# OnLp-Approximation by Iterative Combination of Bernstein-Durrmeyer Polynomials

Authors: T. A. K. Sinha; P. N. Agrawal; Asha Ram Gairola
Journal: ISRN Mathematical Analysis
 (2011)

---

## Abstract

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


---

## Body

## 1. Introduction

The Bernstein-Durrmeyer polynomials(1.1)Mn(f;t)=(n+1)∑ν=0npn,ν(t)∫01pn,ν(u)f(u)du,
where pn,ν(t)=(nv)tν(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 Tn,k(f;t) of the operators Mn(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 Bn(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 Lp-approximation by the operators Tn,k(f;t).Forf∈Lp[0,1], the operators Mn(f;t) can be expressed as (1.2)Mn(f;t)=∫01Wn(u,t)f(u)du,
where (1.3)Wn(u,t)=(n+1)∑ν=0npn,ν(t)pn,ν(u)
is the kernel of the operators.Form∈N0 (the set of nonnegative integers), the mth order moment for the operators Mn is defined as (1.4)μn,m(t)=Mn((u-t)m;t).LetLB[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 combinationTn,k:LB[0,1]→C∞[0,1] of the operators Mn(f;t) is defined as (1.5)Tn,k(f;t)=(I-(I-Mn)k)(f;t)=∑r=1k(-1)r+1(kr)Mnr(f;t),k∈N,
where Mn0=I and Mnr=Mn(Mnr-1) for r∈N.In Section2 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,k(·;t) in terms of Lp-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 that0<a1<a2<a3<b3<b2<b1<1 and Ij=[aj,bj], 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 functionμn,m(t), one has μn,0(t)=1, μn,1(t)=(1-2t)/(n+2), and there holds the recurrence relation
(2.1)(n+m+2)μn,m+1(t)=t(1-t){μn,m′(t)+2mμn,m-1(t)}+(m+1)(1-2t)μn,m(t),
for m⩾1.
Consequently, we have(i)
μn,m(t) is a polynomial in t of degree m,(ii)
for everyt∈[0,1], μn,m(t)=𝒪(n-[(m+1)/2]), where [β] is the integer part of β.Themth order moment for the operator Mnp is defined as μn,m[p](t)=Mnp((u-t)m;t), p∈N (the set of natural numbers). We denote μn,m[1](t) by μn,m(t).Lemma 2.2 (see [1]).
For the functionpn,ν(t), there holds the result
(2.2)tr(1-t)rdrpn,ν(t)dtr=∑2i+j⩽ri,j⩾0⁡ni(ν-nt)jqi,j,r(t)pn,ν(t),
where qi,j,r(t) are certain polynomials in t independent of n and ν.Lemma 2.3 (see [7]).
There holds the recurrence relation(2.3)μn,m[p+1](t)=∑j=0m∑i=0m-j(mj)1i!Di(μn,m-j[p](t))μn,i+j(t).Lemma 2.4 (see [7]).
Fork, l∈N, there holds Tn,k((u-t)l;t)=𝒪(n-k).Using Lemmas2.1 and 2.3, we can prove the following.Lemma 2.5.
Forp∈N, m∈N0, and t∈[0,1], we have
(2.4)μn,m[p](t)=O(n-[(m+1)/2]).Letf∈Lp[a,b], 1⩽p<∞, and I1⊂(a,b). Then, for sufficiently small η>0 the Steklov mean fη,m of mth order corresponding to f is defined as follows:(2.5)fη,m(t)=η-m∫-η/2η/2⋯∫-η/2η/2(f(t)+(-1)m-1Δ∑i=1mtimf(t))∏i=1mdti,t∈I1,
where Δhm is the forward difference operator with step length h.Lemma 2.6.
Letf∈Lp[a,b], 1⩽p<∞, and I1⊂(a,b). Then, for the function fη,m, we have (a)
fη,m has derivatives up to order m over I1,(b)
∥fη,m(r)∥Lp(I1)⩽Cη-rωr(f,η,[a,b]), r=1,2,…,m,(c)
∥f-fη,m∥Lp(I1)⩽Cωm(f,η,[a,b]),(d)
∥fη,m∥Lp(I1)⩽Cη-m∥f∥Lp[a,b],(e)
∥fη,m(r)∥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.Letf∈Lp[0,a], 1⩽p<∞. Then, the Hardy-Littlewood majorant hf(x) of the function f is defined as(2.6)hf(x)=sup⁡ξ≠x1ξ-x∫xξf(t)dt.Lemma 2.7.
If1<p<∞, f∈Lp[0,a], then hf∈Lp[0,a] and
(2.7)‖hf‖Lp[0,a]⩽21/ppp-1‖f‖Lp[0,a].The lemma follows from [10, page 32].The next lemma gives a bound for the intermediate derivatives off in terms of the highest-order derivative and the function in Lp-norm.Lemma 2.8 (see [11]).
Let1⩽p<∞, f∈Lp[a,b]. Suppose f(k)∈AC[a,b] and f(k+1)∈Lp[a,b]. Then,
(2.8)‖f(j)‖Lp[a,b]⩽Kj(‖f(k+1)‖Lp[a,b]+‖f‖Lp[a,b]),j=1,2,…,k,
where Kj are certain constants independent of f.The dual operatorM̂n corresponding to the operator Mn is defined as (2.9)M̂n(f;u)=∫01Wn(u,t)f(t)dt.
Then, the corresponding mth order moment is given by μ̂n,m(u)=M̂n((t-u)m;u)=∫01Wn(u,t)(t-u)mdt.Lemma 2.9.
For the functionμ̂n,m(u), there holds the recurrence relation
(2.10)(n-m-2)μ̂n,m+1(u)=u(1-u)μ̂n,m′(u)+(m+1)(1-2u)μ̂n,m(u)+2m(u-u2)μ̂n,m-1(u).Proof.
In view of the relationu(1-u)pn,k′(u)=(k-nu)pn,k(u), we get
(2.11)u(1-u)μ̂n,m′(u)=(n+1)∑k=0n∫01pn,k(t)u(1-u)pn,k′(u)(t-u)mdt-mu(1-u)μ̂n,m-1(u)=(n+1)∑k=0n(k-nu)pn,k(u)∫01pn,k(t)(t-u)mdt-mu(1-u)μ̂n,m-1(u)=(n+1)∑k=0npn,k(u)∫01((k-nt)+n(t-u))pn,k(t)(t-u)mdt-mu(1-u)μ̂n,m-1(u)=(n+1)∑k=0npn,k(u)∫01(t-t2)pn,k′(t)(t-u)mdt+nμ̂n,m+1(u)-mu(1-u)μ̂n,m-1(u).
Expanding (t-t2) as a polynomial in (t-u) and integrating by parts, we get
(2.12)u(1-u)μ̂n,m′(u)=-(n+1)∑k=0npn,k(u)∫01pn,k(t)[(m+1)(1-2u)(t-u)+m(u-u2)-(m+2)(t-u)2]×(t-u)m-1dt+nμ̂n,m+1(u)-mu(1-u)μ̂n,m-1(u)=-(m+1)(1-2u)μ̂n,m(u)-2m(u-u2)μ̂n,m-1(u)-(m+2)μ̂n,m+1(u)+nμ̂n,m+1(u).
Rearrangement of the terms gives (2.10).Remark 2.10.
From (2.10), it follows that μ̂n,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 ofLp 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.
Ifp>1, f∈Lp[0,1], f has derivatives of order 2k on I1 with f(2k-1)∈AC(I1), and f(2k)∈Lp(I1), then for sufficiently large n(3.1)‖Tn,k(f;⋅)-f(⋅)‖Lp(I2)⩽Cn-k{‖f(2k)‖Lp(I1)+‖f‖Lp[0,1]}.
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
(3.2)‖Tn,k(f;⋅)-f(⋅)‖L1(I2)⩽Cn-k{‖f(2k-1)‖BV(I1)+‖f(2k-1)‖L1(I2)+‖f‖L1(I2)},
where C is a certain constant independent of f and n.Proof.
Letp>1, then for all u∈I1 and t∈I2, we can write
(3.3)f(u)-f(t)=∑j=12k-1f(j)(t)j!(u-t)j+1(2k-1)!∫tuφ(u)(u-v)2k-1f(2k)(v)dv+F(u,t)(1-φ(u)),
where φ(u) is the characteristic function of the interval I1 and
(3.4)F(u,t)=f(u)-∑j=02k-1f(j)(t)j!(u-t)j.
Therefore, operating by Tn,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 Lemmas2.4 and 2.8, we get
(3.5)‖E1‖Lp(I2)⩽Cn-k(‖f(2k)‖Lp(I2)+‖f‖Lp(I2)).
Let hf(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 J1(3.6)J1=Mn(φ(u)(u-t)2k|hf(2k)(u)|;t).
Applying Hölder's inequality, Lemma 2.1, and then Fubini's theorem, we get
(3.7)‖J1‖Lp(I2)p⩽(Mn(φ(u)(u-t)2kq;t))p/q×Mn(φ(u)(u-t)p|hf(2k)(u)|p;t)⩽Cn-kp∫a1b1∫a2b2Wn(t,u)|hf(2k)(u)|pdtdu.
Now, in view of Lemmas 2.1 and 2.7, we have
(3.8)‖J1‖Lp(I2)p⩽Cn-kp∫a1b1|hf(2k)(u)|pdu⩽Cn-kp‖f(2k)‖Lp(I1)p.
Consequently,
(3.9)‖E2‖Lp(I2)⩽Cn-k‖f(2k)‖Lp(I1).
For u∈[0,1]∖I1, t∈I2, we can find a δ>0 such that |u-t|⩾δ. Thus,
(3.10)|Mn(F(u,t)(1-φ(u));t)|=|Mn((f(u)-∑j=02k-1f(j)(t)j!(u-t)j)(1-φ(u));t)|⩽δ-2k{∑mmmmMn(|f(u)|(u-t)2k;t)+∑j=02k-1|f(j)(t)|j!Mn(|u-t|2k+j;t)}=J2+J3,say.
On an application of Hölder's inequality, Lemma 2.1, and Fubini's theorem, we get
(3.11)‖J2‖Lp(I2)⩽Cn-k(∫a2b2∫01|f(u)|pWn(t,u)dtdu)1/p⩽Cn-k‖f‖Lp[0,1].
Now in view of Lemmas 2.1 and 2.8, we have the inequality
(3.12)‖J3‖Lp(I2)⩽Cn-k(‖f‖Lp(I2)+‖f(2k)‖Lp(I2)).
Combining the estimates (3.5)–(3.12), (3.1) follows.
Now, letp=1. Then, we can expand f(u) for almost all t∈I2 and for all u∈I1, as
(3.13)f(u)-f(t)=∑j=12k-1f(j)(t)j!(u-t)j+1(2k-1)!∫tuφ(u)(u-v)2k-1df(2k-1)(v)+F(u,t)(1-φ(u)),
where φ(u) and F(u,t) are defined as above. Therefore, operating by Tn,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 ofE1, we have
(3.14)‖E4‖L1(I2)⩽Cn-k(‖f‖L1(I2)+‖f(2k-1)‖L1(I2)).
It can easily be shown that(3.15)‖Mn(f,t)‖L1(I)⩽‖f(t)‖L1(I).
Consequently, by induction, we get
(3.16)‖Mnr(f,t)‖L1(I)=‖Mn(Mnr-1(f,u),t)‖L1(I)⩽‖f(t)‖L1(I).
Therefore, in order to get an estimate for E5, it is sufficient to consider the estimate for K(3.17)K=‖Mn(∫tuφ(u)|u-v|2k-1df(2k-1)(v);t)‖L1(I2)⩽∫a2b2(∫a1b1Wn(t,u)|u-t|2k-1|∫tu|df2k-1(v)||du)dt.
For each n, there exists the integer r=r(n) s.t. r/n⩽max(b1-a2,b2-a1)⩽(r+1)/n(3.18)K⩽∑l=0r∫a2b2{∫t+l/nt+(l+1)/nWn(t,u)|u-t|2k-1∫tt+(l+1)/nφ(v)|df(2k-1)(v)|du+∫t-(l+1)/nt-l/nWn(t,u)|u-t|2k-1∫t-(l+1)/ntφ(v)|df(2k-1)(v)|du}dt⩽∑l=1r∫a2b2{∫t+l/nt+(l+1)/nWn(t,u)n2l4|u-t|2k+3∫tt+(l+1)/nφ(v)|df(2k-1)(v)|du+∫t-(l+1)/nt-l/nWn(t,u)n2l4|u-t|2k+3∫t-(l+1)/ntφ(v)|df(2k-1)(v)|du}dt+∫a2b2∫a2-1/nb2+1/nWn(t,u)|u-t|2k-1∫t-n-1/2t+n-1/2φ(v)|df(2k-1)|dudt⩽2∑l=1rn2l4l+1n∫a1b1Wn(t,u)|u-t|2k+3du‖f(2k-1)‖BV(I1)+2n∫a1b1Wn(t,u)|u-t|2k-1du‖f(2k-1)‖BV(I1)⩽Cn-k‖f(2k-1)‖BV(I1).(applyingFubini'stheorem).
To estimateE6, for all u∈[0,1]∖I1, t∈I2, we can choose a δ>0 such that |u-t|⩾δ. Therefore, we get the inequality
(3.19)‖E4‖L1(I2)⩽C∫a2b2∫01Wn(t,u)|f(u)(1-φ(u))|dudt+∑i=02k-11i!∫a2b2∫01Wn(t,u)|f(i)(t)(u-t)i(1-φ(u))|dudt=S1+S2,say.
Since, Wn(t,u) is symmetric in t and u, there follows
(3.20)S1⩽Cn-k‖f‖L1(I2).
In view of Lemma 2.8, we obtain
(3.21)S2⩽Cn-k(‖f‖L1(I2)+‖f(2k-1)‖L1(I2)).
From the estimates (3.14)–(3.21) and the definition of Tn,k, we get (3.2).Theorem 3.2.
Ifp⩾1, f∈Lp[0,1]. Then, for all n sufficiently large there holds
(3.22)‖Tn,k(f;⋅)-f‖Lp(I2)⩽C(ω2k(f,1n,p,I1,⋅)+n-k‖f‖Lp[0,1]),
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 functionfg, where g∈C0∞ be such that suppg⊂I1 and g=1 in I2. Let for convenience f̂=fg. Let f̂η,2k be the Steklov mean of order 2k corresponding to the function f̂, where η>0 is sufficiently small. Then, we have
(3.23)‖Tn,k(f̂;⋅)-f̂‖Lp(I2)⩽‖Tn,k(f̂-f̂η,2k;⋅)‖Lp(I2)+‖Tn,k(f̂η,2k;⋅)-f̂η,2k‖Lp(I2)+‖f̂η,2k-f̂‖Lp(I2)=J1+J2+J3,say.
Let φ be the characteristic function of I3. Then,
(3.24)Mn(f̂-f̂η,2k;t)=Mn(φ(u)(f̂-f̂η,2k);t)+Mn((1-φ(u))(f̂-f̂η,2k);t)=J4+J5say.
For p>1 using Hölder's inequality and then applying Fubini's theorem, we get
(3.25)‖J4‖Lp(I2)p⩽∫a2b2∫a3b3Wn(t,u)|f̂-f̂η,2k|pdudt⩽C‖f̂-f̂η,2k‖Lp(I3)p.
This in view of property (c) of Steklov’s mean implies
(3.26)‖J4‖Lp(I2)⩽Cω2k(f̂,η,p,I2).
In the case p=1, (3.22) is obtained by boundedness of the operator Mn. 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
(3.27)J5⩽δ-2k∫I2∖I3Wn(t,u)(f̂-f̂η,2k)(u-t)2kdu.
Therefore,
(3.28)‖J5‖Lp(I2)⩽Cn-k‖f̂-f̂η,2k‖Lp(I2).
Consequently, we have the estimate
(3.29)J1⩽C(ω2k(f̂,η,p,I2)+n-k‖f̂-f̂η,2k‖Lp(I2)).
Now using Theorem 3.1 and Lemma 2.8,
(3.30)J2⩽Cn-k(‖f̂η,2k‖Lp[0,1]+‖f̂η,2k(2k)‖Lp(I1)).
In view of property (c) of Steklov’s means, we get the inequality
(3.31)J3⩽Cω2k(f̂,η,p,I1).
Choosing η=1/n, the result follows from the estimates of J1–J3.



---

*Source: 184374-2010-12-26.xml*