Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 184374, 11 pages
Research Article

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

1Department of Mathematics, SMD College Poonpoon, Patna, Bihar, India
2Department of Mathematics, IIT Roorkee, Roorkee 247667, India

Received 11 November 2010; Accepted 9 December 2010

Academic Editors: O. Guès and R. Stenberg

Copyright Β© 2011 T. A. K. Sinha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 polynomials𝑀𝑛(𝑓;𝑑)=(𝑛+1)π‘›ξ“πœˆ=0𝑝𝑛,πœˆξ€œ(𝑑)10𝑝𝑛,𝜈(𝑒)𝑓(𝑒)𝑑𝑒,(1.1) where 𝑝𝑛,𝜈(𝑑)=(𝑛𝑣)π‘‘πœˆ(1βˆ’π‘‘)π‘›βˆ’πœˆ, π‘‘βˆˆ[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, π’ͺ(π‘›βˆ’1) however smooth the function may be. In order to improve this rate of approximation, we consider an iterative combination 𝑇𝑛,π‘˜(𝑓;𝑑) of the operators 𝑀𝑛(𝑓;𝑑). 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 𝐡𝑛(𝑓;𝑑). 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 𝐿𝑝-approximation by the operators 𝑇𝑛,π‘˜(𝑓;𝑑).

For π‘“βˆˆπΏπ‘[0,1], the operators 𝑀𝑛(𝑓;𝑑) can be expressed as π‘€π‘›ξ€œ(𝑓;𝑑)=10π‘Šπ‘›(𝑒,𝑑)𝑓(𝑒)𝑑𝑒,(1.2) where π‘Šπ‘›(𝑒,𝑑)=(𝑛+1)π‘›ξ“πœˆ=0𝑝𝑛,𝜈(𝑑)𝑝𝑛,𝜈(𝑒)(1.3) is the kernel of the operators.

For π‘šβˆˆπ‘0 (the set of nonnegative integers), the π‘šth order moment for the operators 𝑀𝑛 is defined as πœ‡π‘›,π‘š(𝑑)=𝑀𝑛((π‘’βˆ’π‘‘)π‘š;𝑑).(1.4)

Let 𝐿𝐡[π‘Ž,𝑏], 𝐢∞[π‘Ž,𝑏], 𝐴𝐢[π‘Ž,𝑏], and 𝐡𝑉[π‘Ž,𝑏] denote the classes of bounded Lebesgue integrable, infinitely differentiable, absolutely continuous functions, and functions of bounded variations, respectively, on the interval [π‘Ž,𝑏].

The Iterative combination 𝑇𝑛,π‘˜βˆΆπΏπ΅[0,1]β†’πΆβˆž[0,1] of the operators 𝑀𝑛(𝑓;𝑑) is defined as 𝑇𝑛,π‘˜ξ‚€ξ€·(𝑓;𝑑)=πΌβˆ’πΌβˆ’π‘€π‘›ξ€Έπ‘˜ξ‚(𝑓;𝑑)=π‘˜ξ“π‘Ÿ=1(βˆ’1)π‘Ÿ+1ξ‚΅π‘˜π‘Ÿξ‚Άπ‘€π‘Ÿπ‘›(𝑓;𝑑),π‘˜βˆˆπ‘,(1.5) where 𝑀0𝑛=𝐼 and π‘€π‘Ÿπ‘›=𝑀𝑛(π‘€π‘›π‘Ÿβˆ’1) for π‘Ÿβˆˆπ‘.

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 𝐿𝑝-approximation (1⩽𝑝<∞) by the iterative combination 𝑇𝑛,π‘˜(β‹…;𝑑) in terms of 𝐿𝑝-norm of derivatives of the function. From these estimates, we obtain a general error estimate in terms of 2π‘˜+2th integral modulus of smoothness of the function.

In what follows, we suppose that 0<π‘Ž1<π‘Ž2<π‘Ž3<𝑏3<𝑏2<𝑏1<1 and 𝐼𝑗=[π‘Žπ‘—,𝑏𝑗], 𝑗=1,2,3. Further, 𝐢 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 πœ‡π‘›,π‘š(𝑑), one has πœ‡π‘›,0(𝑑)=1, πœ‡π‘›,1(𝑑)=(1βˆ’2𝑑)/(𝑛+2), and there holds the recurrence relation (𝑛+π‘š+2)πœ‡π‘›,π‘š+1ξ€½πœ‡(𝑑)=𝑑(1βˆ’π‘‘)ξ…žπ‘›,π‘š(𝑑)+2π‘šπœ‡π‘›,π‘šβˆ’1ξ€Ύ+(𝑑)(π‘š+1)(1βˆ’2𝑑)πœ‡π‘›,π‘š(𝑑),(2.1) for π‘šβ©Ύ1.
Consequently, we have (i)πœ‡π‘›,π‘š(𝑑) is a polynomial in 𝑑 of degree π‘š, (ii)for every π‘‘βˆˆ[0,1], πœ‡π‘›,π‘š(𝑑)=π’ͺ(π‘›βˆ’[(π‘š+1)/2]), where [𝛽] is the integer part of 𝛽.

The π‘šth order moment for the operator 𝑀𝑝𝑛 is defined as πœ‡[𝑝]𝑛,π‘š(𝑑)=𝑀𝑝𝑛((π‘’βˆ’π‘‘)π‘š;𝑑), π‘βˆˆπ‘ (the set of natural numbers). We denote πœ‡[1]𝑛,π‘š(𝑑) by πœ‡π‘›,π‘š(𝑑).

Lemma 2.2 (see [1]). For the function 𝑝𝑛,𝜈(𝑑), there holds the result π‘‘π‘Ÿ(1βˆ’π‘‘)π‘Ÿπ‘‘π‘Ÿπ‘π‘›,𝜈(𝑑)π‘‘π‘‘π‘Ÿ=2𝑖+π‘—β©½π‘Ÿπ‘–,𝑗⩾0𝑛𝑖(πœˆβˆ’π‘›π‘‘)π‘—π‘žπ‘–,𝑗,π‘Ÿ(𝑑)𝑝𝑛,𝜈(𝑑),(2.2) where π‘žπ‘–,𝑗,π‘Ÿ(𝑑) are certain polynomials in 𝑑 independent of 𝑛 and 𝜈.

Lemma 2.3 (see [7]). There holds the recurrence relation πœ‡[𝑝+1]𝑛,π‘š(𝑑)=π‘šξ“π‘—=0π‘šβˆ’π‘—ξ“π‘–=0ξ‚΅π‘šπ‘—ξ‚Ά1𝐷𝑖!π‘–ξ‚€πœ‡[𝑝]𝑛,π‘šβˆ’π‘—ξ‚πœ‡(𝑑)𝑛,𝑖+𝑗(𝑑).(2.3)

Lemma 2.4 (see [7]). For π‘˜, π‘™βˆˆπ‘, there holds 𝑇𝑛,π‘˜((π‘’βˆ’π‘‘)𝑙;𝑑)=π’ͺ(π‘›βˆ’π‘˜).

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

Lemma 2.5. For π‘βˆˆπ‘, π‘šβˆˆπ‘0, and π‘‘βˆˆ[0,1], we have πœ‡[𝑝]𝑛,π‘šξ€·π‘›(𝑑)=π’ͺβˆ’[(π‘š+1)/2]ξ€Έ.(2.4)

Let π‘“βˆˆπΏπ‘[π‘Ž,𝑏], 1⩽𝑝<∞, and 𝐼1βŠ‚(π‘Ž,𝑏). Then, for sufficiently small πœ‚>0 the Steklov mean π‘“πœ‚,π‘š of π‘šth order corresponding to 𝑓 is defined as follows:π‘“πœ‚,π‘š(𝑑)=πœ‚βˆ’π‘šξ€œπœ‚/2βˆ’πœ‚/2β‹―ξ€œπœ‚/2βˆ’πœ‚/2𝑓(𝑑)+(βˆ’1)π‘šβˆ’1Ξ”π‘šβˆ‘π‘šπ‘–=1𝑑𝑖𝑓(𝑑)π‘šξ‘π‘–=1𝑑𝑑𝑖,π‘‘βˆˆπΌ1,(2.5) where Ξ”π‘šβ„Ž is the forward difference operator with step length β„Ž.

Lemma 2.6. Let π‘“βˆˆπΏπ‘[π‘Ž,𝑏], 1⩽𝑝<∞, and 𝐼1βŠ‚(π‘Ž,𝑏). Then, for the function π‘“πœ‚,π‘š, we have (a)π‘“πœ‚,π‘š has derivatives up to order π‘š over 𝐼1, (b)‖𝑓(π‘Ÿ)πœ‚,π‘šβ€–πΏπ‘(𝐼1)β©½πΆπœ‚βˆ’π‘Ÿπœ”π‘Ÿ(𝑓,πœ‚,[π‘Ž,𝑏]), π‘Ÿ=1,2,…,π‘š,(c)β€–π‘“βˆ’π‘“πœ‚,π‘šβ€–πΏπ‘(𝐼1)β©½πΆπœ”π‘š(𝑓,πœ‚,[π‘Ž,𝑏]), (d)β€–π‘“πœ‚,π‘šβ€–πΏπ‘(𝐼1)β©½πΆπœ‚βˆ’π‘šβ€–π‘“β€–πΏπ‘[π‘Ž,𝑏], (e)‖𝑓(π‘Ÿ)πœ‚,π‘šβ€–πΏπ‘(𝐼1)⩽𝐢‖𝑓‖𝐿𝑝[π‘Ž,𝑏], where 𝐢 is a constant that depends on 𝑖 but is independent of 𝑓 and πœ‚.

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

Let π‘“βˆˆπΏπ‘[0,π‘Ž], 1⩽𝑝<∞. Then, the Hardy-Littlewood majorant β„Žπ‘“(π‘₯) of the function 𝑓 is defined asβ„Žπ‘“(π‘₯)=supπœ‰β‰ π‘₯1ξ€œπœ‰βˆ’π‘₯πœ‰π‘₯𝑓(𝑑)𝑑𝑑.(2.6)

Lemma 2.7. If 1<𝑝<∞, π‘“βˆˆπΏπ‘[0,π‘Ž], then β„Žπ‘“βˆˆπΏπ‘[0,π‘Ž] and β€–β€–β„Žπ‘“β€–β€–πΏπ‘[0,π‘Ž]β©½21/π‘π‘π‘βˆ’1‖𝑓‖𝐿𝑝[0,π‘Ž].(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 𝐿𝑝-norm.

Lemma 2.8 (see [11]). Let 1⩽𝑝<∞, π‘“βˆˆπΏπ‘[π‘Ž,𝑏]. Suppose 𝑓(π‘˜)∈𝐴𝐢[π‘Ž,𝑏] and 𝑓(π‘˜+1)βˆˆπΏπ‘[π‘Ž,𝑏]. Then, ‖‖𝑓(𝑗)‖‖𝐿𝑝[π‘Ž,𝑏]⩽𝐾𝑗‖‖𝑓(π‘˜+1)‖‖𝐿𝑝[π‘Ž,𝑏]+‖𝑓‖𝐿𝑝[π‘Ž,𝑏],𝑗=1,2,…,π‘˜,(2.8) where 𝐾𝑗 are certain constants independent of 𝑓.

The dual operator ξ‚Šπ‘€π‘› corresponding to the operator 𝑀𝑛 is defined as ξ‚Šπ‘€π‘›ξ€œ(𝑓;𝑒)=10π‘Šπ‘›(𝑒,𝑑)𝑓(𝑑)𝑑𝑑.(2.9) Then, the corresponding π‘šth order moment is given by ξπœ‡π‘›,π‘šξ‚Šπ‘€(𝑒)=𝑛((π‘‘βˆ’π‘’)π‘šβˆ«;𝑒)=10π‘Šπ‘›(𝑒,𝑑)(π‘‘βˆ’π‘’)π‘šπ‘‘π‘‘.

Lemma 2.9. For the function ξπœ‡π‘›,π‘š(𝑒), there holds the recurrence relation (π‘›βˆ’π‘šβˆ’2)ξπœ‡π‘›,π‘š+1(𝑒)=𝑒(1βˆ’π‘’)ξπœ‡ξ…žπ‘›,π‘š(𝑒)+(π‘š+1)(1βˆ’2𝑒)ξπœ‡π‘›,π‘šξ€·(𝑒)+2π‘šπ‘’βˆ’π‘’2ξ€Έξπœ‡π‘›,π‘šβˆ’1(𝑒).(2.10)

Proof. In view of the relation 𝑒(1βˆ’π‘’)π‘ξ…žπ‘›,π‘˜(𝑒)=(π‘˜βˆ’π‘›π‘’)𝑝𝑛,π‘˜(𝑒), we get 𝑒(1βˆ’π‘’)ξπœ‡ξ…žπ‘›,π‘š=(𝑒)(𝑛+1)π‘›ξ“π‘˜=0ξ€œ10𝑝𝑛,π‘˜(𝑑)𝑒(1βˆ’π‘’)π‘ξ…žπ‘›,π‘˜(𝑒)(π‘‘βˆ’π‘’)π‘šπ‘‘π‘‘βˆ’π‘šπ‘’(1βˆ’π‘’)ξπœ‡π‘›,π‘šβˆ’1(𝑒)=(𝑛+1)π‘›ξ“π‘˜=0(π‘˜βˆ’π‘›π‘’)𝑝𝑛,π‘˜ξ€œ(𝑒)10𝑝𝑛,π‘˜(𝑑)(π‘‘βˆ’π‘’)π‘šπ‘‘π‘‘βˆ’π‘šπ‘’(1βˆ’π‘’)ξπœ‡π‘›,π‘šβˆ’1(𝑒)=(𝑛+1)π‘›ξ“π‘˜=0𝑝𝑛,π‘˜ξ€œ(𝑒)10((π‘˜βˆ’π‘›π‘‘)+𝑛(π‘‘βˆ’π‘’))𝑝𝑛,π‘˜(𝑑)(π‘‘βˆ’π‘’)π‘šπ‘‘π‘‘βˆ’π‘šπ‘’(1βˆ’π‘’)ξπœ‡π‘›,π‘šβˆ’1(𝑒)=(𝑛+1)π‘›ξ“π‘˜=0𝑝𝑛,π‘˜ξ€œ(𝑒)10ξ€·π‘‘βˆ’π‘‘2ξ€Έπ‘ξ…žπ‘›,π‘˜(𝑑)(π‘‘βˆ’π‘’)π‘šπ‘‘π‘‘+π‘›ξπœ‡π‘›,π‘š+1(𝑒)βˆ’π‘šπ‘’(1βˆ’π‘’)ξπœ‡π‘›,π‘šβˆ’1(𝑒).(2.11) Expanding (π‘‘βˆ’π‘‘2) as a polynomial in (π‘‘βˆ’π‘’) and integrating by parts, we get 𝑒(1βˆ’π‘’)ξπœ‡ξ…žπ‘›,π‘š(𝑒)=βˆ’(𝑛+1)π‘›ξ“π‘˜=0𝑝𝑛,π‘˜ξ€œ(𝑒)10𝑝𝑛,π‘˜ξ€Ίξ€·(𝑑)(π‘š+1)(1βˆ’2𝑒)(π‘‘βˆ’π‘’)+π‘šπ‘’βˆ’π‘’2ξ€Έβˆ’(π‘š+2)(π‘‘βˆ’π‘’)2ξ€»Γ—(π‘‘βˆ’π‘’)π‘šβˆ’1𝑑𝑑+π‘›ξπœ‡π‘›,π‘š+1(𝑒)βˆ’π‘šπ‘’(1βˆ’π‘’)ξπœ‡π‘›,π‘šβˆ’1(𝑒)=βˆ’(π‘š+1)(1βˆ’2𝑒)ξπœ‡π‘›,π‘šξ€·(𝑒)βˆ’2π‘šπ‘’βˆ’π‘’2ξ€Έξπœ‡π‘›,π‘šβˆ’1(𝑒)βˆ’(π‘š+2)ξπœ‡π‘›,π‘š+1(𝑒)+π‘›ξπœ‡π‘›,π‘š+1(𝑒).(2.12)
Rearrangement of the terms gives (2.10).

Remark 2.10. From (2.10), it follows that ξπœ‡π‘›,π‘š(𝑒)=π’ͺ(π‘›βˆ’[(π‘š+1)/2]), where [𝛽] is the integer part of 𝛽.

3. Main Result

In this section, we obtain an error estimate in terms of 𝐿𝑝 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, π‘“βˆˆπΏπ‘[0,1], 𝑓 has derivatives of order 2π‘˜ on 𝐼1 with 𝑓(2π‘˜βˆ’1)∈𝐴𝐢(𝐼1), and 𝑓(2π‘˜)βˆˆπΏπ‘(𝐼1), then for sufficiently large 𝑛‖‖𝑇𝑛,π‘˜β€–β€–(𝑓;β‹…)βˆ’π‘“(β‹…)𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜ξ‚†β€–β€–π‘“(2π‘˜)‖‖𝐿𝑝(𝐼1)+‖𝑓‖𝐿𝑝[0,1].(3.1) Moreover, if π‘“βˆˆπΏ1[0,1], 𝑓 has derivatives up to the order (2π‘˜βˆ’1) on 𝐼1 with 𝑓(2π‘˜βˆ’2)∈𝐴𝐢(𝐼1), and 𝑓(2π‘˜βˆ’1)βˆˆπ΅π‘‰(𝐼1), then for sufficiently large 𝑛 there holds ‖‖𝑇𝑛,π‘˜β€–β€–(𝑓;β‹…)βˆ’π‘“(β‹…)𝐿1(𝐼2)β©½πΆπ‘›βˆ’π‘˜ξ‚†β€–β€–π‘“(2π‘˜βˆ’1)‖‖𝐡𝑉(𝐼1)+‖‖𝑓(2π‘˜βˆ’1)‖‖𝐿1(𝐼2)+‖𝑓‖𝐿1(𝐼2),(3.2) where 𝐢 is a certain constant independent of 𝑓 and 𝑛.

Proof. Let 𝑝>1, then for all π‘’βˆˆπΌ1 and π‘‘βˆˆπΌ2, we can write 𝑓(𝑒)βˆ’π‘“(𝑑)=2π‘˜βˆ’1𝑗=1𝑓(𝑗)(𝑑)𝑗!(π‘’βˆ’π‘‘)𝑗+1ξ€œ(2π‘˜βˆ’1)!π‘’π‘‘πœ‘(𝑒)(π‘’βˆ’π‘£)2π‘˜βˆ’1𝑓(2π‘˜)(𝑣)𝑑𝑣+𝐹(𝑒,𝑑)(1βˆ’πœ‘(𝑒)),(3.3) where πœ‘(𝑒) is the characteristic function of the interval 𝐼1 and 𝐹(𝑒,𝑑)=𝑓(𝑒)βˆ’2π‘˜βˆ’1𝑗=0𝑓(𝑗)(𝑑)𝑗!(π‘’βˆ’π‘‘)𝑗.(3.4) Therefore, operating by 𝑇𝑛,π‘˜ 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)β©½πΆπ‘›βˆ’π‘˜ξ‚€β€–β€–π‘“(2π‘˜)‖‖𝐿𝑝(𝐼2)+‖𝑓‖𝐿𝑝(𝐼2).(3.5) Let β„Žπ‘“(2π‘˜) be the Hardy-Littleood majorant of 𝑓(2π‘˜) on 𝐼1. Then, in order to estimate 𝐸2, it is sufficient to consider the estimate for 𝐽1𝐽1=π‘€π‘›ξ€·πœ‘(𝑒)(π‘’βˆ’π‘‘)2π‘˜||β„Žπ‘“(2π‘˜)||ξ€Έ.(𝑒);𝑑(3.6) Applying HΓΆlder's inequality, Lemma 2.1, and then Fubini's theorem, we get ‖‖𝐽1‖‖𝑝𝐿𝑝(𝐼2)β©½ξ€·π‘€π‘›ξ€·πœ‘(𝑒)(π‘’βˆ’π‘‘)2π‘˜π‘ž;𝑑𝑝/π‘žΓ—π‘€π‘›ξ€·πœ‘(𝑒)(π‘’βˆ’π‘‘)𝑝||β„Žπ‘“(2π‘˜)||(𝑒)𝑝;π‘‘β©½πΆπ‘›βˆ’π‘˜π‘ξ€œπ‘1π‘Ž1ξ€œπ‘2π‘Ž2π‘Šπ‘›||β„Ž(𝑑,𝑒)𝑓(2π‘˜)||(𝑒)𝑝𝑑𝑑𝑑𝑒.(3.7) Now, in view of Lemmas 2.1 and 2.7, we have ‖‖𝐽1‖‖𝑝𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜π‘ξ€œπ‘1π‘Ž1||β„Žπ‘“(2π‘˜)||(𝑒)π‘π‘‘π‘’β©½πΆπ‘›βˆ’π‘˜π‘β€–β€–π‘“(2π‘˜)‖‖𝑝𝐿𝑝(𝐼1).(3.8) Consequently, ‖‖𝐸2‖‖𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜β€–β€–π‘“(2π‘˜)‖‖𝐿𝑝(𝐼1).(3.9) For π‘’βˆˆ[0,1]⧡𝐼1, π‘‘βˆˆπΌ2, we can find a 𝛿>0 such that |π‘’βˆ’π‘‘|⩾𝛿. Thus, ||𝑀𝑛||=|||||𝑀(𝐹(𝑒,𝑑)(1βˆ’πœ‘(𝑒));𝑑)𝑛𝑓(𝑒)βˆ’2π‘˜βˆ’1𝑗=0𝑓(𝑗)(𝑑)𝑗!(π‘’βˆ’π‘‘)𝑗ξƒͺξƒͺ|||||(1βˆ’πœ‘(𝑒));π‘‘β©½π›Ώβˆ’2π‘˜ξƒ―π‘€π‘›ξ€·||||𝑓(𝑒)(π‘’βˆ’π‘‘)2π‘˜ξ€Έ+;𝑑2π‘˜βˆ’1𝑗=0||𝑓(𝑗)||(𝑑)𝑀𝑗!𝑛|π‘’βˆ’π‘‘|2π‘˜+𝑗;𝑑=𝐽2+𝐽3,say.(3.10) On an application of HΓΆlder's inequality, Lemma 2.1, and Fubini's theorem, we get ‖‖𝐽2‖‖𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜ξ‚΅ξ€œπ‘2π‘Ž2ξ€œ10||||𝑓(𝑒)π‘π‘Šπ‘›ξ‚Ά(𝑑,𝑒)𝑑𝑑𝑑𝑒1/π‘β©½πΆπ‘›βˆ’π‘˜β€–π‘“β€–πΏπ‘[0,1].(3.11) Now in view of Lemmas 2.1 and 2.8, we have the inequality ‖‖𝐽3‖‖𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜ξ‚€β€–π‘“β€–πΏπ‘(𝐼2)+‖‖𝑓(2π‘˜)‖‖𝐿𝑝(𝐼2).(3.12) Combining the estimates (3.5)–(3.12), (3.1) follows.
Now, let 𝑝=1. Then, we can expand 𝑓(𝑒) for almost all π‘‘βˆˆπΌ2 and for all π‘’βˆˆπΌ1, as 𝑓(𝑒)βˆ’π‘“(𝑑)=2π‘˜βˆ’1𝑗=1𝑓(𝑗)(𝑑)𝑗!(π‘’βˆ’π‘‘)𝑗+1ξ€œ(2π‘˜βˆ’1)!π‘’π‘‘πœ‘(𝑒)(π‘’βˆ’π‘£)2π‘˜βˆ’1𝑑𝑓(2π‘˜βˆ’1)(𝑣)+𝐹(𝑒,𝑑)(1βˆ’πœ‘(𝑒)),(3.13) where πœ‘(𝑒) and 𝐹(𝑒,𝑑) are defined as above. Therefore, operating by 𝑇𝑛,π‘˜ 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)β©½πΆπ‘›βˆ’π‘˜ξ‚€β€–π‘“β€–πΏ1(𝐼2)+‖‖𝑓(2π‘˜βˆ’1)‖‖𝐿1(𝐼2).(3.14)
It can easily be shown that ‖‖𝑀𝑛‖‖(𝑓,𝑑)𝐿1(𝐼)⩽‖𝑓(𝑑)‖𝐿1(𝐼).(3.15) Consequently, by induction, we get β€–β€–π‘€π‘Ÿπ‘›β€–β€–(𝑓,𝑑)𝐿1(𝐼)=‖‖𝑀𝑛(π‘€π‘›π‘Ÿβˆ’1β€–β€–(𝑓,𝑒),𝑑)𝐿1(𝐼)⩽‖𝑓(𝑑)‖𝐿1(𝐼).(3.16) Therefore, in order to get an estimate for 𝐸5, it is sufficient to consider the estimate for 𝐾‖‖‖𝑀𝐾=π‘›ξ‚΅ξ€œπ‘’π‘‘πœ‘(𝑒)|π‘’βˆ’π‘£|2π‘˜βˆ’1𝑑𝑓(2π‘˜βˆ’1)ξ‚Άβ€–β€–β€–(𝑣);𝑑𝐿1(𝐼2)β©½ξ€œπ‘2π‘Ž2ξƒ©ξ€œπ‘1π‘Ž1π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜βˆ’1||||ξ€œπ‘’π‘‘||𝑑𝑓2π‘˜βˆ’1||||||ξƒͺ(𝑣)𝑑𝑒𝑑𝑑.(3.17) For each 𝑛, there exists the integer π‘Ÿ=π‘Ÿ(𝑛) s.t. βˆšπ‘Ÿ/𝑛⩽max(𝑏1βˆ’π‘Ž2,𝑏2βˆ’π‘Ž1√)β©½(π‘Ÿ+1)/π‘›πΎβ©½π‘Ÿξ“π‘™=0ξ€œπ‘2π‘Ž2ξƒ―ξ€œβˆšπ‘‘+(𝑙+1)/π‘›βˆšπ‘‘+𝑙/π‘›π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜βˆ’1ξ€œβˆšπ‘‘+(𝑙+1)/𝑛𝑑||πœ‘(𝑣)𝑑𝑓(2π‘˜βˆ’1)||+ξ€œ(𝑣)π‘‘π‘’βˆšπ‘‘βˆ’π‘™/π‘›βˆšπ‘‘βˆ’(𝑙+1)/π‘›π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜βˆ’1ξ€œπ‘‘βˆšπ‘‘βˆ’(𝑙+1)/π‘›πœ‘||(𝑣)𝑑𝑓(2π‘˜βˆ’1)||ξƒ°β©½(𝑣)π‘‘π‘’π‘‘π‘‘π‘Ÿξ“π‘™=1ξ€œπ‘2π‘Ž2ξƒ―ξ€œβˆšπ‘‘+(𝑙+1)/π‘›βˆšπ‘‘+𝑙/π‘›π‘Šπ‘›π‘›(𝑑,𝑒)2𝑙4|π‘’βˆ’π‘‘|2π‘˜+3ξ€œβˆšπ‘‘+(𝑙+1)/𝑛𝑑||πœ‘(𝑣)𝑑𝑓(2π‘˜βˆ’1)||+ξ€œ(𝑣)π‘‘π‘’βˆšπ‘‘βˆ’π‘™/π‘›βˆšπ‘‘βˆ’(𝑙+1)/π‘›π‘Šπ‘›π‘›(𝑑,𝑒)2𝑙4|π‘’βˆ’π‘‘|2π‘˜+3ξ€œπ‘‘βˆšπ‘‘βˆ’(𝑙+1)/𝑛||πœ‘(𝑣)𝑑𝑓(2π‘˜βˆ’1)||ξƒ°+ξ€œ(𝑣)𝑑𝑒𝑑𝑑𝑏2π‘Ž2ξ€œπ‘2√+1/π‘›π‘Ž2βˆšβˆ’1/π‘›π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜βˆ’1ξ€œπ‘‘+π‘›βˆ’1/2π‘‘βˆ’π‘›βˆ’1/2||πœ‘(𝑣)𝑑𝑓(2π‘˜βˆ’1)||𝑑𝑒𝑑𝑑⩽2π‘Ÿξ“π‘™=1𝑛2𝑙4𝑙+1βˆšπ‘›ξ€œπ‘1π‘Ž1π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜+3‖‖𝑓𝑑𝑒(2π‘˜βˆ’1)‖‖𝐡𝑉(𝐼1)+2βˆšπ‘›ξ€œπ‘1π‘Ž1π‘Šπ‘›(𝑑,𝑒)|π‘’βˆ’π‘‘|2π‘˜βˆ’1‖‖𝑓𝑑𝑒(2π‘˜βˆ’1)‖‖𝐡𝑉(𝐼1)β©½πΆπ‘›βˆ’π‘˜β€–β€–π‘“(2π‘˜βˆ’1)‖‖𝐡𝑉(𝐼1).(applyingFubini'stheorem).(3.18)
To estimate 𝐸6, for all π‘’βˆˆ[0,1]⧡𝐼1, π‘‘βˆˆπΌ2, we can choose a 𝛿>0 such that |π‘’βˆ’π‘‘|⩾𝛿. Therefore, we get the inequality ‖‖𝐸4‖‖𝐿1(𝐼2)ξ€œβ©½πΆπ‘2π‘Ž2ξ€œ10π‘Šπ‘›||||+(𝑑,𝑒)𝑓(𝑒)(1βˆ’πœ‘(𝑒))𝑑𝑒𝑑𝑑2π‘˜βˆ’1𝑖=01ξ€œπ‘–!𝑏2π‘Ž2ξ€œ10π‘Šπ‘›||𝑓(𝑑,𝑒)(𝑖)(𝑑)(π‘’βˆ’π‘‘)𝑖||(1βˆ’πœ‘(𝑒))𝑑𝑒𝑑𝑑=𝑆1+𝑆2,say.(3.19) Since, π‘Šπ‘›(𝑑,𝑒) is symmetric in 𝑑 and 𝑒, there follows 𝑆1β©½πΆπ‘›βˆ’π‘˜β€–π‘“β€–πΏ1(𝐼2).(3.20) In view of Lemma 2.8, we obtain 𝑆2β©½πΆπ‘›βˆ’π‘˜ξ‚€β€–π‘“β€–πΏ1(𝐼2)+‖‖𝑓(2π‘˜βˆ’1)‖‖𝐿1(𝐼2).(3.21) From the estimates (3.14)–(3.21) and the definition of 𝑇𝑛,π‘˜, we get (3.2).

Theorem 3.2. If 𝑝⩾1, π‘“βˆˆπΏπ‘[0,1]. Then, for all 𝑛 sufficiently large there holds ‖‖𝑇𝑛,π‘˜β€–β€–(𝑓;β‹…)βˆ’π‘“πΏπ‘(𝐼2)ξƒ©πœ”β©½πΆ2π‘˜ξƒ©1𝑓,βˆšπ‘›,𝑝,𝐼1ξƒͺ,β‹…+π‘›βˆ’π‘˜β€–π‘“β€–πΏπ‘[0,1]ξƒͺ,(3.22) where 𝐢 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 π‘”βˆˆπΆβˆž0 be such that suppπ‘”βŠ‚πΌ1 and 𝑔=1 in 𝐼2. Let for convenience 𝑓=𝑓𝑔. Let ξπ‘“πœ‚,2π‘˜ be the Steklov mean of order 2π‘˜ corresponding to the function 𝑓, where πœ‚>0 is sufficiently small. Then, we have ‖‖𝑇𝑛,π‘˜(𝑓‖‖𝑓;β‹…)βˆ’πΏπ‘(𝐼2)⩽‖‖𝑇𝑛,π‘˜(ξξπ‘“π‘“βˆ’πœ‚,2π‘˜β€–β€–;β‹…)𝐿𝑝(𝐼2)+‖‖𝑇𝑛,π‘˜(ξπ‘“πœ‚,2π‘˜ξπ‘“;β‹…)βˆ’πœ‚,2π‘˜β€–β€–πΏπ‘(𝐼2)+β€–β€–ξπ‘“πœ‚,2π‘˜βˆ’ξπ‘“β€–β€–πΏπ‘(𝐼2)=𝐽1+𝐽2+𝐽3,say.(3.23) Let πœ‘ be the characteristic function of 𝐼3. Then, π‘€π‘›ξ‚€ξξπ‘“π‘“βˆ’πœ‚,2π‘˜ξ‚;𝑑=π‘€π‘›ξ‚€ξ‚€ξξπ‘“πœ‘(𝑒)π‘“βˆ’πœ‚,2π‘˜ξ‚ξ‚;𝑑+𝑀𝑛𝑓(1βˆ’πœ‘(𝑒))π‘“βˆ’πœ‚,2π‘˜ξ‚ξ‚;𝑑=𝐽4+𝐽5say.(3.24) For 𝑝>1 using HΓΆlder's inequality and then applying Fubini's theorem, we get ‖‖𝐽4‖‖𝑝𝐿𝑝(𝐼2)β©½ξ€œπ‘2π‘Ž2ξ€œπ‘3π‘Ž3π‘Šπ‘›||𝑓(𝑑,𝑒)π‘“βˆ’πœ‚,2π‘˜||π‘β€–β€–ξξπ‘“π‘‘π‘’π‘‘π‘‘β©½πΆπ‘“βˆ’πœ‚,2π‘˜β€–β€–π‘πΏπ‘(𝐼3).(3.25) This in view of property (c) of Steklov’s mean implies ‖‖𝐽4‖‖𝐿𝑝(𝐼2)β©½πΆπœ”2π‘˜ξ‚€ξπ‘“,πœ‚,𝑝,𝐼2.(3.26) In the case 𝑝=1, (3.22) is obtained by boundedness of the operator 𝑀𝑛. Now, as above in Theorem 3.1 for π‘’βˆˆπΌ2⧡𝐼3, we can choose a 𝛿>0 such that |π‘’βˆ’π‘‘|⩾𝛿. Then, from Fubini's theorem and moment estimates 1 of dual operator, we get 𝐽5β©½π›Ώβˆ’2π‘˜ξ€œπΌ2⧡𝐼3π‘Šπ‘›(𝑓𝑑,𝑒)π‘“βˆ’πœ‚,2π‘˜ξ‚(π‘’βˆ’π‘‘)2π‘˜π‘‘π‘’.(3.27) Therefore, ‖‖𝐽5‖‖𝐿𝑝(𝐼2)β©½πΆπ‘›βˆ’π‘˜β€–β€–ξξπ‘“π‘“βˆ’πœ‚,2π‘˜β€–β€–πΏπ‘(𝐼2).(3.28) Consequently, we have the estimate 𝐽1ξ‚€πœ”β©½πΆ2π‘˜ξ‚€ξπ‘“,πœ‚,𝑝,𝐼2+π‘›βˆ’π‘˜β€–β€–ξξπ‘“π‘“βˆ’πœ‚,2π‘˜β€–β€–πΏπ‘(𝐼2).(3.29) Now using Theorem 3.1 and Lemma 2.8, 𝐽2β©½πΆπ‘›βˆ’π‘˜ξ‚΅β€–β€–ξπ‘“πœ‚,2π‘˜β€–β€–πΏπ‘[0,1]+‖‖𝑓(2π‘˜)πœ‚,2π‘˜β€–β€–πΏπ‘(𝐼1)ξ‚Ά.(3.30) In view of property (c) of Steklov’s means, we get the inequality 𝐽3β©½πΆπœ”2π‘˜ξ‚€ξπ‘“,πœ‚,𝑝,𝐼1.(3.31) Choosing βˆšπœ‚=1/𝑛, the result follows from the estimates of 𝐽1-𝐽3.


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.


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