On 𝐿_𝑝-Approximation by Iterative Combination of Bernstein-Durrmeyer Polynomials

Abstract

We improve the degree of approximation by Bernstein-Durrmeyer polynomials taking their iterates and obtain error estimate in higher-order approximation.

1. Introduction

The Bernstein-Durrmeyer polynomials𝑀_𝑛(𝑓;𝑡)=(𝑛+1)^𝑛_𝜈=0𝑝_𝑛,𝜈(𝑡)¹⁰𝑝_𝑛,𝜈(𝑢)𝑓(𝑢)𝑑𝑢,(1.1) where 𝑝_𝑛,𝜈(𝑡)=(^𝑛𝑣)𝑡^𝜈(1−𝑡)^𝑛−𝜈, 𝑡∈[0,1], were introduced by Durrmeyer [1] and extensively studied by Derriennic [2] and several other researchers. It turns out that the order of approximation by these operators is, at best, 𝒪(𝑛⁻¹) however smooth the function may be. In order to improve this rate of approximation, we consider an iterative combination 𝑇_𝑛,𝑘(𝑓;𝑡) of the operators 𝑀_𝑛(𝑓;𝑡). This technique of improving the rate of convergence was given by Micchelli [3] who first used it to improve the order of approximation by Bernstein polynomials 𝐵_𝑛(𝑓;𝑡). Recently, this technique has been applied to obtain some direct and inverse theorems in ordinary and simultaneous approximation by several sequences of linear positive operators in uniform norm (c.f., e.g., [4–6]). The object of this paper is to study some direct theorems in 𝐿_𝑝-approximation by the operators 𝑇_𝑛,𝑘(𝑓;𝑡).

For 𝑓∈𝐿_𝑝[0,1], the operators 𝑀_𝑛(𝑓;𝑡) can be expressed as 𝑀_𝑛(𝑓;𝑡)=¹⁰𝑊_𝑛(𝑢,𝑡)𝑓(𝑢)𝑑𝑢,(1.2) where 𝑊_𝑛(𝑢,𝑡)=(𝑛+1)^𝑛_𝜈=0𝑝_𝑛,𝜈(𝑡)𝑝_𝑛,𝜈(𝑢)(1.3) is the kernel of the operators.

For 𝑚∈𝑁⁰ (the set of nonnegative integers), the 𝑚th order moment for the operators 𝑀_𝑛 is defined as 𝜇_𝑛,𝑚(𝑡)=𝑀_𝑛((𝑢−𝑡)^𝑚;𝑡).(1.4)

Let 𝐿_𝐵[𝑎,𝑏], 𝐶^∞[𝑎,𝑏], 𝐴𝐶[𝑎,𝑏], and 𝐵𝑉[𝑎,𝑏] denote the classes of bounded Lebesgue integrable, infinitely differentiable, absolutely continuous functions, and functions of bounded variations, respectively, on the interval [𝑎,𝑏].

The Iterative combination 𝑇_𝑛,𝑘∶𝐿_𝐵[0,1]→𝐶^∞[0,1] of the operators 𝑀_𝑛(𝑓;𝑡) is defined as 𝑇_𝑛,𝑘(𝑓;𝑡)=𝐼−𝐼−𝑀_𝑛^𝑘(𝑓;𝑡)=^𝑘_𝑟=1(−1)^𝑟+1𝑘𝑟𝑀^𝑟𝑛(𝑓;𝑡),𝑘∈𝑁,(1.5) where 𝑀^0𝑛=𝐼 and 𝑀^𝑟𝑛=𝑀_𝑛(𝑀^𝑛𝑟−1) for 𝑟∈𝑁.

In Section 2 of this paper, we give some definitions and auxiliary results which will be needed to prove the main results. In Section 3, we obtain an estimate of error in 𝐿_𝑝-approximation (1⩽𝑝<∞) by the iterative combination 𝑇_𝑛,𝑘(⋅;𝑡) in terms of 𝐿_𝑝-norm of derivatives of the function. From these estimates, we obtain a general error estimate in terms of 2𝑘+2th integral modulus of smoothness of the function.

In what follows, we suppose that 0<𝑎₁<𝑎₂<𝑎₃<𝑏₃<𝑏₂<𝑏₁<1 and 𝐼_𝑗=[𝑎_𝑗,𝑏_𝑗], 𝑗=1,2,3. Further, 𝐶 is a constant not always the same.

2. Preliminaries and Auxiliary Results

In the sequel, we will require the following results.

Lemma 2.1 (see [5]). For the function 𝜇_𝑛,𝑚(𝑡), one has 𝜇_𝑛,0(𝑡)=1, 𝜇_𝑛,1(𝑡)=(1−2𝑡)/(𝑛+2), and there holds the recurrence relation (𝑛+𝑚+2)𝜇_𝑛,𝑚+1(𝑡)=𝑡(1−𝑡)𝜇^𝑛,𝑚(𝑡)+2𝑚𝜇_{𝑛,𝑚−1}(𝑡)+(𝑚+1)(1−2𝑡)𝜇_𝑛,𝑚(𝑡),(2.1) for 𝑚⩾1.
Consequently, we have (i)𝜇_𝑛,𝑚(𝑡) is a polynomial in 𝑡 of degree 𝑚, (ii)for every 𝑡∈[0,1], 𝜇_𝑛,𝑚(𝑡)=𝒪(𝑛^{−[(𝑚+1)/2]}), where [𝛽] is the integer part of 𝛽.

The 𝑚th order moment for the operator 𝑀^𝑝𝑛 is defined as 𝜇^{[𝑝]𝑛,𝑚}(𝑡)=𝑀^𝑝𝑛((𝑢−𝑡)^𝑚;𝑡), 𝑝∈𝑁 (the set of natural numbers). We denote 𝜇^[1]𝑛,𝑚(𝑡) by 𝜇_𝑛,𝑚(𝑡).

Lemma 2.2 (see [1]). For the function 𝑝_𝑛,𝜈(𝑡), there holds the result 𝑡^𝑟(1−𝑡)^𝑟𝑑^𝑟𝑝_𝑛,𝜈(𝑡)𝑑𝑡^𝑟=_{2𝑖+𝑗⩽𝑟𝑖,𝑗⩾0}𝑛^𝑖(𝜈−𝑛𝑡)^𝑗𝑞_{𝑖,𝑗,𝑟}(𝑡)𝑝_𝑛,𝜈(𝑡),(2.2) where 𝑞_{𝑖,𝑗,𝑟}(𝑡) are certain polynomials in 𝑡 independent of 𝑛 and 𝜈.

Lemma 2.3 (see [7]). There holds the recurrence relation 𝜇^{[𝑝+1]𝑛,𝑚}(𝑡)=^𝑚_{𝑗=0𝑚−𝑗𝑖=0}𝑚𝑗𝑖1!𝐷^𝑖𝜇^{[𝑝]𝑛,𝑚−𝑗}(𝑡)𝜇_{𝑛,𝑖+𝑗}(𝑡).(2.3)

Lemma 2.4 (see [7]). For 𝑘, 𝑙∈𝑁, there holds 𝑇_𝑛,𝑘((𝑢−𝑡)^𝑙;𝑡)=𝒪(𝑛^−𝑘).

Using Lemmas 2.1 and 2.3, we can prove the following.

Lemma 2.5. For 𝑝∈𝑁, 𝑚∈𝑁⁰, and 𝑡∈[0,1], we have 𝜇^{[𝑝]𝑛,𝑚}(𝑡)=𝒪𝑛^{−[(𝑚+1)/2]}.(2.4)

Let 𝑓∈𝐿_𝑝[𝑎,𝑏], 1⩽𝑝<∞, and 𝐼₁⊂(𝑎,𝑏). Then, for sufficiently small 𝜂>0 the Steklov mean 𝑓_𝜂,𝑚 of 𝑚th order corresponding to 𝑓 is defined as follows:𝑓_𝜂,𝑚(𝑡)=𝜂^−𝑚^{𝜂/2−𝜂/2}⋯^{𝜂/2−𝜂/2}𝑓(𝑡)+(−1)^𝑚−1Δ^𝑚∑_{𝑚𝑖=1𝑡𝑖}𝑓(𝑡)^𝑚_𝑖=1𝑑𝑡_𝑖,𝑡∈𝐼₁,(2.5) where Δ^𝑚ℎ is the forward difference operator with step length ℎ.

Lemma 2.6. Let 𝑓∈𝐿_𝑝[𝑎,𝑏], 1⩽𝑝<∞, and 𝐼₁⊂(𝑎,𝑏). Then, for the function 𝑓_𝜂,𝑚, we have (a)𝑓_𝜂,𝑚 has derivatives up to order 𝑚 over 𝐼₁, (b)‖𝑓^{(𝑟)𝜂,𝑚}‖_{𝐿𝑝(𝐼1)}⩽𝐶𝜂^−𝑟𝜔_𝑟(𝑓,𝜂,[𝑎,𝑏]), 𝑟=1,2,…,𝑚,(c)‖𝑓−𝑓_𝜂,𝑚‖_{𝐿𝑝(𝐼1)}⩽𝐶𝜔_𝑚(𝑓,𝜂,[𝑎,𝑏]), (d)‖𝑓_𝜂,𝑚‖_{𝐿𝑝(𝐼1)}⩽𝐶𝜂^−𝑚‖𝑓‖_{𝐿𝑝[𝑎,𝑏]}, (e)‖𝑓^{(𝑟)𝜂,𝑚}‖_{𝐿𝑝(𝐼1)}⩽𝐶‖𝑓‖_{𝐿𝑝[𝑎,𝑏]}, where 𝐶 is a constant that depends on 𝑖 but is independent of 𝑓 and 𝜂.

Following [8, Theorem  18.17] or [9, pages 163–165], the proof of the above lemma easily follows hence the details are omitted.

Let 𝑓∈𝐿_𝑝[0,𝑎], 1⩽𝑝<∞. Then, the Hardy-Littlewood majorant ℎ_𝑓(𝑥) of the function 𝑓 is defined asℎ_𝑓(𝑥)=sup_𝜉≠𝑥1𝜉−𝑥^𝜉𝑥𝑓(𝑡)𝑑𝑡.(2.6)

Lemma 2.7. If 1<𝑝<∞, 𝑓∈𝐿_𝑝[0,𝑎], then ℎ_𝑓∈𝐿_𝑝[0,𝑎] and ‖‖ℎ_𝑓‖‖_{𝐿𝑝[0,𝑎]}⩽2^1/𝑝𝑝𝑝−1‖𝑓‖^{𝐿𝑝[0,𝑎]}.(2.7)

The lemma follows from [10, page 32].

The next lemma gives a bound for the intermediate derivatives of 𝑓 in terms of the highest-order derivative and the function in 𝐿_𝑝-norm.

Lemma 2.8 (see [11]). Let 1⩽𝑝<∞, 𝑓∈𝐿_𝑝[𝑎,𝑏]. Suppose 𝑓^(𝑘)∈𝐴𝐶[𝑎,𝑏] and 𝑓^(𝑘+1)∈𝐿_𝑝[𝑎,𝑏]. Then, ‖‖𝑓^(𝑗)‖‖_{𝐿𝑝[𝑎,𝑏]}⩽𝐾_𝑗‖‖𝑓^(𝑘+1)‖‖_{𝐿𝑝[𝑎,𝑏]}+‖𝑓‖_{𝐿𝑝[𝑎,𝑏]},𝑗=1,2,…,𝑘,(2.8) where 𝐾_𝑗 are certain constants independent of 𝑓.

The dual operator 𝑀_𝑛 corresponding to the operator 𝑀_𝑛 is defined as 𝑀_𝑛(𝑓;𝑢)=¹⁰𝑊_𝑛(𝑢,𝑡)𝑓(𝑡)𝑑𝑡.(2.9) Then, the corresponding 𝑚th order moment is given by 𝜇_𝑛,𝑚(𝑢)=𝑀_𝑛((𝑡−𝑢)^𝑚;𝑢)=∫¹⁰𝑊_𝑛(𝑢,𝑡)(𝑡−𝑢)^𝑚𝑑𝑡.

Lemma 2.9. For the function 𝜇_𝑛,𝑚(𝑢), there holds the recurrence relation (𝑛−𝑚−2)𝜇_𝑛,𝑚+1(𝑢)=𝑢(1−𝑢)𝜇^𝑛,𝑚(𝑢)+(𝑚+1)(1−2𝑢)𝜇_𝑛,𝑚(𝑢)+2𝑚𝑢−𝑢²𝜇_{𝑛,𝑚−1}(𝑢).(2.10)

Proof. In view of the relation 𝑢(1−𝑢)𝑝^𝑛,𝑘(𝑢)=(𝑘−𝑛𝑢)𝑝_𝑛,𝑘(𝑢), we get 𝑢(1−=𝑢()𝑛𝜇+^𝑛,𝑚1)(𝑢)_𝑛_𝑘=0¹⁰=(𝑛+1)_𝑛𝑝_𝑛,𝑘(𝑡)𝑢(1−𝑢)𝑝^𝑛,𝑘(𝑢)(𝑡−𝑢)^𝑚𝑑𝑡−𝑚𝑢(1−𝑢)𝜇_{𝑛,𝑚−1}(𝑢)_𝑘=0=(𝑛+1)_𝑛(𝑘−𝑛𝑢)𝑝_𝑛,𝑘(𝑢)¹⁰𝑝_𝑛,𝑘(𝑡)(𝑡−𝑢)^𝑚𝑑𝑡−𝑚𝑢(1−𝑢)𝜇_{𝑛,𝑚−1}(𝑢)_𝑘=0=(𝑛+1)_𝑛𝑝_𝑛,𝑘(𝑢)¹⁰((𝑘−𝑛𝑡)+𝑛(𝑡−𝑢))𝑝_𝑛,𝑘(𝑡)(𝑡−𝑢)^𝑚𝑑𝑡−𝑚𝑢(1−𝑢)𝜇_{𝑛,𝑚−1}(𝑢)_𝑘=0𝑝_𝑛,𝑘(𝑢)¹⁰𝑡−𝑡²𝑝^𝑛,𝑘(𝑡)(𝑡−𝑢)^𝑚𝑑𝑡+𝑛𝜇_𝑛,𝑚+1(𝑢)−𝑚𝑢(1−𝑢)𝜇_{𝑛,𝑚−1}(𝑢).(2.11) Expanding (𝑡−𝑡²) as a polynomial in (𝑡−𝑢) and integrating by parts, we get 𝑢(1=−−𝑢)(𝜇𝑛+^𝑛,𝑚1(𝑢))_𝑛_𝑘=0×(𝑡−𝑢)_𝑚−1𝑝_𝑛,𝑘(𝑢)¹⁰𝑝_𝑛,𝑘(𝑡)(𝑚+1)(1−2𝑢)(𝑡−𝑢)+𝑚𝑢−𝑢²−(𝑚+2)(𝑡−𝑢)²=−(𝑚+1)(1𝑑−𝑡+2𝑢𝑛)𝜇𝜇_{𝑛,𝑚+1𝑛,𝑚}(𝑢)−𝑚𝑢(1−𝑢)𝜇_{𝑛,𝑚−1}(𝑢)−(𝑚+2)𝜇_𝑛,𝑚+1(𝑢)−2𝑚𝑢−𝑢²𝜇_{𝑛,𝑚−1}(𝑢)(𝑢)+𝑛𝜇_𝑛,𝑚+1(𝑢).(2.12)
Rearrangement of the terms gives (2.10).

Remark 2.10. From (2.10), it follows that 𝜇_𝑛,𝑚(𝑢)=𝒪(𝑛^{−[(𝑚+1)/2]}), where [𝛽] is the integer part of 𝛽.

3. Main Result

In this section, we obtain an error estimate in terms of 𝐿_𝑝 norm. The proof of the case 𝑝>1 makes use of Lemma 2.7 regarding Hardy-Littlewood majorant and Lemma 2.8, while for 𝑝=1, we require only Lemma 2.8.

Theorem 3.1. If 𝑝>1, 𝑓∈𝐿_𝑝[0,1], 𝑓 has derivatives of order 2𝑘 on 𝐼₁ with 𝑓^(2𝑘−1)∈𝐴𝐶(𝐼₁), and 𝑓^(2𝑘)∈𝐿_𝑝(𝐼₁), then for sufficiently large 𝑛‖‖𝑇_𝑛,𝑘(𝑓;⋅)−𝑓(⋅)‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘‖‖𝑓^(2𝑘)‖‖_{𝐿𝑝(𝐼1)}+‖𝑓‖_{𝐿𝑝[0,1]}.(3.1) Moreover, if 𝑓∈𝐿₁[0,1], 𝑓 has derivatives up to the order (2𝑘−1) on 𝐼₁ with 𝑓^(2𝑘−2)∈𝐴𝐶(𝐼₁), and 𝑓^(2𝑘−1)∈𝐵𝑉(𝐼₁), then for sufficiently large 𝑛 there holds ‖‖𝑇_𝑛,𝑘(𝑓;⋅)−𝑓(⋅)‖‖_𝐿1(𝐼2)⩽𝐶𝑛^−𝑘‖‖𝑓^(2𝑘−1)‖‖_{𝐵𝑉(𝐼1)}+‖‖𝑓^(2𝑘−1)‖‖_𝐿1(𝐼2)+‖𝑓‖_𝐿1(𝐼2),(3.2) where 𝐶 is a certain constant independent of 𝑓 and 𝑛.

Proof. Let 𝑝>1, then for all 𝑢∈𝐼₁ and 𝑡∈𝐼₂, we can write 𝑓(𝑢)−𝑓(𝑡)=^2𝑘−1_𝑗=1𝑓^(𝑗)(𝑡)𝑗!(𝑢−𝑡)^𝑗+1(2𝑘−1)!^𝑢𝑡𝜑(𝑢)(𝑢−𝑣)^2𝑘−1𝑓^(2𝑘)(𝑣)𝑑𝑣+𝐹(𝑢,𝑡)(1−𝜑(𝑢)),(3.3) where 𝜑(𝑢) is the characteristic function of the interval 𝐼₁ and 𝐹(𝑢,𝑡)=𝑓(𝑢)−^2𝑘−1_𝑗=0𝑓^(𝑗)(𝑡)𝑗!(𝑢−𝑡)^𝑗.(3.4) Therefore, operating by 𝑇_𝑛,𝑘 on both sides of (3.3), we obtain three terms, say 𝐸₁, 𝐸₂, and 𝐸₃ corresponding to the three terms in the right-hand side of (3.3).
In view of Lemmas 2.4 and 2.8, we get ‖‖𝐸₁‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘‖‖𝑓^(2𝑘)‖‖_{𝐿𝑝(𝐼2)}+‖𝑓‖_{𝐿𝑝(𝐼2)}.(3.5) Let ℎ_𝑓^(2𝑘) be the Hardy-Littleood majorant of 𝑓^(2𝑘) on 𝐼₁. Then, in order to estimate 𝐸₂, it is sufficient to consider the estimate for 𝐽₁𝐽₁=𝑀_𝑛𝜑(𝑢)(𝑢−𝑡)^2𝑘||ℎ_𝑓(2𝑘)(𝑢)||;𝑡.(3.6) Applying Hölder's inequality, Lemma 2.1, and then Fubini's theorem, we get ‖‖𝐽₁‖‖^𝑝𝐿_𝑝(𝐼2)⩽⩽𝐶𝑀𝑛_{𝑛−𝑘𝑝}𝜑(𝑢)(𝑢−𝑡)^2𝑘𝑞;𝑡^𝑝/𝑞×𝑀_𝑛𝜑(𝑢)(𝑢−𝑡)^𝑝||ℎ_𝑓(2𝑘)(𝑢)||^𝑝;𝑡^𝑏1_𝑎1^𝑏2_𝑎2𝑊_𝑛(𝑡,𝑢)||ℎ_𝑓^(2𝑘)(𝑢)||^𝑝𝑑𝑡𝑑𝑢.(3.7) Now, in view of Lemmas 2.1 and 2.7, we have ‖‖𝐽₁‖‖^𝑝𝐿_𝑝(𝐼2)⩽𝐶𝑛^−𝑘𝑝^𝑏1_𝑎1||ℎ_𝑓(2𝑘)(𝑢)||^𝑝𝑑𝑢⩽𝐶𝑛^−𝑘𝑝‖‖𝑓^(2𝑘)‖‖^𝑝𝐿_𝑝(𝐼1).(3.8) Consequently, ‖‖𝐸₂‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘‖‖𝑓^(2𝑘)‖‖_{𝐿𝑝(𝐼1)}.(3.9) For 𝑢∈[0,1]⧵𝐼₁, 𝑡∈𝐼₂, we can find a 𝛿>0 such that |𝑢−𝑡|⩾𝛿. Thus, ||𝑀_𝑛=(𝐹|||||𝑀(𝑢,𝑡)(1−𝜑(𝑢));𝑡)_𝑛||𝑓(𝑢)−^2𝑘−1_𝑗=0𝑓^(𝑗)(𝑡)⩽𝛿_−2𝑘𝑗!(𝑢−𝑡)^𝑗(1−𝜑(𝑢));𝑡|||||𝑀^𝑛||𝑓(𝑢)||(𝑢−𝑡)^2𝑘;𝑡+^2𝑘−1_𝑗=0||𝑓^(𝑗)(𝑡)||𝑗!𝑀^𝑛|𝑢−𝑡|^2𝑘+𝑗;𝑡=𝐽₂+𝐽₃,say.(3.10) On an application of Hölder's inequality, Lemma 2.1, and Fubini's theorem, we get ‖‖𝐽₂‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘^𝑏2_𝑎2¹⁰||𝑓(𝑢)||^𝑝𝑊_𝑛(𝑡,𝑢)𝑑𝑡𝑑𝑢^1/𝑝⩽𝐶𝑛^−𝑘‖𝑓‖_{𝐿𝑝[0,1]}.(3.11) Now in view of Lemmas 2.1 and 2.8, we have the inequality ‖‖𝐽₃‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘‖𝑓‖^𝐿_𝑝^(𝐼₂⁾+‖‖𝑓^(2𝑘)‖‖_{𝐿𝑝(𝐼2)}.(3.12) Combining the estimates (3.5)–(3.12), (3.1) follows.
Now, let 𝑝=1. Then, we can expand 𝑓(𝑢) for almost all 𝑡∈𝐼₂ and for all 𝑢∈𝐼₁, as 𝑓(𝑢)−𝑓(𝑡)=^2𝑘−1_𝑗=1𝑓^(𝑗)(𝑡)𝑗!(𝑢−𝑡)^𝑗+1(2𝑘−1)!^𝑢𝑡𝜑(𝑢)(𝑢−𝑣)^2𝑘−1𝑑𝑓^(2𝑘−1)(𝑣)+𝐹(𝑢,𝑡)(1−𝜑(𝑢)),(3.13) where 𝜑(𝑢) and 𝐹(𝑢,𝑡) are defined as above. Therefore, operating by 𝑇_𝑛,𝑘 on both sides of (3.13), we obtain three terms 𝐸₄, 𝐸₅, and 𝐸₆, say corresponding to the three terms in the right-hand side of (3.13).
Now proceeding as in the case of the estimate of 𝐸₁, we have ‖‖𝐸₄‖‖_𝐿1(𝐼2)⩽𝐶𝑛^−𝑘‖𝑓‖^𝐿₁^(𝐼₂⁾+‖‖𝑓^(2𝑘−1)‖‖_𝐿1(𝐼2).(3.14)
It can easily be shown that ‖‖𝑀_𝑛(𝑓,𝑡)‖‖_𝐿1(𝐼)⩽‖𝑓(𝑡)‖_𝐿1(𝐼).(3.15) Consequently, by induction, we get ‖‖𝑀^𝑟𝑛(𝑓,𝑡)‖‖_𝐿1(𝐼)=‖‖𝑀_𝑛(𝑀^𝑛𝑟−1(𝑓,𝑢),𝑡)‖‖_𝐿1(𝐼)⩽‖𝑓(𝑡)‖_𝐿1(𝐼).(3.16) Therefore, in order to get an estimate for 𝐸₅, it is sufficient to consider the estimate for 𝐾𝐾=‖‖‖𝑀_𝑛^𝑢𝑡𝜑(𝑢)|𝑢−𝑣|^2𝑘−1𝑑𝑓^(2𝑘−1)(𝑣);𝑡‖‖‖_𝐿1(𝐼2)⩽^𝑏2_𝑎2^𝑏1_𝑎1𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘−1||||^𝑢𝑡||𝑑𝑓^2𝑘−1(𝑣)||||||𝑑𝑢𝑑𝑡.(3.17) For each 𝑛, there exists the integer 𝑟=𝑟(𝑛) s.t. 𝑟/√𝑛⩽max(𝑏₁−𝑎₂,𝑏₂−𝑎₁)⩽(𝑟+1)/√𝑛𝐾⩽^𝑟_𝑙=0^𝑏2_𝑎2^{√𝑡+(𝑙+1)/𝑛√𝑡+𝑙/}_𝑛𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘−1^{√𝑡+(𝑙+1)/𝑛𝑡}+𝜑(𝑣)||𝑑𝑓^(2𝑘−1)(𝑣)||𝑑𝑢_{√𝑡−𝑙/𝑛√𝑡−(𝑙+1)/𝑛}𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘−1^{𝑡√𝑡−(𝑙+1)/}_𝑛⩽𝜑(𝑣)||𝑑𝑓^(2𝑘−1)(𝑣)||𝑑𝑢𝑑𝑡_𝑟_𝑙=1^𝑏2_𝑎2^{√𝑡+(𝑙+1)/𝑛√𝑡+𝑙/}_𝑛𝑊_𝑛(𝑡,𝑢)𝑛²𝑙⁴|𝑢−𝑡|^2𝑘+3^{√𝑡+(𝑙+1)/𝑛𝑡}+𝜑(𝑣)||𝑑𝑓^(2𝑘−1)(𝑣)||𝑑𝑢_{√𝑡−𝑙/𝑛√𝑡−(𝑙+1)/𝑛}𝑊_𝑛(𝑡,𝑢)𝑛²𝑙⁴|𝑢−𝑡|^2𝑘+3^{𝑡√𝑡−(𝑙+1)/}_𝑛+𝜑(𝑣)||𝑑𝑓^(2𝑘−1)(𝑣)||𝑑𝑢𝑑𝑡_𝑏2𝑎2^{𝑏2√+1/𝑛𝑎}_{2√−1/𝑛}𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘−1^{𝑡+𝑛−1/2}_{𝑡−𝑛−1/2}⩽2_𝑟𝜑(𝑣)||𝑑𝑓^(2𝑘−1)||𝑑𝑢𝑑𝑡_𝑙=1𝑛²𝑙⁴𝑙+1√𝑛^𝑏1𝑎₁𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘+3𝑑𝑢‖‖𝑓^(2𝑘−1)‖‖_{𝐵𝑉(𝐼1)}+2√𝑛^𝑏1𝑎₁𝑊_𝑛(𝑡,𝑢)|𝑢−𝑡|^2𝑘−1𝑑𝑢‖‖𝑓^(2𝑘−1)‖‖_{𝐵𝑉(𝐼1)}⩽𝐶𝑛^−𝑘‖‖𝑓^(2𝑘−1)‖‖_{𝐵𝑉(𝐼1).}(applyingFubini'stheorem).(3.18)
To estimate 𝐸₆, for all 𝑢∈[0,1]⧵𝐼₁, 𝑡∈𝐼₂, we can choose a 𝛿>0 such that |𝑢−𝑡|⩾𝛿. Therefore, we get the inequality ‖‖𝐸₄‖‖_𝐿1(𝐼2)⩽𝐶^𝑏2_𝑎2¹⁰+𝑊_𝑛(𝑡,𝑢)||𝑓(𝑢)(1−𝜑(𝑢))||𝑑𝑢𝑑𝑡_2𝑘−1_𝑖=01𝑖!^𝑏2_𝑎2¹⁰=𝑆₁𝑊_𝑛(𝑡,𝑢)||𝑓^(𝑖)(𝑡)(𝑢−𝑡)^𝑖(1−𝜑(𝑢))||𝑑𝑢𝑑𝑡+𝑆₂,say.(3.19) Since, 𝑊_𝑛(𝑡,𝑢) is symmetric in 𝑡 and 𝑢, there follows 𝑆₁⩽𝐶𝑛^−𝑘‖𝑓‖_𝐿1(𝐼2).(3.20) In view of Lemma 2.8, we obtain 𝑆₂⩽𝐶𝑛^−𝑘‖𝑓‖^𝐿₁^(𝐼₂⁾+‖‖𝑓^(2𝑘−1)‖‖_𝐿1(𝐼2).(3.21) From the estimates (3.14)–(3.21) and the definition of 𝑇_𝑛,𝑘, we get (3.2).

Theorem 3.2. If 𝑝⩾1, 𝑓∈𝐿_𝑝[0,1]. Then, for all 𝑛 sufficiently large there holds ‖‖𝑇_𝑛,𝑘(𝑓;⋅)−𝑓‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝜔_2𝑘𝑓,√1𝑛,𝑝,𝐼¹,⋅+𝑛^−𝑘‖𝑓‖_{𝐿𝑝[0,1]},(3.22) where 𝐶 is a constant independent of 𝑓 and 𝑛.

Proof. In order to prove the theorem, it is sufficient to prove it for the function 𝑓𝑔, where 𝑔∈𝐶^∞0 be such that supp𝑔⊂𝐼₁ and 𝑔=1 in 𝐼₂. Let for convenience 𝑓=𝑓𝑔. Let 𝑓_𝜂,2𝑘 be the Steklov mean of order 2𝑘 corresponding to the function 𝑓, where 𝜂>0 is sufficiently small. Then, we have ‖‖𝑇_𝑛,𝑘(𝑓;⋅)−𝑓‖‖_{𝐿𝑝(𝐼2)}⩽‖‖𝑇_𝑛,𝑘(𝑓−𝑓_𝜂,2𝑘;⋅)‖‖_{𝐿𝑝(𝐼2)}+‖‖𝑇_𝑛,𝑘(𝑓_𝜂,2𝑘;⋅)−𝑓_𝜂,2𝑘‖‖_{𝐿𝑝(𝐼2)}+‖‖𝑓_𝜂,2𝑘−𝑓‖‖_{𝐿𝑝(𝐼2)}=𝐽₁+𝐽₂+𝐽₃,say.(3.23) Let 𝜑 be the characteristic function of 𝐼₃. Then, 𝑀_𝑛𝑓−𝑓^𝜂,2𝑘;𝑡==𝑀𝐽^𝑛₄𝜑(𝑢)𝑓−𝑓^𝜂,2𝑘;𝑡+𝑀^𝑛(1−𝜑(𝑢))𝑓−𝑓^𝜂,2𝑘;𝑡+𝐽₅say.(3.24) For 𝑝>1 using Hölder's inequality and then applying Fubini's theorem, we get ‖‖𝐽₄‖‖^𝑝𝐿_𝑝(𝐼2)⩽^𝑏2_𝑎2^𝑏3_𝑎3⩽𝐶‖‖𝑓−𝑊𝑓_𝑛(_𝜂,2𝑘𝑡,𝑢)||𝑓−𝑓_𝜂,2𝑘||^𝑝𝑑𝑢𝑑𝑡‖‖^𝑝𝐿_𝑝(𝐼3).(3.25) This in view of property (c) of Steklov’s mean implies ‖‖𝐽₄‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝜔_2𝑘𝑓,𝜂,𝑝,𝐼².(3.26) In the case 𝑝=1, (3.22) is obtained by boundedness of the operator 𝑀_𝑛. Now, as above in Theorem 3.1 for 𝑢∈𝐼₂⧵𝐼₃, we can choose a 𝛿>0 such that |𝑢−𝑡|⩾𝛿. Then, from Fubini's theorem and moment estimates 1 of dual operator, we get 𝐽₅⩽𝛿^−2𝑘_{𝐼2⧵𝐼3}𝑊_𝑛(𝑡,𝑢)𝑓−𝑓_𝜂,2𝑘(𝑢−𝑡)^2𝑘𝑑𝑢.(3.27) Therefore, ‖‖𝐽₅‖‖_{𝐿𝑝(𝐼2)}⩽𝐶𝑛^−𝑘‖‖𝑓−𝑓_𝜂,2𝑘‖‖_{𝐿𝑝(𝐼2)}.(3.28) Consequently, we have the estimate 𝐽₁⩽𝐶𝜔_2𝑘𝑓,𝜂,𝑝,𝐼²+𝑛^−𝑘‖‖𝑓−𝑓_𝜂,2𝑘‖‖_{𝐿𝑝(𝐼2)}.(3.29) Now using Theorem 3.1 and Lemma 2.8, 𝐽₂⩽𝐶𝑛^−𝑘‖‖𝑓^𝜂,2𝑘‖‖_{𝐿𝑝[0,1]}+‖‖𝑓^{(2𝑘)𝜂,2𝑘}‖‖_{𝐿𝑝(𝐼1)}.(3.30) In view of property (c) of Steklov’s means, we get the inequality 𝐽₃⩽𝐶𝜔_2𝑘𝑓,𝜂,𝑝,𝐼¹.(3.31) Choosing 𝜂=1/√𝑛, the result follows from the estimates of 𝐽₁-𝐽₃.

Acknowledgment

A. R. Gairola is thankful to the Council of Scientific and Industrial Research, New Delhi, India for financial support to carry out the above work.