The Average Errors for the Grünwald Interpolation in the Wiener Space

Abstract

We determine the weakly asymptotically orders for the average errors of the Grünwald interpolation sequences based on the Tchebycheff nodes in the Wiener space. By these results we know that for the 𝐿_𝑝-norm (2≤𝑞≤4) approximation, the 𝑝-average (1≤𝑝≤4) error of some Grünwald interpolation sequences is weakly equivalent to the 𝑝-average errors of the best polynomial approximation sequence.

1. Introduction and Main Results

Let 𝐹 be a real separable Banach space equipped with a probability measure 𝜇 on the Borel sets of 𝐹. Let 𝐻 be another normed space such that 𝐹 is continuously embedded in 𝐻. By ‖⋅‖ we denote the norm in 𝐻. Any 𝐴∶𝐹→𝐻 such that 𝑓↦‖𝑓−𝐴(𝑓)‖ is a measurable mapping is called an approximation operator (or just approximation). The 𝑝-average error of 𝐴 is defined as 𝑒_𝑝(𝐴,𝐻)=(_𝐹‖𝑓−𝐴(𝑓)‖^𝑝𝜇(𝑑𝑓))^1/𝑝. (1.1)

Since in practice the underlying function is usually given via its (exact or noisy) values at finitely many points, the approximation operator 𝐴(𝑓) is often considered depending on some function values about 𝑓 only. Many papers such as [1–4] studied the complexity of computing an 𝜀-approximation in average case setting. Noticed that the polynomial interpolation operators are important approximation tool in the continuous functions space, and they are depending on some function values about 𝑓 only, we want to know the average error for some polynomial interpolation operators in the Wiener measure. Now we turn to describe the contents in detail.

Let 𝑋 be the space of continuous function 𝑓 defined on [0,1] such that 𝑓(0)=0. The space 𝑋 is equipped with the sup norm. The Wiener measure 𝜔 is uniquely defined by the following property: 𝜔(𝑓∈𝑋∶(𝑓(𝑡₁),…,𝑓(𝑡_𝑛))∈𝐵)=^𝑛_𝑗=112𝜋(𝑡_𝑗−𝑡_𝑗−1)^𝐵𝑒^{∑𝑛𝑗=1(−(𝑢𝑗−𝑢𝑗−1)2/2(𝑡𝑗−𝑡𝑗−1))}𝑑𝑢₁⋯𝑑𝑢_𝑛, (1.2) for every 𝑛≥1,𝐵∈ℬ(ℛ^𝑛), where ℬ(ℛ^𝑛) denote the set class of all Borel measurable subsets of ℛ^𝑛, and 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛≤1 with 𝑢₀=0. Its mean is zero, and its correlation operator is given by 𝐿_𝑥1(𝐶_𝜔𝐿_𝑥2)=min{𝑥₁,𝑥₂} for 𝐿_𝑥𝑖(𝑓)=𝑓(𝑥_𝑖), that is, _𝑋𝑓(𝑥₁)𝑓(𝑥₂)𝜔(𝑑𝑓)=min{𝑥₁,𝑥₂},∀𝑥₁,𝑥₂∈[0,1]. (1.3)

In this paper, we specify 𝐹={𝑓∈𝐶[−1,1]∶𝑔(𝑡)=𝑓(2𝑡−1)∈𝑋}, and for every measurable subset 𝐴⊂𝐹, we define 𝜇(𝐴)=𝜔({𝑔(𝑡)=𝑓(2𝑡−1),𝑓∈𝐴}). (1.4)

For 1≤𝑝<∞, denote by 𝐿_𝑝 the linear normed space of 𝐿_𝑝-integrable function 𝑓 on [−1,1] with the following finite norm: ‖𝑓‖_𝑝=(¹₋₁|𝑓(𝑥)|^𝑝𝑑𝑥)^1/𝑝. (1.5)

Let 𝑡_𝑘=𝑡_𝑘𝑛=cos2𝑘−12𝑛𝜋,𝑘=1,…,𝑛 (1.6) be the zeros of 𝑇_𝑛(𝑥)=cos𝑛𝜃(𝑥=cos𝜃), the 𝑛th degree Tchebycheff polynomial of the first kind. The well-known Grünwald interpolation polynomial of 𝑓 based on {𝑡_𝑘}^𝑛_𝑘=1 is given by (see [5]) 𝐺_𝑛(𝑓,𝑥)=^𝑛_𝑘=1𝑓(𝑡_𝑘)𝑙²_𝑘(𝑥), (1.7) where 𝑙_𝑘(𝑥)=(−1)^𝑘+11−𝑡²_𝑘𝑇_𝑛(𝑥)𝑛(𝑥−𝑡_𝑘),𝑘=1,…,𝑛. (1.8)

Theorem 1.1. Let 𝐺_𝑛(𝑓,𝑥) be defined as above. Then 𝑒_𝑝(𝐺_𝑛,𝐿_𝑝1)≍√1𝑛,1≤𝑝≤4,𝑛^2/𝑝,𝑝≥4, (1.9) where in the following 𝐴(𝑛)≍𝐵(𝑛) means that there exists 𝐶 independent of 𝑛 such that 𝐴(𝑛)/𝐶≤𝐵(𝑛)≤𝐶𝐴(𝑛), and the constant 𝐶 may be different in the same expression.

By Hölder inequality, combining Theorem 1.1 with paper [2] we know that for 1≤𝑝,𝑞≤4, 𝑒_𝑝(𝐺_𝑛,𝐿_𝑞1)≍√𝑛. (1.10)

Remark 1.2. Denote by 𝒫_𝑛 the set of algebraic polynomials of degree ≤𝑛. For 𝑓∈𝐹, let 𝑇_𝑛𝑓 denote the best 𝐿_𝑞-approximation polynomial of 𝑓 from 𝒫_𝑛. Then the 𝑝-average error of the best 𝐿_𝑞-approximation of continuous functions by polynomials from 𝒫_𝑛 over the Wiener space is given by 𝑒_𝑝(𝑇_𝑛,𝐿_𝑞)=(_𝐹‖𝑓−𝑇_𝑛𝑓‖^𝑝_𝑞𝜇(𝑑𝑓))^1/𝑝. (1.11) By Theorem 1.1 and paper [6] we can obtain that for 2≤𝑞≤4 and 1≤𝑝≤4, we have 𝑒_𝑝(𝐺_𝑛,𝐿_𝑞)≍𝑒_𝑝(𝑇_𝑛,𝐿_𝑞1)≍√𝑛. (1.12)

Remark 1.3. Let us recall some fundamental notions about the information-based complexity in the average case setting. Let 𝐹 be a set with a probability measure 𝜇, and let 𝐺 be a normed linear space with norm ‖⋅‖. Let 𝑆 be a measurable mapping from 𝐹 into 𝐺 which is called a solution operator. Let 𝑁 be a measurable mapping from 𝐹 into ℛ^𝑛, and let 𝜙 be a measurable mapping from ℛ^𝑛 into 𝐺 which are called an information operator and an algorithm, respectively. For 1≤𝑝<+∞, the 𝑝-average error of the approximation 𝜙∘𝑁 with respect to the measure 𝜇 is defined by 𝑒_𝑝(𝑆,𝑁,𝜙,‖⋅‖)∶=(_𝐹‖𝑆𝑓−𝜙(𝑁(𝑓))‖^𝑝𝜇(𝑑𝑓))^1/𝑝, (1.13) and the 𝑝-average radius of information 𝑁 with respect to 𝜇 is defined by 𝑟_𝑝(𝑆,𝑁,‖⋅‖)∶=inf_𝜙𝑒_𝑝(𝑆,𝑁,𝜙,‖⋅‖), (1.14) where 𝜙 ranges over the set of all algorithms. Futhermore, let Λ denote a class of permissible information functional and denote 𝒩^Λ_𝑛 the set of nonadaptive information operators 𝑁 from Λ of cardinality 𝑛, that is, 𝑁(𝑓)=(𝐿₁(𝑓),𝐿₂(𝑓),…,𝐿_𝑛(𝑓)),𝐿_𝑖∈Λ,𝑖=1,…,𝑛. (1.15) Let 𝑟_𝑝(𝑛,𝑆,Λ,‖⋅‖)=inf_{𝑁∈𝒩Λ𝑛}𝑟_𝑝(𝑆,𝑁,‖⋅‖), (1.16) denote the 𝑛th minimal 𝑝-average radius of nonadaptive information in the class Λ.

For example, if 𝐹 and 𝜇 are defined as above, 𝑆 is the identity mapping 𝐼, and Λ is consist of function evaluations at fixed point; then by [2] we know that for 𝐿_𝑞-norm approximation, if 1≤𝑝,𝑞<∞, we have 𝑟_𝑝(𝑛,𝐼,Λ,𝐿_𝑞1)≍√𝑛. (1.17)

It is easy to understand that 𝐺_𝑛(𝑓,𝑥) can be viewed as a composition of a nonadaptive information operator from 𝒩^Λ_𝑛 and a linear algorithm, and for 1≤𝑝,𝑞≤4, 𝑒_𝑝(𝐺_𝑛,𝐿_𝑞)≍𝑟_𝑝(𝑛,𝐼,Λ,𝐿_𝑞). (1.18)

In comparison with the result of Theorem 1.1, we consider the following Grünwald interpolation. Let 𝑥_𝑘=𝑥_𝑘𝑛=cos𝑘𝜋𝑛+1,𝑘=1,…,𝑛 (1.19) be the zeros of 𝑢_𝑛(𝑥)=sin(𝑛+1)𝜃/sin𝜃,(𝑥=cos𝜃), the 𝑛th Tchebycheff polynomial of the second kind. The Grünwald interpolation polynomial of 𝑓 based on {𝑥_𝑘}^𝑛_𝑘=1 is given by 𝐺_𝑛(𝑓,𝑥)=^𝑛_𝑘=1𝑓(𝑥_𝑘)𝑙²_𝑘(𝑥), (1.20) where 𝑙_𝑘𝑢(𝑥)=^𝑛(𝑥)𝑢^′_𝑛(𝑥_𝑘)(𝑥−𝑥_𝑘)=(−1)^𝑘+1(1−𝑥²_𝑘)𝑢_𝑛(𝑥)(𝑛+1)(𝑥−𝑥_𝑘),𝑘=1,…,𝑛. (1.21)

Theorem 1.4. Let 𝐺_𝑛(𝑓,𝑥) be defined as above. Then 𝑒₂(𝐺_𝑛,𝐿₂)≍1. (1.22)

2. The Proof of Theorem 1.1

We consider the upper estimate first. From [7, page 107, (28)] we obtain 𝑒^𝑝_𝑝(𝐺_𝑛,‖⋅‖_𝑝)=𝑣^𝑝_𝑝⋅¹₋₁(_𝐹|𝑓(𝑥)−𝐺_𝑛(𝑓,𝑥)|²𝜇(𝑑𝑓))^𝑝/2𝑑𝑥, (2.1) where 𝑣^𝑝_𝑝 is the 𝑝th absolute moment of the standard normal distribution. It is easy to verify 𝑓(𝑥)−𝐺_𝑛(𝑓,𝑥)=(1−^𝑛_𝑘=1𝑙²_𝑘(𝑥))𝑓(𝑥)+^𝑛_𝑘=1(𝑓(𝑥)−𝑓(𝑡_𝑘))𝑙²_𝑘(𝑥). (2.2) From (2.2) and Hölder inequality we can obtain _𝐹|𝑓(𝑥)−𝐺_𝑛(𝑓,𝑥)|²𝜇(𝑑𝑓)≤2(1−^𝑛_𝑘=1𝑙²_𝑘(𝑥))²_𝐹𝑓²(𝑥)𝜇(𝑑𝑓)+2_𝐹(^𝑛_𝑘=1(𝑓(𝑥)−𝑓(𝑡_𝑘))𝑙²_𝑘(𝑥))²𝜇(𝑑𝑓)=2𝐼₁(𝑥)+2𝐼₂(𝑥). (2.3) By (1.3) we obtain _𝐹𝑓²(𝑥)𝜇(𝑑𝑓)=_𝑋𝑔²(1+𝑥2)𝜔(𝑑𝑔)=1+𝑥2. (2.4) Let 𝑥=cos𝜃, then it is easy to verify ^𝑛_𝑘=1𝑙²_𝑘𝑇(𝑥)−1=^𝑛(𝑥)𝑛²(𝑥𝑇^′_𝑛(𝑥)−𝑛𝑇_𝑛(𝑥))=cos𝑛𝜃sin(𝑛−1)𝜃𝑛sin𝜃. (2.5) By (2.4), (2.5), and a simple computation we can obtain ¹₋₁|𝐼₁(𝑥)|^𝑝/21𝑑𝑥≤𝑛^𝑝𝜋₀|cos𝑛𝜃sin(𝑛−1)𝜃|^𝑝|sin𝜃|^𝑝−1≤1𝑑𝜃𝑛^𝑝,1≤𝑝≤2,ln𝑛𝑛^𝑝1,𝑝=2,𝑛²,𝑝>2. (2.6) By (1.3), it is easy to verify that for 𝑘≥𝑗, _𝐹(𝑓(𝑥)−𝑓(𝑡_𝑘))(𝑓(𝑥)−𝑓(𝑡_𝑗𝑡))𝜇(𝑑𝑓)=^𝑘−𝑥2,𝑥<𝑡_𝑘,0,𝑡_𝑘≤𝑥≤𝑡_𝑗,𝑥−𝑡_𝑗2,𝑥>𝑡_𝑗. (2.7) Let 𝑡₀=1,𝑡_𝑛+1=−1. From (2.7) and a simple computation we know that for 𝑥∈[𝑡_𝑚+1,𝑡_𝑚], 𝑚=0,…,𝑛, 𝐼₂1(𝑥)=2^𝑛_𝑘=1|𝑥−𝑡_𝑘|𝑙⁴_𝑘(𝑥)+^𝑛−1_𝑘=𝑠+1(𝑥−𝑡_𝑘)𝑙²_𝑘(𝑥)^𝑛_𝑗=𝑘+1𝑙²_𝑗(𝑥)+^𝑚_𝑘=1(𝑡_𝑘−𝑥)𝑙²_𝑘(𝑥)^𝑘−1_𝑗=1𝑙²_𝑗(𝑥)=𝐽₁(𝑥)+𝐽₂(𝑥)+𝐽₃(𝑥). (2.8) From [8] we know ∑^𝑛_𝑘=1𝑙²_𝑘(𝑥)≤2, hence ^𝑛_𝑘=1𝑙^𝑝_𝑘(𝑥)≤𝐶,∀𝑝≥2. (2.9) From (1.8) it follows that |(𝑥−𝑥_𝑘)𝑙_𝑘1(𝑥)|≤𝑛,𝑘=1,…,𝑛. (2.10) From (2.7) and (2.10) it follows that |𝐽₁1(𝑥)|≤2𝑛^𝑛_𝑘=1¹₋₁|𝑙³_𝑘𝐶(𝑥)|𝑑𝑥≤𝑛. (2.11) From (2.10) it follows that 𝐽₂1(𝑥)≤𝑛^𝑛−1_𝑗=𝑚+1|𝑙_𝑗(𝑥)|^𝑛_𝑘=𝑗+1𝑙²_𝑘(𝑥). (2.12) Let 𝑥=cos𝜃, we have ^𝑛−1_𝑗=𝑚+1|𝑙_𝑗(𝑥)|^𝑛_𝑘=𝑗+1𝑙²_𝑘(𝑥)=^𝑛−1_𝑗=𝑚+1|sin((2𝑗−1)𝜋/2𝑛)cos𝑛𝜃|×𝑛(cos𝜃−cos((2𝑗−1)𝜋/2𝑛))_{𝑛𝑘=𝑗+1}sin²((2𝑘−1)𝜋/2𝑛)cos²𝑛𝜃𝑛²(cos𝜃−cos((2𝑘−1)𝜋/2𝑛))²≤1𝑛^3𝑛−1_𝑗=𝑚+1|sin((2𝑗−1)𝜋/2𝑛)|×(cos((2𝑚−1)𝜋/2𝑛)−cos((2𝑗−1)𝜋/2𝑛))_{𝑛𝑘=𝑗+1}sin²((2𝑘−1)𝜋/2𝑛)(cos((2𝑚−1)𝜋/2𝑛)−cos((2𝑘−1)𝜋/2𝑛))²=14𝑛^3𝑛−1_𝑗=𝑚+1|sin((2𝑗−1)𝜋/2𝑛)|×(sin((𝑗−𝑚)𝜋/2𝑛)sin((𝑗+𝑚−1)𝜋/2𝑛))_{𝑛𝑘=𝑗+1}sin²((2𝑘−1)𝜋/2𝑛)(sin((𝑘−𝑚)𝜋/2𝑛)sin((𝑘+𝑚−1)𝜋/2𝑛))². (2.13) By sin𝑥+sin𝑦=2sin((𝑥+𝑦)/2)cos(𝑥−𝑦)/2 we know that for 0<𝑥,𝑦<𝜋, thus 0<sin𝑥≤2sin𝑥+𝑦2. (2.14) It is easy to know ^𝑛_𝑘=𝑗+11(𝑘−𝑚)²<^𝑛_𝑘=𝑗+11=1(𝑘−𝑚)(𝑘−𝑚−1)−1𝑗−𝑚≤1𝑛−𝑚𝑗−𝑚. (2.15) By 2𝑥/𝜋≤sin𝑥,∀𝑥∈(0,𝜋/2], (2.16), (2.17), and (2.18) we can obtain ^𝑛−1_𝑗=𝑚|𝑙_𝑗(𝑥)|^𝑛_𝑘=𝑗+1𝑙²_𝑘(𝑥)=|𝑙_𝑚(𝑥)|^𝑛_𝑘=𝑚+1𝑙²_𝑘(𝑥)+^𝑛−1_𝑗=𝑚+1|𝑙_𝑗(𝑥)|^𝑛_𝑘=𝑗+1𝑙²_𝑘(𝑥)≤𝐶. (2.16) From (2.12) and (2.16) we can obtain |𝐽₂𝐶(𝑥)|≤𝑛. (2.17) Similarly |𝐽₃𝐶(𝑥)|≤𝑛. (2.18) From (2.3), (2.8), (2.11), (2.17), and (2.18) we can obtain ¹₋₁|𝐼₂(𝑥)|^𝑝/2𝐶𝑑𝑥≤𝑛^𝑝/2. (2.19) By (2.1), (2.3), (2.6), and (2.19) we can obtain the upper estimate.

Now we consider the lower estimate. For 1≤𝑝≤4, we can obtain the lower estimate from [2]. For 𝑝>4, from (2.4) we know that ¹₋₁(_𝐹|(1−^𝑛_𝑘=1𝑙²_𝑘(𝑥))𝑓(𝑥)|²𝜇(𝑑𝑓))^𝑝/2𝑑𝑥=¹₋₁|1+𝑥2|^𝑝/2|(1−^𝑛_𝑘=1𝑙²_𝑘(𝑥))|^𝑝𝑑𝑥. (2.20) Let 𝑥=cos𝜃, then from (2.5) we know that ^𝑛_𝑘=1𝑙²_𝑘(𝑥)−1=cos𝜃sin2𝑛𝜃−2𝑛sin𝜃cos²𝑛𝜃𝑛. (2.21) Hence we can verify that for 𝜃∈[5𝜋/8𝑛,7𝜋/8𝑛],|^𝑛_𝑘=1𝑙²_𝑘(𝑥)−1|≥|sin2𝑛𝜃|≥14𝑛sin𝜃7. (2.22) From (2.20) and (2.22) and a simple computation we can obtain ¹₋₁(_𝐹|(1−^𝑛_𝑘=1𝑙²_𝑘(𝑥))𝑓(𝑥)|²𝜇(𝑑𝑓))^𝑝/2𝑑𝑥≥cos5𝜋/8𝑛−cos7𝜋/8𝑛14^𝑝≥38⋅14^𝑝𝑛². (2.23) From (2.2), (2.3), and (2.19) it follows that ¹₋₁(_𝐹|^𝑛_𝑘=1(𝑓(𝑥)−𝑓(𝑡_𝑘))𝑙²_𝑘(𝑥)|²𝜇(𝑑𝑓))^𝑝/2𝐶𝑑𝑥≤𝑛^𝑝/2. (2.24) From (2.1), (2.2), (2.23), and (2.24) we can obtain the lower estimate for 𝑝>4.

3. The Proof of Theorem 1.4

Let 𝑄_𝑛(𝑓,𝑥)=(1+𝑥2𝑓(1)+1−𝑥2𝑢𝑓(−1))^2𝑛(𝑥)(𝑛+1)²+^𝑛_𝑘=1𝑓(𝑥_𝑘)(1−𝑥²)(1−𝑥𝑥_𝑘𝑢)(^𝑛(𝑥)(𝑛+1)(𝑥−𝑥_𝑘))² (3.1) be the quasi-Hermite-Fejer interpolation polynomial of degree ≤2𝑛+1 based on the extended Tchebycheff nodes of the second kind (see [8]); then by a simple computation we obtain 𝐺_𝑛(𝑓,𝑥_𝑘)−𝑄_𝑛(𝑓,𝑥_𝑘)=0,𝑘=1,…,𝑛,𝐺^′_𝑛(𝑓,𝑥_𝑘)−𝑄^′_𝑛(𝑓,𝑥_𝑘)=𝑓(𝑥_𝑘)3𝑥_𝑘1−𝑥²_𝑘,𝑘=1,…,𝑛,𝐺_𝑛(𝑓,1)−𝑄_𝑛(𝑓,1)=^𝑛_𝑘=1𝑓(𝑥_𝑘)(1+𝑥_𝑘)²−𝑓(1),𝐺_𝑛(𝑓,−1)−𝑄_𝑛(𝑓,−1)=^𝑛_𝑘=1𝑓(𝑥_𝑘)(1−𝑥_𝑘)²−𝑓(−1). (3.2) Denote 𝜑_𝑘(𝑥)=(1−𝑥²)(1−𝑥𝑥_𝑘𝑢)(^𝑛(𝑥)(𝑛+1)(𝑥−𝑥_𝑘))²𝜑,𝑘=1,…,𝑛,₀(𝑥)=1+𝑥2𝑢²_𝑛(𝑥)(𝑛+1)²,𝜑_𝑛+1(𝑥)=1−𝑥2𝑢²_𝑛(𝑥)(𝑛+1)²,𝜙_𝑘(𝑥)=(1−𝑥²)(1−𝑥²_𝑘)𝑢²_𝑛(𝑥)(𝑛+1)²(𝑥−𝑥_𝑘),𝑘=1,…,𝑛. (3.3) By (3.2) and the unique of the Hermite interpolation polynomial 𝐻_𝑛(𝑓,𝑥) which satisfies interpolation conditions, 𝐻_𝑛(𝑓,𝑥_𝑘)=𝑓(𝑥_𝑘𝐻),𝑘=0,…,𝑛+1,_′𝑛(𝑓,𝑥_𝑘)=𝑓^′(𝑥_𝑘),𝑘=1,…,𝑛, (3.4) we obtain 𝐺_𝑛(𝑓,𝑥)−𝑄_𝑛(𝑓,𝑥)=𝜑₀(𝑥)[^𝑛_𝑘=1𝑓(𝑥_𝑘)(1+𝑥_𝑘)²−𝑓(1)]+𝜑_𝑛+1(𝑥)[^𝑛_𝑘=1𝑓(𝑥_𝑘)(1−𝑥_𝑘)²+−𝑓(−1)]_𝑛𝑘=1𝑓(𝑥_𝑘)3𝑥_𝑘1−𝑥²_𝑘𝜙_𝑘=𝑢(𝑥)_2𝑛(𝑥)(𝑛+1)^2𝑛_𝑘=1𝑓(𝑥_𝑘)(1+𝑥²_𝑘)+2𝑥𝑢²_𝑛(𝑥)(𝑛+1)^2𝑛_𝑘=1𝑓(𝑥_𝑘𝑢)−^2𝑛(𝑥)2(𝑛+1)²−[𝑓(1)+𝑓(−1)]𝑥𝑢²_𝑛(𝑥)2(𝑛+1)²[𝑓(1)−𝑓(−1)]+^𝑛_𝑘=1𝑓(𝑥_𝑘)3𝑥_𝑘1−𝑥²_𝑘𝜙_𝑘(𝑥)=𝐴₁(𝑥)+𝐴₂(𝑥)+𝐴₃(𝑥)+𝐴₄(𝑥)+𝐴₅(𝑥). (3.5) By (3.5) and (𝑎+𝑏+𝑐+𝑑)²≤4(𝑎²+𝑏²+𝑐²+𝑑²) we know that 𝑒₂(𝐺_𝑛,𝐿₂)=_𝐹¹₋₁|𝑓(𝑥)−𝐺_𝑛(𝑓,𝑥)|²𝑑𝑥𝜇(𝑑𝑓)≤4_𝐹¹₋₁|𝑓(𝑥)−𝑄_𝑛(𝑓,𝑥)|²𝑑𝑥𝜇(𝑑𝑓)+4_𝐹¹₋₁[𝐴₁(𝑥)+𝐴₂(𝑥)]²𝑑𝑥𝜇(𝑑𝑓)+4_𝐹¹₋₁[𝐴₃(𝑥)+𝐴₄(𝑥)]²𝑑𝑥𝜇(𝑑𝑓)+4_𝐹¹₋₁[𝐴₅(𝑥)]²𝑑𝑥𝜇(𝑑𝑓)=4𝐼₁+4𝐼₂+4𝐼₃+4𝐼₄. (3.6) From [8] we know that for every 𝑓∈𝐶[−1,1], ¹₋₁|𝑓(𝑥)−𝑄_𝑛(𝑓,𝑥)|²𝑑𝑥≤¹₋₁|𝑓(𝑥)−𝑄_𝑛(𝑓,𝑥)|²(1−𝑥²)^−1/2𝑑𝑥≤𝐶𝜔²1(𝑓,𝑛), (3.7) where 𝜔(𝑓,𝑡) is the modulus of continuity of 𝑓 on [−1,1] defined for every 𝑡≥0, and 𝐶 is independent of 𝑛 and 𝑓. By (3.7) and [6] we can obtain 𝐼₁≤𝐶_𝐹𝜔²1(𝑓,𝑛)𝜇(𝑑𝑓)≤𝐶ln𝑛𝑛. (3.8) By using 𝐴₁(𝑥) and 𝐴₂(𝑥) we obtain 𝐼₂=_𝐹¹₋₁[𝐴₁(𝑥)]²𝑑𝑥𝜇(𝑑𝑓)+_𝐹¹₋₁[𝐴₂(𝑥)]²=1𝑑𝑥𝜇(𝑑𝑓)(𝑛+1)⁴¹⁻¹𝑢⁴_𝑛(𝑥)𝑑𝑥_𝐹[^𝑛_𝑘=1𝑓(𝑥_𝑘)(1+𝑥²_𝑘)]²+4𝜇(𝑑𝑓)(𝑛+1)⁴¹⁻¹𝑥²𝑢⁴_𝑛(𝑥)𝑑𝑥_𝐹[^𝑛_𝑘=1𝑓(𝑥_𝑘)]²𝜇(𝑑𝑓). (3.9) By (1.3) we obtain _𝐹[^𝑛_𝑘=1𝑓(𝑥_𝑘)]²𝜇(𝑑𝑓)=_𝑋[^𝑛_𝑘=1𝑔(1+𝑥_𝑘2)]²𝜔(𝑑𝑔)=^𝑛𝑛_𝑘=1𝑗=1min{1+𝑥_𝑘2,1+𝑥_𝑗2}≍𝑛²._𝐹[^𝑛_𝑘=1𝑓(𝑥_𝑘)(1+𝑥²_𝑘)]²𝜇(𝑑𝑓)≍𝑛². (3.10) From (3.9) and (3.10) we obtain 𝐼₂≍1𝑛²¹₋₁𝑢⁴_𝑛(𝑥)𝑑𝑥≍1. (3.11) Similar to (3.11), we have |𝐼₃1|=4(𝑛+1)⁴¹⁻¹(1+𝑥²)𝑢⁴_𝑛𝐶(𝑥)𝑑𝑥≤𝑛². (3.12) By (3.3) and 0≤(1−𝑥²)𝑢²_𝑛(𝑥)≤1 we obtain 0≤𝐼₄=9(𝑛+1)^4𝐹1−1(1−𝑥²)²𝑢²_𝑛(𝑥)[^𝑛_𝑘=1𝑓(𝑥_𝑘)𝑥_𝑘𝑢_𝑛(𝑥)𝑥−𝑥_𝑘]²≤9𝑑𝑥𝜇(𝑑𝑓)(𝑛+1)^4𝐹1−1(1−𝑥²)^1/2[^𝑛_𝑘=1𝑓(𝑥_𝑘)𝑥_𝑘𝑢_𝑛(𝑥)𝑥−𝑥_𝑘]²=9𝑑𝑥𝜇(𝑑𝑓)(𝑛+1)^4𝑛𝑛_𝑘=1𝑗=1¹⁻¹(1−𝑥²)^1/2𝑢²_𝑛(𝑥)(𝑥−𝑥_𝑘)(𝑥−𝑥_𝑗)𝑑𝑥_𝐹𝑥_𝑘𝑥_𝑗𝑓(𝑥_𝑘)𝑓(𝑥_𝑗)𝜇(𝑑𝑓). (3.13) By [8] we know that ¹₋₁(1−𝑥²)𝑢²_𝑛(𝑥)(𝑥−𝑥_𝑘)(𝑥−𝑥_𝑗)1√1−𝑥²𝑑𝑥=(𝑛+1)𝜋1−𝑥²_𝑗,𝑗=𝑘,0,𝑗≠𝑘. (3.14) By (1.3), (3.13), (3.14), and (2/𝜋)𝑥<sin𝑥<𝑥,0<𝑥<𝜋/2, we obtain 0≤𝐼₄≤9𝜋2(𝑛+1)^3𝑛_𝑘=1𝑥²_𝑘(1+𝑥_𝑘)1−𝑥²_𝑘≤9𝜋(𝑛+1)^3𝑛_𝑘=11sin²≤𝐶𝑘𝜋/(𝑛+1)𝑛. (3.15) By (3.6), (3.8), (3.11), (3.12), and (3.15) we can obtain the upper estimate of Theorem 1.4. On the other hand, by (3.5) we can verify that |𝑓(𝑥)−𝐺_𝑛(𝑓,𝑥)|²≥[𝐴₁(𝑥)+𝐴₂(𝑥)]²4−|𝑓(𝑥)−𝑄_𝑛(𝑓,𝑥)|²−[𝐴₃(𝑥)+𝐴₄(𝑥)]²−[𝐴₅(𝑥)]². (3.16) From (3.16), (3.8), (3.11), (3.12), and (3.15) we can obtain the lower estimate of Theorem 1.4.