𝜔(𝑥)𝜔(𝑥)∈𝐺𝐽,^1/𝑝−1/21−𝑥^21/4∈𝐿^𝑝, (∗) then, for 𝑓^(𝑟)(𝑥)∈𝐶[−1,1], we have ¹₋₁𝑓^(𝑟)(𝑥)−𝐿_𝑛^(𝑟)𝑓,𝑥^𝑝1−𝑥^2𝑟𝑝/2𝜔(𝑥)𝑑𝑥≤𝐶_𝑟𝐸^𝑝_{𝑛−𝑟−1}𝑓^(𝑟)(𝑛≥𝑟+1). (1.3)

Here and in the following, the constant 𝐶_𝑟 (may be different in the same expression) is independent of 𝑛 and 𝑓 but depends on 𝑟, and 𝐸_𝑛(⋅) denotes the error of the best polynomial approximation of degree 𝑛 of the corresponding function in the 𝐿_∞ metric.

Mastroianni and Nevai [4] get sharper estimates in terms of modulus of continuity instead of the best approximation. It improves some old results. But its proof also needs weighted Markov-Bernstein type inequality in 𝐿^𝑝 metric and the idea of additional points. For the weight functions not satisfying (*), it is not possible to discuss by their method. To deal with these case, Du and Xu [7] consider the most important special case 𝜔(𝑥)=1/√1−𝑥². Let 𝑡_𝑘=𝑡_𝑘𝑛=cos2𝑘−12𝑛𝜋,𝑘=1,…,𝑛, (1.4) be the zeros of 𝑇_𝑛(𝑥)=cos𝑛𝜃,𝑥=cos𝜃, the 𝑛th degree Chebyshev polynomial of the first kind. If 𝑓∈𝐶[−1,1], then the well-known Lagrange interpolation polynomial of 𝑓 based on {𝑡_𝑘}^𝑛_𝑘=1 is given by (see [8]) 𝐿_𝑛𝑓,𝑥=^𝑛_𝑘=1𝑓𝑡_𝑘ℓ_𝑘(𝑥), (1.5) where ℓ_𝑘(𝑥)=(−1)^𝑘+11−𝑡²_𝑘𝑇_𝑛(𝑥)𝑛𝑥−𝑡_𝑘,𝑘=1,…,𝑛. (1.6)

Du and Xu [7] obtained the following.

Theorem A. Let 𝐿_𝑛(𝑓,𝑥) be as defined as above. Then, for 𝑓∈𝐶¹_[−1,1], we have ¹₋₁𝑓^′(𝑥)−𝐿^′_𝑛𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−2𝑓^′𝑝,𝛼>2𝐶()−1,ln𝑛^1/𝑝𝐸_𝑛−2𝑓^′𝑝,𝛼=2−1,𝐶𝑛^{1−((2𝛼+2)/𝑝)}𝐸_𝑛−2𝑓^′𝑝,−1<𝛼<2−1, (1.7) and the estimation for −1<𝛼≤(𝑝/2)−1 is sharp.

We notice that although the sharp estimate is obtained, the upper bound is not 𝐸_𝑛−2(𝑓^′) for −1≤𝛼≤(𝑝/2)−1. Now we will give a Lagrange interpolation to improve their results.

Let 𝑥_𝑘=𝑥_𝑘𝑛=cos𝑘𝜋𝑛+1,𝑘=1,…,𝑛, (1.8) be the zeros of 𝑈_𝑛(𝑥)=sin(𝑛+1)𝜃/sin𝜃,𝑥=cos𝜃, the 𝑛th degree Chebyshev polynomial of the second kind. If 𝑓∈𝐶[−1,1], then the well-known Lagrange interpolation polynomial of 𝑓 based on {𝑥_𝑘}^𝑛_𝑘=1∪{𝑥₀=1,𝑥_𝑛+1=−1} is given by (see [9]) 𝑄_𝑛+2𝑓,𝑥=^𝑛+1_𝑘=0𝑓𝑥_𝑘𝜑_𝑘(𝑥), (1.9) where 𝜑₀(𝑥)=(1+𝑥)𝑈_𝑛(𝑥)2(𝑛+1),𝜑_𝑛+1(𝑥)=(−1)^𝑛(𝑥−1)𝑈_𝑛(𝑥)2(),𝜑𝑛+1_𝑘(𝑥)=(−1)^𝑘+11−𝑥²𝑈_𝑛(𝑥)(𝑛+1)𝑥−𝑥_𝑘,𝑘=1,…,𝑛. (1.10)

Firstly, we obtain the following.

Theorem 1.1. Let 𝑄_𝑛(𝑓,𝑥) be as defined as above, 0<𝑝<+∞,𝛼>−1. Then, for 𝑓∈𝐶¹_[−1,1], we have ¹₋₁𝑓^′(𝑥)−𝑄^′_𝑛𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−2𝑓^′. (1.11)

By Theorem A and Theorem 1.1, we know that 𝑄_𝑛+2(𝑓,𝑥) have better convergence rate than 𝐿_𝑛(𝑓,𝑥) in the case −1≤𝛼≤(𝑝/2)−1. But for continuous function approximation, we noticed that 𝑄_𝑛 have the same approximation order with 𝐿_𝑛, that is, if 0<𝑝<+∞,𝛼>−1, then, for 𝑓∈𝐶_[−1,1], from Hölder inequality [8, 9], it follows that ¹₋₁𝑓(𝑥)−𝐿_𝑛𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓,₁₋₁𝑓(𝑥)−𝑄_𝑛𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓. (1.12)

For high derivatives approximation, how the cases are? Secondly, we will consider second derivative approximation by 𝐿_𝑛 and 𝑄_𝑛 and obtain the following.

Theorem 1.2. Let 𝑄_𝑛(𝑓,𝑥) and 𝐿_𝑛(𝑓,𝑥) be as defined as above. Then, for 𝑓∈𝐶²_[−1,1], we have ¹₋₁𝑓^′′(𝑥)−𝑄_𝑛^′′𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−3𝑓^′′𝑝,𝛼>2𝐶()−1,ln𝑛^1/𝑝𝐸_𝑛−3𝑓^′′𝑝,𝛼=2−1,𝐶𝑛^{1−(2𝛼+2)/𝑝}𝐸_𝑛−3𝑓^′′𝑝,−1<𝛼<2−1,₁₋₁𝑓^′′(𝑥)−𝐿_𝑛^′′𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−3𝑓^′′𝐶(),𝛼>𝑝−1,ln𝑛^1/𝑝𝐸_𝑛−3𝑓^′′,𝛼=𝑝−1,𝐶𝑛^{2−(2𝛼+2)/𝑝}𝐸_𝑛−3𝑓^′′,−1<𝛼<𝑝−1, (1.13) and the estimation for −1<𝛼≤(𝑝/2)−1 or (−1<𝛼≤𝑝−1) is sharp.

From Theorem 1.2, we know that for the second derivative approximation, 𝑄_𝑛 have better approximation orders than 𝐿_𝑛 in the case −1<𝛼≤𝑝−1.

Using the same way as in the proof of Theorem 1.2, we can consider the 𝑟 order derivatives approximation for 𝑟≥3, but the computation is more complicated, and we omit the detail.

2. Some Lemmas

We introduce some lemmas which are the main tools in our proof.

Lemma 2.1 (see [10, p. 519]). If 𝑓∈𝐶^𝑟_[−1,1], then there exists an algebraic polynomial 𝑝_𝑛(𝑥) of degree at most 𝑛 such that 𝑓^(𝑗)(𝑥)−𝑝_𝑛^(𝑗)(𝑥√)≤𝐶1−𝑥²𝑛^𝑟−𝑗𝐸_𝑛−𝑟𝑓^(𝑗),𝑗=0,1,…,𝑟. (2.1)

In the past, the error estimate depended on the Markov-Bernstein type inequalities in 𝐿_𝑝 metric. In this paper, we will use the inequality in 𝐿_∞ metric.

Lemma 2.2 (see [7, p. 50]). Let 𝜑_𝑘(𝑥) be as defined by (1.10), 𝛼>−1. Then, for any fixed 𝑝>0, ¹₋₁^𝑛_𝑘=1𝐴_𝑘𝜑_𝑘(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶max_{1≤𝑘≤𝑛}𝐴_𝑘. (2.2)

To prove our results, we need to build another polynomial integral inequality in 𝐿_∞ metric. For its proof, we introduce two lemmas.

Lemma 2.3 (see [8, p. 914]). Let 𝑣₁,𝑣₂,…,𝑣_2𝑁 be distinct integers between 1 and 𝑛. Then, we have ¹₋₁ℓ_𝑣1(𝑥)ℓ_𝑣2(𝑥)⋯ℓ_𝑣2𝑁(𝑥)𝑑𝑥√1−𝑥²=0, (2.3) and it is well known that ^𝑛_𝑘=1ℓ²_𝑘(𝑥)≤2. (2.4)

Let 𝑥₁,…,𝑥_𝑛 be independent variables, 𝑠 are positive integers, and 𝑉_𝑠=^𝑛_𝑘=1𝑥^𝑠_𝑘^1/𝑠. (2.5)

By the mathematical induction we can obtain the following.

Lemma 2.4. If 𝑁 is a positive integer, 𝑛>2𝑁, then, the homogeneous symmetrical polynomial of degree 2𝑁: 𝐵_2𝑁=^𝑛_𝑖=1𝑥_𝑖^2𝑁−(2𝑁)!_{𝑘1<𝑘2<⋯<𝑘2𝑁}𝑥_𝑘1⋯𝑥_𝑘2𝑁, (2.6) can be represented as a homogeneous polynomial of degree 2𝑁 about 𝑉₁,…,𝑉_2𝑁: 𝐵_2𝑁=_{𝑡1≤2𝑁−2,𝑡𝑖≥0}𝐵_{𝑡1⋯𝑡2𝑁}𝑉^𝑡1₁⋯𝑉^𝑡2𝑁_2𝑁. (2.7)

Now we give the inequality in 𝐿_∞ metric which plays a key role in our paper.

Lemma 2.5. Let ℓ_𝑘(𝑥) be as defined by (2.1), 𝛼>−1. Then, for any fixed 𝑝>0, ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶max_{1≤𝑘≤𝑛+1}𝐴_𝑘. (2.8)

Proof. Firstly, we will consider the special case 𝑝=2𝑁,𝛼=−1/2 by induction on 𝑁. For 𝑁=1, by (2.3) and (2.4), we obtain ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)²𝑑𝑥√1−𝑥²=^𝑛_𝑘=1𝐴²_𝑘¹₋₁ℓ²_𝑘(𝑥)𝑑𝑥√1−𝑥²+2^𝑛_𝑘=1^𝑘−1_𝑗=1𝐴_𝑘𝐴_𝑗¹₋₁ℓ_𝑘(𝑥)ℓ_𝑗(𝑥)𝑑𝑥√1−𝑥²≤max_{1≤𝑘≤𝑛}𝐴_𝑘^21𝑛_−1𝑘=1ℓ²_𝑘(𝑥)𝑑𝑥√1−𝑥²≤2𝜋max_{1≤𝑘≤𝑛}𝐴_𝑘². (2.9)
Suppose that for 0<𝑝≤2(𝑁−1), we have ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^𝑝𝑑𝑥√1−𝑥²≤𝐶_𝑝max_{1≤𝑘≤𝑛}𝐴_𝑘^𝑝. (2.10)
For 𝑝=2𝑁, if 𝑛≤2𝑁, then, (2.4) gives ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^2𝑁𝑑𝑥√1−𝑥²≤𝜋(4𝑁)^2𝑁max_{1≤𝑘≤𝑛}𝐴_𝑘^2𝑁. (2.11) If 𝑛>2𝑁, then by Lemma 2.4, we know ^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^2𝑁=(2𝑁)!_{𝑘1<𝑘2<⋯<𝑘2𝑁}𝐴_𝑘1⋯𝐴_𝑘2𝑁ℓ_𝑘1(𝑥)⋯ℓ_𝑘2𝑁(𝑥)+_{𝑡1≤2𝑁−2,𝑡𝑖≥0}𝐵_{𝑡1⋯𝑡2𝑁}𝑉^𝑡1₁(𝑥)⋯𝑉^𝑡2𝑁_2𝑁(𝑥)=𝐼₁(𝑥)+𝐼₂(𝑥), (2.12) where 𝑉_𝑠(𝑥)=^𝑛_𝑘=1𝐴^𝑠_𝑘ℓ^𝑠_𝑘(𝑥)^1/𝑠. (2.13) From (2.3), it follows that ¹₋₁𝐼₁(𝑥)𝑑𝑥√1−𝑥²=0. (2.14) From (2.4), we know that, for 𝑠≥2, 𝑉_𝑠(𝑥)≤max_{1≤𝑘≤𝑛}𝐴_𝑘^𝑛_𝑘=1ℓ_𝑘(𝑥)^𝑠1/𝑠≤√2max_{1≤𝑘≤𝑛}𝐴_𝑘. (2.15) By virtue of (2.12) and (2.15), we have ¹₋₁𝐼₂(𝑥)𝑑𝑥√1−𝑥²≤_{𝑡1≤2𝑁−2,𝑡𝑖≥0}𝐵_{𝑡1⋯𝑡2𝑁}¹₋₁𝑉^𝑡1₁(𝑥)⋯𝑉^𝑡2𝑁_2𝑁(𝑥)𝑑𝑥√1−𝑥²≤_{𝑡1≤2𝑁−2,𝑡𝑖≥0}2^𝑁𝐵_{𝑡1⋯𝑡2𝑁}max_{1≤𝑘≤𝑛}𝐴_𝑘^{2𝑁−𝑡11}₋₁𝑉^𝑡1₁(𝑥)𝑑𝑥√1−𝑥²≤_{𝑡1≤2𝑁−2,𝑡𝑖≥0}2^𝑁𝐵_{𝑡1⋯𝑡2𝑁}𝜋+^2𝑁−2_𝑖=1𝐶_𝑖max_{1≤𝑘≤𝑛}𝐴_𝑘^2𝑁. (2.16) From (2.11), (2.12), (2.14), and (2.16), it follows that ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^2𝑁𝑑𝑥√1−𝑥²≤𝐶_2𝑁max_{1≤𝑘≤𝑛}𝐴_𝑘^2𝑁. (2.17)
Now we consider the general case. For arbitrary 𝑝>0 and 𝛼>−1, it is easy to see that we can choose a positive integer 𝑁 satisfying 𝑝/4𝑁<1 and (𝛼+(𝑝/4𝑁))/(1−(𝑝/2𝑁))>−1. By Hölder inequality and (2.17), we can obtain ¹₋₁^𝑛_𝑘=1𝐴_𝑘ℓ_𝑘(𝑥)^𝑝1−𝑥^2𝛼≤𝑑𝑥_{1−1𝑛𝑘=1}𝐴_𝑘ℓ_𝑘(𝑥)^2𝑁𝑑𝑥√1−𝑥^{2𝑝/2𝑁1−1}1−𝑥^{2(𝛼+(𝑝/4𝑁))/(1−(𝑝/2𝑁))}𝑑𝑥^{1−(𝑝/2𝑁)}≤𝐶_𝑝max_{1≤𝑘≤𝑛}𝐴_𝑘^𝑝. (2.18)

Remark 2.6. P. Erdös and E. Feldheim [8] give a proof for 𝑝=2,4 and 𝛼=−1/2. We give a mathematical induction proof for completion.

3. Proof of Theorem 1.1

We will consider 𝑄_𝑛+2(𝑓,𝑥) instead of 𝑄_𝑛(𝑓,𝑥) for simplicity. For 𝑓∈𝐶¹_[−1,1], let 𝑝_𝑛+1(𝑥) be the polynomial of degree at most 𝑛+1 satisfying (2.1). It is easily checked that for −1≤𝑥≤1,𝑓(𝑥)−𝑄_𝑛+2𝑓,𝑥=𝑓(𝑥)−𝑝_𝑛+1(𝑥)+𝑄_𝑛+2𝑝_𝑛+1−𝑓,𝑥. (3.1) From (3.1), we can conclude that 𝑓^′(𝑥)−𝑄^′_𝑛+2𝑓,𝑥=𝑓^′(𝑥)−𝑝^′_𝑛+1(𝑥)+𝑄^′_𝑛+2𝑝_𝑛+1−𝑓,𝑥=𝐼₁(𝑥)+𝐼₂(𝑥). (3.2) From (2.1), we can derive ¹₋₁𝐼₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛𝑓^′1₋₁1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛𝑓^′. (3.3) It is easy to see that 𝐼₂(𝑥) is a polynomial of degree at most 𝑛. Hence, 𝐼₂(𝑥)=^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝜑^′_𝑘(𝑥)=𝐿_𝑛+1𝐼₂=,𝑥_{𝑛+1𝑠=1𝑛𝑘=1}𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝜑^′_𝑘𝑡_𝑠ℓ_𝑠(𝑥). (3.4)

By a direct computation, we know 𝜑^′_𝑘𝑡_𝑠=(−1)^𝑘+𝑠+11−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘²+(−1)^𝑘+𝑠+1𝑡_𝑠(𝑛+1)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘. (3.5) Combining (3.4) and (3.5), we derive 𝐼₂(𝑥)=^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(−1)^𝑘+𝑠+11−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘²ℓ_𝑠(𝑥)+^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(−1)^𝑘+𝑠+1𝑡_𝑠(𝑛+1)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘ℓ_𝑠(𝑥)=𝐽₁(𝑥)+𝐽₂(𝑥). (3.6) We consider 𝐽₁(𝑥) first. For an arbitrary 1≤𝑠≤𝑛+1, ^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(−1)^𝑘+𝑠+11−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛𝑓^′𝑛^2𝑛_𝑘=11−𝑥²_𝑘1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛𝑓^′𝑛^2𝑛_𝑘=11−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²+^𝑛_𝑘=11−𝑥²_𝑘𝑡_𝑠−𝑥_𝑘². (3.7) Similar to [9, p. 71], we have ^𝑛_𝑘=11−𝑥²𝑈²_𝑛(𝑥)𝑥−𝑥_𝑘²=1−𝑥²𝑈^′_𝑛(𝑥)²−𝑈_𝑛(𝑥)𝑈_𝑛^′′(𝑥)_𝑛𝑘=1,1−𝑥²_𝑘𝑈²_𝑛(𝑥)𝑥−𝑥_𝑘²=^𝑛_𝑘=11−𝑥²𝑈²_𝑛(𝑥)𝑥−𝑥_𝑘²+2𝑥𝑈_𝑛(𝑥)𝑈^′_𝑛(𝑥)−𝑛𝑈²_𝑛(𝑥). (3.8) By [9, p. 71], we know 1−𝑥²𝑈^′_𝑛(𝑥)=𝑥𝑈_𝑛(𝑥)−(𝑛+1)𝑇_𝑛+1(𝑥),1−𝑥²𝑈_𝑛^′′(𝑥)=3𝑥𝑈^′_𝑛(𝑥)−𝑛(𝑛+2)𝑈_𝑛(𝑥). (3.9)(3.10) Let 𝑥=𝑡_𝑠, then by (3.8), (3.9), and (3.10), we obtain ^𝑛_𝑘=11−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²=𝑛(𝑛+2)−2𝑡²_𝑠1−𝑡²_𝑠≤𝑛(𝑛+2_𝑛𝑘=1),1−𝑥²_𝑘𝑡_𝑠−𝑥_𝑘²=𝑛²+𝑛. (3.11)(3.12) From (3.7), (3.11), and (3.12), we obtain that for an arbitrary 1≤𝑠≤𝑛+1, ^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(−1)^𝑘+𝑠+11−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛𝑓^′. (3.13) From (2.8) and (3.13), we can obtain ¹₋₁𝐽₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛𝑓^′. (3.14) Now we consider 𝐽₂(𝑥). Exchanging the summation order, we have 𝐽₂(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝑛+1^𝑛+1_𝑠=1(−1)^𝑠+1𝑡_𝑠1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘ℓ_𝑠(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(𝑛+1)^2𝑛+1_𝑠=1𝑡_𝑠𝑇_𝑛+1(𝑥)𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠. (3.15) It is easy to know ^𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)𝑥−𝑡_𝑠=𝑇^′_𝑛+1(𝑥)=(𝑛+1)𝑈_𝑛(𝑥). (3.16) Let 𝑥=𝑥_𝑘, then, we have ^𝑛+1_𝑠=11𝑥_𝑘−𝑡_𝑠=0. (3.17) By (3.16), (3.17), and the identity 𝑡_𝑠𝑇_𝑛+1(𝑥)𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠=𝑥𝑇_𝑛+1(𝑥)𝑥−𝑥_𝑘𝑥−𝑡_𝑠−𝑥_𝑘𝑇_𝑛+1(𝑥)𝑥−𝑥_𝑘𝑥_𝑘−𝑡_𝑠, (3.18) we conclude that ^𝑛+1_𝑠=1𝑡_𝑠𝑇_𝑛+1(𝑥)𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠=(𝑛+1)𝑥𝑈_𝑛(𝑥)𝑥−𝑥_𝑘. (3.19) From (3.15) and (3.19), it follows that 𝐽₂(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘⋅𝑛+1𝑥𝑈_𝑛(𝑥)𝑥−𝑥_𝑘. (3.20) For an arbitrary 1≤𝑘≤𝑛, by (3.20), (2.1), |𝑈^′_𝑛(𝑥_𝑘)|=(𝑛+1)/(1−𝑥²_𝑘),𝑘=1,2,…,𝑛, and a simple computation, we can obtain 𝐽₂𝑥_𝑘=(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝑛+1⋅𝑥_𝑘𝑈^′_𝑛𝑥_𝑘≤𝐶𝐸_𝑛𝑓^′. (3.21) For 𝑘=0, by (2.1), 𝑈_𝑛(1)=𝑛+1 and a simple computation we obtain 𝐽₂(1)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘1−𝑥_𝑘≤𝐶𝐸_𝑛𝑓^′𝑛^𝑛_𝑘=111−𝑥_𝑘. (3.22) From 2𝑥/𝜋≤sin𝑥≤𝑥,forall𝑥∈[0,𝜋/2], we derive ^𝑛_𝑘=111−𝑥_𝑘=^𝑛_𝑘=11√()≤2sin𝑘𝜋/2𝑛+1^𝑛_𝑘=1𝑛+1𝑘≤𝐶𝑛ln𝑛. (3.23) Hence, 𝐽₂(1)≤𝐶ln𝑛𝐸_𝑛𝑓^′. (3.24) Similarly, 𝐽₂(−1)≤𝐶ln𝑛𝐸_𝑛𝑓^′. (3.25) The fact that 𝐽₂(𝑥) is an algebraic polynomial of degree at most 𝑛 implies 𝐽₂(𝑥)=𝑄_𝑛+2𝐽₂,𝑥=𝐽₂(1)𝜑₀(𝑥)+𝐽₂(−1)𝜑_𝑛+1(𝑥)+^𝑛_𝑘=1𝐽₂𝑥_𝑘𝜑_𝑘(𝑥). (3.26) Let 𝑥=cos𝜃. By (3.24) and a simple computation similar to [11, p. 204], we obtain that, for 𝑝>0 and 𝛼>−1, ¹₋₁𝐽₂(1)𝜑₀(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶ln^𝑝𝑛𝐸^𝑝_𝑛𝑓^′(𝑛+1)^𝑝𝜋₀|sin𝑛𝜃|^𝑝sin^{𝑝−2𝛼−1}𝜃𝑑𝜃≤𝐶𝐸^𝑝_𝑛𝑓^′. (3.27) Similarly, ¹₋₁𝐽₂(−1)𝜑_𝑛+1(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛𝑓^′. (3.28) By virtue of (2.2) and (3.21), we have ¹₋₁^𝑛_𝑘=1𝐽₂𝑥_𝑘𝜑_𝑘(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛𝑓^′. (3.29) From (3.26), (3.27), (3.28), and (3.29), it follows that ¹₋₁𝐽₂(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛𝑓^′. (3.30) By (3.2), (3.3), (3.6), (3.14), and (3.30), we obtain the upper estimate.

4. Proof of Theorem 1.2

We consider 𝑄_𝑛 first. We will consider 𝑄_𝑛+2(𝑓,𝑥) instead of 𝑄_𝑛(𝑓,𝑥) for simplicity. For 𝑓∈𝐶²_[−1,1], let 𝑝_𝑛+1(𝑥) be the polynomial of degree at most 𝑛+1 satisfying (2.1). From (3.1), it follows that 𝑓^′′(𝑥)−𝑄^′′_𝑛+2𝑓,𝑥=𝑓^′′(𝑥)−𝑝^′′_𝑛+1(𝑥)+𝑄^′′_𝑛+2𝑝_𝑛+1−𝑓,𝑥=𝑀₁(𝑥)+𝑀₂(𝑥). (4.1)

From (2.1), we can derive ¹₋₁𝑀₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛−1𝑓^′′. (4.2)

Similar to (3.4), 𝑀₂(𝑥)=^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝜑_𝑘^′′𝑡_𝑠ℓ_𝑠(𝑥). (4.3)

By a direct computation, we get 𝜑_𝑘^′′𝑡_𝑠=2(−1)^𝑘+𝑠1−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘³+2(−1)^𝑘+𝑠𝑡_𝑠(𝑛+1)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²+(𝑛+1)(−1)^𝑘+𝑠+11−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘+(−1)^𝑘+𝑠+1(𝑛+1)1−𝑡²_𝑠^3/2𝑡_𝑠−𝑥_𝑘. (4.4)

Equations (4.3) and (4.4) yield 𝑀₂(𝑥)=^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘2(−1)^𝑘+𝑠1−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘³ℓ_𝑠(𝑥)+^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘2(−1)^𝑘+𝑠𝑡_𝑠(𝑛+1)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²ℓ_𝑠(𝑥)+^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(𝑛+1)(−1)^𝑘+𝑠+11−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘ℓ_𝑠(𝑥)+^𝑛+1_𝑠=1^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(−1)^𝑘+𝑠+1(𝑛+1)1−𝑡²_𝑠^3/2𝑡_𝑠−𝑥_𝑘ℓ_𝑠(𝑥)=𝑁₁(𝑥)+𝑁₂(𝑥)+𝑁₃(𝑥)+𝑁₄(𝑥). (4.5) We consider 𝑁₁(𝑥) now. For an arbitrary 1≤𝑠≤𝑛+1, from (2.1), (3.12), and ∑^𝑛_𝑘=1|𝜑_𝑘(𝑥)|²≤2 (see [9]), it follows that ^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘2(−1)^𝑘+𝑠1−𝑡²_𝑠(𝑛+1)𝑡_𝑠−𝑥_𝑘³≤𝐶𝐸_𝑛−1𝑓^′′(𝑛+1)^3𝑛_𝑘=11−𝑥²_𝑘1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘³=𝐶𝐸_𝑛−1𝑓^′′(𝑛+1)^2𝑛_𝑘=11−𝑥²_𝑘𝜑_𝑘𝑡_𝑠𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛−1𝑓^′′(𝑛+1)^2𝑛_𝑘=11−𝑥²_𝑘𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛−1𝑓^′′. (4.6) From (2.8) and (4.6), we can obtain ¹₋₁𝑁₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′. (4.7) Now we consider 𝑁₂(𝑥). From 2𝑥/𝜋≤sin𝑥≤𝑥,forall𝑥∈[0,𝜋/2], it follows that 1−𝑡²_𝑠≥sin(𝜋/2(𝑛+1))≥1/(𝑛+1). By (2.1) and (3.12), we have that, for an arbitrary 1≤𝑠≤𝑛+1, ^𝑛_𝑘=1𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘2(−1)^𝑘+𝑠𝑡_𝑠(𝑛+1)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛−1𝑓^′′(𝑛+1)³1−𝑡²_𝑠^𝑛𝑘=11−𝑥²_𝑘𝑡_𝑠−𝑥_𝑘²≤𝐶𝐸_𝑛−1𝑓^′′. (4.8) From (2.8) and (4.8), we can obtain ¹₋₁𝑁₂(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′. (4.9) For the 𝑁₃(𝑥), similar to (3.15), we have 𝑁₃(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘^𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠=(𝑛+1)^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘𝑈_𝑛(𝑥)𝑥−𝑥_𝑘. (4.10) For an arbitrary 1≤𝑘≤𝑛, by (2.1) and a simple computation, we can obtain 𝑁₃𝑥_𝑘=(𝑛+1)𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘⋅𝑈^′_𝑛𝑥_𝑘≤𝐶𝐸_𝑛−1𝑓^′′. (4.11) For 𝑘=0, (2.1) leads to 𝑁₃(1)=(𝑛+1)^2𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘1−𝑥_𝑘≤𝐶𝑛𝐸_𝑛−1𝑓^′′. (4.12) Similarly, 𝑁₃(−1)≤𝐶𝑛𝐸_𝑛−1𝑓^′′. (4.13) Similar to (3.30), from (4.10), (4.11), (4.12), and (4.13), it follows that ¹₋₁𝑁₃(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′𝑝,𝛼>2𝐶()−1,ln𝑛^1/𝑝𝐸_𝑛−1𝑓^′′𝑝,𝛼=2−1,𝐶𝑛^{1−(2𝛼+2)/𝑝}𝐸_𝑛−1𝑓^′′𝑝,−1<𝛼<2−1. (4.14) For the 𝑁₄(𝑥), similar to (3.15), we have 𝑁₄(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(𝑛+1)^2𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠. (4.15) It is easy to verify 𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠=1𝑥−𝑥_𝑘𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑥−𝑡_𝑠+𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘. (4.16) For 𝑎≠±1, it is easy to verify that 11−𝑥²()1𝑥−𝑎=−2()+11+𝑎)(1+𝑥2()+11−𝑎)(1−𝑥1−𝑎²(𝑥−𝑎). (4.17) From (4.17), (3.16) and ^𝑛+1_𝑠=111+𝑡_𝑠=^𝑛+1_𝑠=111−𝑡_𝑠=(𝑛+1)², (4.18)we obtain ^𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑥−𝑡_𝑠=𝑇_𝑛+1(𝑥)2(1+𝑥)^𝑛+1_𝑠=111+𝑡_𝑠−𝑇_𝑛+1(𝑥)2(1−𝑥)^𝑛+1_𝑠=111−𝑡_𝑠+11−𝑥^2𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)𝑥−𝑡_𝑠()=−𝑛+1²𝑥𝑇_𝑛+1(𝑥)1−𝑥²+(𝑛+1)𝑈_𝑛(𝑥)1−𝑥²_𝑛+1𝑠=1,𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘=(𝑛+1)²𝑥_𝑘𝑇_𝑛+1(𝑥)1−𝑥²_𝑘. (4.19) From (4.16), (4.19), (61), (3.9), and a direct computation, we get ^𝑛+1_𝑠=1𝑇_𝑛+1(𝑥)1−𝑡²_𝑠𝑡_𝑠−𝑥_𝑘𝑥−𝑡_𝑠()=−𝑛+1²1+𝑥𝑥_𝑘𝑇_𝑛+1(𝑥)1−𝑥²1−𝑥²_𝑘+(𝑛+1)𝑈_𝑛(𝑥)1−𝑥²𝑥−𝑥_𝑘=(𝑛+1)1+𝑥𝑥_𝑘𝑈^′_𝑛(𝑥)1−𝑥²_𝑘+(𝑛+1)𝑥_𝑘𝑈_𝑛(𝑥)1−𝑥²_𝑘+(𝑛+1)𝑈_𝑛(𝑥)1−𝑥²_𝑘𝑥−𝑥_𝑘. (4.20) From (4.15) and (4.20), we obtain 𝑁₄(𝑥)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘(𝑛+1)1+𝑥𝑥_𝑘𝑈^′_𝑛(𝑥)1−𝑥²_𝑘+𝑥_𝑘𝑈_𝑛(𝑥)1−𝑥²_𝑘+^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘()𝑈𝑛+1^𝑛(𝑥)1−𝑥²_𝑘𝑥−𝑥_𝑘=𝑁₄₁(𝑥)+𝑁₄₂(𝑥). (4.21) For 𝑁₄₁(𝑥), from (2.1), we can obtain 𝑁₄₁(𝑥≤𝑈)^′𝑛(𝑥)+𝑈_𝑛(𝑥)(𝑛+1)²𝐸_𝑛−1𝑓^′′. (4.22) By (3.9), Markov inequality, and ‖𝑈_𝑛(𝑥)‖_∞=𝑛+1, we obtain 𝑈^′_𝑛(𝑥≤2())𝑛+11−𝑥²,𝑈^′_𝑛(𝑥)≤𝑛²(𝑛+1). (4.23) So for an arbitrary 0≤𝐴≤1, 𝑈^′_𝑛(𝑥≤2)^1−𝐴(𝑛+1)^1+2𝐴1−𝑥^21−𝐴≤8𝑛^1+2𝐴1−𝑥^21−𝐴. (4.24) Let 𝑥=cos𝜃. From 2𝛼+1>−1, we can choose 𝐴 such that 0<𝐴<1 and 2𝛼+1−2𝑝+2𝑝𝐴>−1. Then by (4.24) and 2𝑥/𝜋≤sin𝑥≤𝑥,forall𝑥∈[0,𝜋/2], we can obtain ¹₋₁𝑈^′_𝑛(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥=2⁰₋₁𝑈^′_𝑛(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤2^3𝑝+1₀^{𝜋/2(𝑛+1)}𝑛^{𝑝(1+2𝐴)}sin^{2𝛼+1−2𝑝+2𝑝𝐴}𝜃𝑑𝜃+^𝜋/2_{𝜋/2(𝑛+1)}𝑛^𝑝sin^{1+2𝛼−2𝑝}≤𝜃𝑑𝜃𝐶𝑛^𝑝,𝛼>𝑝−1,𝐶𝑛^𝑝ln𝑛,𝛼=𝑝−1,𝐶𝑛^{3𝑝−2𝛼−2},−1<𝛼<𝑝−1. (4.25) From ‖𝑈_𝑛(𝑥)‖_∞=𝑛+1, it follows that ¹₋₁𝑈_𝑛(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝑛^𝑝. (4.26) From (4.22), (4.25), and (4.26), it follows that ¹₋₁𝑁₄₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′𝑝,𝛼≥2−1,𝐶𝑛^{1−(2𝛼+2)/𝑝}𝐸_𝑛−1𝑓^′′𝑝,−1<𝛼<2−1. (4.27) For 𝑁₄₂(𝑥), from (2.1) and a simple computation, we can obtain that, for 1≤𝑘≤𝑛, 𝑁₄₂𝑥_𝑘=1(𝑛+1)1−𝑥²_𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘⋅𝑈^′_𝑛𝑥_𝑘≤𝐶𝐸_𝑛−1𝑓^′′. (4.28) Let 𝑥=1. Then, from (3.10) and ^𝑛−1_𝑠=1𝑈_𝑛(𝑥)𝑥−𝑥_𝑠=𝑈^′_𝑛(𝑥), (4.29) we obtain ^𝑛_𝑘=111−𝑥_𝑘=𝑛(𝑛+2)3. (4.30) From (2.1), it follows that 𝑁₄₂(1)=^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛+1𝑥_𝑘−𝑓𝑥_𝑘1−𝑥_𝑘1−𝑥²_𝑘≤𝐶𝐸_𝑛−1𝑓^′′(𝑛+1)^2𝑛_𝑘=111−𝑥_𝑘≤𝐶𝐸_𝑛−1𝑓^′′. (4.31) Similarly, 𝑁₄₂(−1)≤𝐶𝐸_𝑛−1𝑓^′′. (4.32) Similar to (3.30), from (4.28), (4.31), and (4.32), we can obtain ¹₋₁𝑁₄₂(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′. (4.33) From (4.1), (4.2), (4.5), (4.7), (4.9), (4.14), (4.21), (4.27), and (4.33), we obtain the upper estimate.

On the other hand, for 𝑝≥2𝛼+2, let 𝑓(𝑥)=(1−𝑥²)𝑈_𝑛(𝑥). Then,𝑓^′′(𝑥)=−2(𝑛+2)(𝑛+1)𝑇_𝑛(𝑥)+𝑞_𝑛−1(𝑥), (4.34) here, 𝑞_𝑛−1(𝑥) is a polynomial of degree at most 𝑛−1. Hence, 𝐸_𝑛−1𝑓^′′=2(𝑛+2)(𝑛+1). (4.35) It is easy to verify that 𝑄^′′_𝑛+2𝑓𝑓,𝑥=0,_′′(𝑥)=−(𝑛+1)²𝑈_𝑛(𝑥)+−𝑈_𝑛(𝑥)+(𝑛+1)𝑥𝑇_𝑛+1(𝑥)1−𝑥². (4.36)

Let 𝑥=cos𝜃, then, (2𝑘𝜋+𝜋)/2(𝑛+1)≤𝜃≤(2𝑘𝜋+2𝜋)/2(𝑛+1) implies that 𝑇_𝑛+1(𝑥)𝑈_𝑛(𝑥)≤0. Therefore, ¹₋₁𝑓^′′(𝑥)−𝑄_𝑛^′′𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥≥¹₀𝑓^′′(𝑥)−𝑄_𝑛^′′𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥≥𝑛^{2𝑝[(𝑛+1)/2]}_𝑘=0^{(2𝑘𝜋+𝜋)/2(𝑛+1)}_{2𝑘𝜋/2(𝑛+1)}|sin(𝑛+1)𝜃|^𝑝sin^{𝑝−2𝛼−1}𝜃≥𝑑𝜃𝐶𝑛^2𝑝𝑝ln𝑛,𝛼=2−1,𝐶𝑛^{3𝑝−2𝛼−2}𝑝,−1<𝛼<2≥−1,𝐶ln𝑛𝐸^𝑝_𝑛−1𝑓^′′𝑝,𝛼=2−1,𝐶𝑛^{𝑝−2𝛼−2}𝐸^𝑝_𝑛−1𝑓^′′𝑝,−1<𝛼<2−1, (4.37)

We consider 𝐿_𝑛 in the following. For 𝑓∈𝐶²_[−1,1], let 𝑝_𝑛−1(𝑥) be the polynomial of degree at most 𝑛−1 satisfying (2.1). Then, 𝑓^′′(𝑥)−𝐿_𝑛^′′𝑓,𝑥=𝑓^′′(𝑥)−𝑝^′′_𝑛−1(𝑥)+𝐿_𝑛^′′𝑝_𝑛−1−𝑓,𝑥=𝐾₁(𝑥)+𝐾₂(𝑥). (4.38) From (2.1), we can derive ¹₋₁𝐾₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥≤𝐶𝐸^𝑝_𝑛−3𝑓^′′. (4.39) If 𝑓∈𝐶[−1,1], then the well-known Lagrange interpolation polynomial of 𝑓 based on {𝑥_𝑘}^𝑛_𝑘=1 is given by 𝑅_𝑛𝑓,𝑥=^𝑛_𝑘=1𝑓𝑥_𝑘𝜙_𝑘(𝑥), (4.40) where 𝜙_𝑘(𝑥)=(−1)^𝑘+11−𝑥²_𝑘𝑈_𝑛(𝑥)(𝑛+1)𝑥−𝑥_𝑘,𝑘=1,…,𝑛. (4.41) Similar to (3.4), we have 𝐾₂(𝑥)=𝑅_𝑛−1𝐾₂,𝑥=^𝑛−1_𝑠=1^𝑛_𝑘=1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘ℓ_𝑘^′′𝑥_𝑠𝜙_𝑠(𝑥). (4.42) By a direct computation, we obtain ℓ_𝑘^′′𝑥_𝑠=𝑛(−1)^𝑘+𝑠1−𝑡²_𝑘1−𝑥²_𝑠𝑥_𝑠−𝑡_𝑘+2(−1)^𝑘+𝑠+11−𝑡²_𝑘𝑛𝑥_𝑠−𝑡_𝑘³,𝑠=1,…,𝑛−1. (4.43) From (4.42) and (4.43), it follows that 𝐾₂(𝑥)=^𝑛−1_𝑠=1^𝑛_𝑘=1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘𝑛(−1)^𝑘+𝑠1−𝑡²_𝑘1−𝑥²_𝑠𝑥_𝑠−𝑡_𝑘𝜙_𝑠(𝑥)+^𝑛−1_𝑠=1^𝑛_𝑘=1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘2(−1)^𝑘+𝑠+11−𝑡²_𝑘𝑛𝑥_𝑠−𝑡_𝑘³𝜙_𝑠(𝑥)=𝐴₁(𝑥)+𝐴₂(𝑥). (4.44) Exchanging the summation order, we have 𝐴₁(𝑥)=^𝑛_𝑘=1(−1)^𝑘+1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘1−𝑡²_𝑘^𝑛−1_𝑠=1𝑈_𝑛−1(𝑥)𝑥_𝑠−𝑡_𝑘𝑥−𝑥_𝑠. (4.45) For an arbitrary 1≤𝑠≤𝑛−1, 𝑈_𝑛−1(𝑥)𝑥_𝑠−𝑡_𝑘𝑥−𝑥_𝑠=1𝑥−𝑡_𝑘𝑈_𝑛−1(𝑥)𝑥_𝑠−𝑡_𝑘+𝑈_𝑛−1(𝑥)𝑥−𝑥_𝑠. (4.46) It is well known that ^𝑛−1_𝑠=1𝑈_𝑛−1(𝑥)𝑥−𝑥_𝑠=𝑈^′_𝑛−1(𝑥). (4.47) Let 𝑥=𝑡_𝑘. Then, from (4.47) and (3.9), it follows that ^𝑛−1_𝑠=11𝑡_𝑘−𝑥_𝑠=𝑈^′_𝑛−1𝑡_𝑘𝑈_𝑛−1𝑡_𝑘=𝑡_𝑘1−𝑡²_𝑘. (4.48) Combining (4.47), (4.48), and (3.9), we obtain ^𝑛−1_𝑠=11𝑥−𝑡_𝑘𝑈_𝑛−1(𝑥)𝑥_𝑠−𝑡_𝑘+𝑈_𝑛−1(𝑥)𝑥−𝑥_𝑠=1𝑥−𝑡_𝑘−𝑡_𝑘𝑈_𝑛−1(𝑥)+1−𝑡²_𝑘𝑈^′_𝑛−1(𝑥)1−𝑡²_𝑘=𝑈_𝑛−1(𝑥)+𝑥+𝑡_𝑘𝑈^′_𝑛−1(𝑥)1−𝑡²_𝑘+1𝑥−𝑡_𝑘−𝑥𝑈_𝑛−1(𝑥)+1−𝑥²𝑈^′_𝑛−1(𝑥)1−𝑡²_𝑘=𝑈_𝑛−1(𝑥)+𝑥+𝑡_𝑘𝑈^′_𝑛−1(𝑥)1−𝑡²_𝑘−𝑛𝑇_𝑛(𝑥)𝑥−𝑡_𝑘1−𝑡²_𝑘. (4.49) From (4.45) and (4.49), it follows that 𝐴₁(𝑥)=^𝑛_𝑘=1(−1)^𝑘+1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘𝑈_𝑛−1(𝑥)+𝑥+𝑡_𝑘𝑈^′_𝑛−1(𝑥)1−𝑡²_𝑘+^𝑛_𝑘=1(−1)^𝑘𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘𝑛𝑇_𝑛(𝑥)𝑥−𝑡_𝑘1−𝑡²_𝑘=𝐴₁₁(𝑥)+𝐴₁₂(𝑥). (4.50) By (2.1), we have 𝐴₁₁(𝑥)≤𝐶𝐸_𝑛−3𝑓^′′𝑈^′_𝑛−1(𝑥)+𝑈_𝑛(𝑥)𝑛. (4.51) From (4.51), (4.25), and (4.26), it follows that ¹₋₁𝐴₁₁(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−3𝑓^′′𝐶(),𝛼>𝑝−1,ln𝑛^1/𝑝𝐸_𝑛−3𝑓^′′,𝛼=𝑝−1,𝐶𝑛^{2−(2𝛼+2)/𝑝}𝐸_𝑛−3𝑓^′′,−1<𝛼<𝑝−1. (4.52) From (2.1) and |𝑇^′_𝑛(𝑡_𝑘)|=𝑛/1−𝑡²_𝑘, it follows that, for 1≤𝑘≤𝑛, 𝐴₁₂𝑡_𝑘=(−1)^𝑘+1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘𝑛𝑇^′_𝑛𝑡_𝑘1−𝑡²_𝑘≤𝐶𝐸_𝑛−3𝑓^′′. (4.53) From (4.53), (2.8), and 𝐴₁₂(𝑥)=^𝑛_𝑘=1𝐴₁₂𝑡_𝑘ℓ_𝑘(𝑥), (4.54) we know ¹₋₁𝐴₁₂(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−3𝑓^′′. (4.55)

We consider 𝐴₂(𝑥) now. For an arbitrary 1≤𝑠≤𝑛−1, by (2.1) and (2.4), we obtain ^𝑛_𝑘=1𝑝_𝑛−1𝑡_𝑘−𝑓𝑡_𝑘2(−1)^𝑘+𝑠+11−𝑡²_𝑘𝑛𝑥_𝑠−𝑡_𝑘³≤𝐶𝐸_𝑛−3𝑓^′′𝑛_𝑘=11−𝑡²_𝑘^3/2𝑛³𝑥_𝑠−𝑡_𝑘³=𝐶𝐸_𝑛−3𝑓^′′𝑛_𝑘=1ℓ³_𝑘𝑥_𝑠≤𝐶𝐸_𝑛−3𝑓^′′. (4.56) From (4.56), (4.44), and (4.41), we can obtain 𝐴₂𝑥_𝑠≤𝐶𝐸_𝑛−3𝑓^′′𝐴,1≤𝑠≤𝑛−1.₂(1)≤𝐶𝑛𝐸_𝑛−3𝑓^′′,𝐴₂(−1)≤𝐶𝑛𝐸_𝑛−3𝑓^′′. (4.57) Similar to the proof of (3.30), by (4.57), (95), and 𝐴₂(𝑥)=^𝑛_𝑠=0𝐴₂𝑥_𝑠𝜑_𝑠(𝑥), (4.58) we can obtain ¹₋₁𝐴₂(𝑥)^𝑝1−𝑥^2𝛼𝑑𝑥^1/𝑝≤𝐶𝐸_𝑛−1𝑓^′′𝑝,𝛼>2𝐶()−1,ln𝑛^1/𝑝𝐸_𝑛−1𝑓^′′𝑝,𝛼=2−1,𝐶𝑛^{1−(2𝛼+2)/𝑝}𝐸_𝑛−1𝑓^′′𝑝,−1<𝛼<2−1. (4.59) From (4.38), (4.39), (4.44), (4.50), (4.52), (4.55), and (4.59), we can obtain the upper estimate.

On the other hand, let 𝑓(𝑥)=𝑇_𝑛(𝑥). Then, it is easy to see 𝐿_𝑛𝑓𝑓,𝑥=0,_′′(𝑥)=4𝑛(𝑛−1)𝑇_𝑛−2(𝑥)+𝑞_𝑛−3(𝑥), (4.60)(4.61) here, 𝑞_𝑛−3(𝑥) is a polynomial of degree at most 𝑛−3. Consequently, due to (4.61), we get 𝐸_𝑛−3𝑓^′′=4𝑛(𝑛−1). (4.62) Let 𝑥=cos𝜃. From (4.60), (4.62), (3.9), 𝑇_𝑛^′′(𝑥)=𝑛𝑈^′_𝑛−1(𝑥), and the odevity of 𝑈^′_𝑛−1(𝑥), it follows that ¹₋₁𝑓^′′(𝑥)−𝐿_𝑛^′′𝑓,𝑥^𝑝1−𝑥^2𝛼𝑑𝑥=2𝐸^𝑝_𝑛−3𝑓^′′4^𝑝(𝑛−1)^𝑝1₀𝑈^′_𝑛−1(𝑥)^𝑝1−𝑥^2𝛼≥𝐸𝑑𝑥_𝑝𝑛−3𝑓^′′8^𝑝𝑛/2_𝑘=1()^{𝑘𝜋/𝑛}_{𝑘𝜋−𝜋/4/𝑛}1|sin𝜃|^{2𝑝−2𝛼−1}≥𝑑𝜃𝐶𝐸^𝑝_𝑛−3𝑓^′′,𝛼>𝑝−1,𝐶ln𝑛𝐸^𝑝_𝑛−3𝑓^′′,𝛼=𝑝−1,𝐶𝑛^{2𝑝−2𝛼−2}𝐸^𝑝_𝑛−3𝑓^′′,−1<𝛼<𝑝−1. (4.63)