On L p-approximation by iterative combination of Bernstein–Durrmeyer polynomials

International Scholarly Research NetworkISRN Mathematical AnalysisVolume 2011, Article ID 184374, 11 pagesdoi:10.5402/2011/184374

Research ArticleOn Lp-Approximation by Iterative Combination ofBernstein-Durrmeyer Polynomials

T. A. K. Sinha,1 P. N. Agrawal,2 and Asha Ram Gairola2

1 Department of Mathematics, SMD College Poonpoon, Patna, Bihar, India2 Department of Mathematics, IIT Roorkee, Roorkee 247667, India

Correspondence should be addressed to P. N. Agrawal, pna [email protected]

Received 11 November 2010; Accepted 9 December 2010

Academic Editors: O. Gues and R. Stenberg

Copyright q 2011 T. A. K. Sinha et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

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

1. Introduction

The Bernstein-Durrmeyer polynomials

Mn

(f ; t)= (n + 1)

n∑

ν=0

pn,ν(t)∫1

0pn,ν(u)f(u)du, (1.1)

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 thatthe order of approximation by these operators is, at best, O(n−1) however smooth thefunction may be. In order to improve this rate of approximation, we consider an iterativecombination Tn,k(f ; t) of the operators Mn(f ; t). This technique of improving the rateof convergence was given by Micchelli [3] who first used it to improve the order ofapproximation by Bernstein polynomials Bn(f ; t). Recently, this technique has been appliedto obtain some direct and inverse theorems in ordinary and simultaneous approximationby several sequences of linear positive operators in uniform norm (c.f., e.g., [4–6]). Theobject of this paper is to study some direct theorems in Lp-approximation by the operatorsTn,k(f ; t).

2 ISRN Mathematical Analysis

For f ∈ Lp[0, 1], the operators Mn(f ; t) can be expressed as

Mn

(f ; t)=∫1

0Wn(u, t)f(u)du, (1.2)

where

Wn(u, t) = (n + 1)n∑

ν=0

pn,ν(t)pn,ν(u) (1.3)

is the kernel of the operators.Form ∈ N0 (the set of nonnegative integers), themth order moment for the operators

Mn is defined as

μn,m(t) = Mn

((u − t)m; t

). (1.4)

Let LB[a, b], C∞[a, b], AC[a, b], and BV [a, b] denote the classes of bounded Lebesgueintegrable, infinitely differentiable, absolutely continuous functions, and functions ofbounded variations, respectively, on the interval [a, b].

The Iterative combination Tn,k : LB[0, 1] → C∞[0, 1] of the operators Mn(f ; t) isdefined as

Tn,k(f ; t)=(I − (I −Mn)k

)(f ; t)=

k∑

r=1

(−1)r+1(k

r

)

Mrn

(f ; t), k ∈ N, (1.5)

where M0n = I andMr

n = Mn(Mr−1n ) for r ∈ N.

In Section 2 of this paper, we give some definitions and auxiliary results which willbe 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 ofderivatives of the function. From these estimates, we obtain a general error estimate in termsof 2k + 2th integral modulus of smoothness of the function.

In what follows, we suppose that 0 < 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 function μn,m(t), one has μn,0(t) = 1, μn,1(t) = (1 − 2t)/(n + 2), andthere holds the recurrence relation

(n +m + 2)μn,m+1(t) = t(1 − t){μ′n,m(t) + 2mμn,m−1(t)

}+ (m + 1)(1 − 2t)μn,m(t), (2.1)

form � 1.

ISRN Mathematical Analysis 3

Consequently, we have

(i) μn,m(t) is a polynomial in t of degree m,

(ii) for every t ∈ [0, 1], μn,m(t) = O(n−[(m+1)/2]), where [β] is the integer part of β.

The mth order moment for the operator Mpn 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 μn,m(t).

Lemma 2.2 (see [1]). For the function pn,ν(t), there holds the result

tr(1 − t)rdrpn,ν(t)

dtr=∑

2i+j�ri,j�0

ni(ν − nt)jqi,j,r(t)pn,ν(t), (2.2)

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

μ[p+1]n,m (t) =

m∑

j=0

m−j∑

i=0

(m

j

)1i!Di(μ[p]n,m−j(t)

)μn,i+j(t). (2.3)

Lemma 2.4 (see [7]). For k, l ∈ N, there holds Tn,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 ∈ Lp[a, b], 1 � p < ∞, and I1 ⊂ (a, b). Then, for sufficiently small η > 0 theSteklov mean fη,m of mth order corresponding to f is defined as follows:

fη,m(t) = η−m∫η/2

−η/2· · ·∫η/2

−η/2

(f(t) + (−1)m−1Δm∑m

i=1 tif(t)) m∏

i=1

dti, t ∈ I1, (2.5)

where Δmhis the forward difference operator with step length h.

Lemma 2.6. Let f ∈ Lp[a, b], 1 � p < ∞, and I1 ⊂ (a, b). Then, for the function fη,m, we have

(a) fη,m has derivatives up to orderm over I1,

(b) ‖f (r)η,m‖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) ‖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 lemmaeasily follows hence the details are omitted.

Let f ∈ Lp[0, a], 1 � p < ∞. Then, theHardy-Littlewoodmajorant hf(x) of the functionf is defined as

hf(x) = supξ /=x

1ξ − x

∫ ξ

x

f(t)dt. (2.6)

Lemma 2.7. If 1 < p < ∞, f ∈ Lp[0, a], then hf ∈ Lp[0, a] and

∥∥hf

∥∥Lp[0,a]

� 21/pp

p − 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 Lp-norm.

Lemma 2.8 (see [11]). Let 1 � p < ∞, f ∈ Lp[a, b]. Suppose f (k) ∈ AC[a, b] and f (k+1) ∈Lp[a, b]. Then,

∥∥∥f (j)∥∥∥Lp[a,b]

� Kj

(∥∥∥f (k+1)∥∥∥Lp[a,b]

+∥∥f∥∥Lp[a,b]

), j = 1, 2, . . . , k, (2.8)

where Kj are certain constants independent of f .

The dual operator Mn corresponding to the operator Mn is defined as

Mn

(f ;u)=∫1

0Wn(u, t)f(t)dt. (2.9)

Then, the corresponding mth order moment is given by μn,m(u) = Mn((t − u)m;u) =∫10 Wn(u, t)(t − u)mdt.

Lemma 2.9. For the function μn,m(u), there holds the recurrence relation

(n −m − 2)μn,m+1(u) = u(1 − u)μ′n,m(u) + (m + 1)(1 − 2u)μn,m(u) + 2m

(u − u2

)μn,m−1(u).

(2.10)


Proof. In view of the relation u(1 − u)p′n,k(u) = (k − nu)pn,k(u), we get

u(1 − u)μ′n,m(u)

= (n + 1)n∑

k=0

∫1

0pn,k(t)u(1 − u)p′n,k(u)(t − u)mdt −mu(1 − u)μn,m−1(u)

= (n + 1)n∑

k=0

(k − nu)pn,k(u)∫1

0pn,k(t)(t − u)mdt −mu(1 − u)μn,m−1(u)

= (n + 1)n∑

k=0

pn,k(u)∫1

0((k − nt) + n(t − u))pn,k(t)(t − u)mdt −mu(1 − u)μn,m−1(u)

= (n + 1)n∑

k=0

pn,k(u)∫1

0

(t − t2

)p′n,k(t)(t − u)mdt + nμn,m+1(u) −mu(1 − u)μn,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,m(u)

= −(n + 1)n∑

k=0

pn,k(u)∫1

0pn,k(t)

[(m + 1)(1 − 2u)(t − u) +m

(u − u2

)− (m + 2)(t − u)2

]

× (t − u)m−1dt + nμn,m+1(u) −mu(1 − u)μn,m−1(u)

= −(m + 1)(1 − 2u)μn,m(u) − 2m(u − u2

)μn,m−1(u)

− (m + 2)μn,m+1(u) + nμn,m+1(u).(2.12)

Rearrangement of the terms gives (2.10).

Remark 2.10. From (2.10), it follows that μn,m(u) = O(n−[(m+1)/2]), where [β] is the integer partof β.

3. Main Result

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


Theorem 3.1. If p > 1, f ∈ Lp[0, 1], f has derivatives of order 2k on I1 with f (2k−1) ∈ AC(I1), andf (2k) ∈ Lp(I1), then for sufficiently large n

∥∥Tn,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), andf (2k−1) ∈ BV (I1), then for sufficiently large n there holds

∥∥Tn,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=1

f (j)(t)j!

(u − t)j +1

(2k − 1)!

∫u

t

ϕ(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=0

f (j)(t)j!

(u − t)j . (3.4)

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 Lemmas 2.4 and 2.8, we get

‖E1‖Lp(I2) � Cn−k(∥∥∥f (2k)

∥∥∥Lp(I2)

+∥∥f∥∥Lp(I2)

). (3.5)

Let hf (2k) be the Hardy-Littleood majorant of f (2k) on I1. Then, in order to estimate E2, it issufficient to consider the estimate for J1

J1 = Mn

(ϕ(u)(u − t)2k

∣∣∣hf (2k) (u)∣∣∣; t). (3.6)

Applying Holder’s inequality, Lemma 2.1, and then Fubini’s theorem, we get

‖J1‖pLp(I2)�(Mn

(ϕ(u)(u − t)2kq; t

))p/q ×Mn

(ϕ(u)(u − t)p

∣∣∣hf (2k) (u)∣∣∣p; t)

� Cn−kp∫b1

a1

∫b2

a2

Wn(t, u)∣∣∣hf (2k) (u)

∣∣∣pdt du.

(3.7)


Now, in view of Lemmas 2.1 and 2.7, we have

‖J1‖pLp(I2)� Cn−kp

∫b1

a1

∣∣∣hf (2k) (u)

∣∣∣pdu � Cn−kp

∥∥∥f (2k)

∥∥∥p

Lp(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,

∣∣Mn

(F(u, t)

(1 − ϕ(u)

); t)∣∣

=

∣∣∣∣∣∣Mn

⎛

⎝

⎛

⎝f(u) −2k−1∑

j=0

f (j)(t)j!

(u − t)j⎞

⎠(1 − ϕ(u)); t

⎞

⎠

∣∣∣∣∣∣

� δ−2k

⎧⎨

⎩Mn

(∣∣f(u)∣∣(u − t)2k; t

)+

2k−1∑

j=0

∣∣f (j)(t)∣∣

j!Mn

(|u − t|2k+j ; t

)⎫⎬

⎭= J2 + J3, say.

(3.10)

On an application of Holder’s inequality, Lemma 2.1, and Fubini’s theorem, we get

‖J2‖Lp(I2) � Cn−k(∫b2

a2

∫1

0

∣∣f(u)∣∣pWn(t, u)dt du

)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

∥∥Lp(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=1

f (j)(t)j!

(u − t)j +1

(2k − 1)!

∫u

t

ϕ(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 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 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

∥∥Mn(f, t)

∥∥L1(I)

�∥∥f(t)

∥∥L1(I)

. (3.15)

Consequently, by induction, we get

∥∥Mr

n(f, t)∥∥L1(I)

=∥∥∥Mn(Mr−1

n (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 K

K =∥∥∥∥Mn

(∫u

t

ϕ(u)|u − v|2k−1df (2k−1)(v); t)∥∥∥∥

L1(I2)

�∫b2

a2

(∫b1

a1

Wn(t, u)|u − t|2k−1∣∣∣∣

∫u

t

∣∣∣df2k−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)/

√n

K �r∑

l=0

∫b2

a2

{∫ t+(l+1)/√n

t+l/√n

Wn(t, u)|u − t|2k−1∫ t+(l+1)/

√n

t

ϕ(v)∣∣∣df (2k−1)(v)

∣∣∣du

+∫ t−l/√n

t−(l+1)/√n

Wn(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/√n

Wn(t, u)n2

l4|u − t|2k+3

∫ t+(l+1)/√n

t

ϕ(v)∣∣∣df (2k−1)(v)

∣∣∣du

+∫ t−l/√n

t−(l+1)/√n

Wn(t, u)n2

l4|u − t|2k+3

∫ t

t−(l+1)/√n

ϕ(v)∣∣∣df (2k−1)(v)

∣∣∣du

}

dt

+∫b2

a2

∫b2+1/√n

a2−1/√n

Wn(t, u)|u − t|2k−1∫ t+n−1/2

t−n−1/2ϕ(v)

∣∣∣df (2k−1)∣∣∣dudt

� 2r∑

l=1

n2

l4l + 1√

n

∫b1

a1

Wn(t, u)|u − t|2k+3du∥∥∥f (2k−1)

∥∥∥BV (I1)

+2√n

∫b1

a1

Wn(t, u)|u − t|2k−1du∥∥∥f (2k−1)

∥∥∥BV (I1)

� Cn−k∥∥∥f (2k−1)

∥∥∥BV (I1).

(applying Fubini’s theorem

).

(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

∫1

0Wn(t, u)

∣∣f(u)

(1 − ϕ(u)

)∣∣dudt

+2k−1∑

i=0

1i!

∫b2

a2

∫1

0Wn(t, u)

∣∣∣f (i)(t)(u − t)i

(1 − ϕ(u)

)∣∣∣dudt

= S1 + S2, say.

(3.19)

Since, Wn(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 Tn,k, we get (3.2).

Theorem 3.2. If p � 1, f ∈ Lp[0, 1]. Then, for all n sufficiently large there holds

∥∥Tn,k(f ; ·) − f∥∥Lp(I2)

� C

(ω2k

(f,

1√n, 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, whereg ∈ C∞

0 be such that supp g ⊂ I1 and g = 1 in I2. Let for convenience f = fg. Let fη,2k be theSteklov mean of order 2k corresponding to the function f , where η > 0 is sufficiently small.Then, we have

∥∥∥Tn,k(f ; ·) − f∥∥∥Lp(I2)

�∥∥∥Tn,k(f − fη,2k; ·)

∥∥∥Lp(I2)

+∥∥∥Tn,k(fη,2k; ·) − fη,2k

∥∥∥Lp(I2)

+∥∥∥fη,2k − f

∥∥∥Lp(I2)

= J1 + J2 + J3, say.(3.23)

Let ϕ be the characteristic function of I3. Then,

Mn

(f − fη,2k; t

)= Mn

(ϕ(u)

(f − fη,2k

); t)+Mn

((1 − ϕ(u)

)(f − fη,2k

); t)

= J4 + J5 say.(3.24)


For p > 1 using Holder’s inequality and then applying Fubini’s theorem, we get

‖J4‖pLp(I2)�∫b2

a2

∫b3

a3

Wn(t, u)∣∣∣f − fη,2k

∣∣∣pdudt

� C∥∥∥f − fη,2k

∥∥∥p

Lp(I3).

(3.25)

This in view of property (c) of Steklov’s mean implies

‖J4‖Lp(I2) � Cω2k

(f , η, p, I2

). (3.26)

In the case p = 1, (3.22) is obtained by boundedness of the operator Mn. Now, as above inTheorem 3.1 for u ∈ I2 \ I3, we can choose a δ > 0 such that |u − t| � δ. Then, from Fubini’stheorem and moment estimates 1 of dual operator, we get

J5 � δ−2k∫

I2\I3Wn(t, u)

(f − fη,2k

)(u − t)2kdu. (3.27)

Therefore,

‖J5‖Lp(I2) � Cn−k∥∥∥f − fη,2k

∥∥∥Lp(I2)

. (3.28)

Consequently, we have the estimate

J1 � C

(ω2k

(f , η, p, I2

)+ n−k

∥∥∥f − fη,2k∥∥∥Lp(I2)

). (3.29)

Now using Theorem 3.1 and Lemma 2.8,

J2 � Cn−k(∥∥∥fη,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 � Cω2k

(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, Indiafor financial support to carry out the above work.


References

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On L p-approximation by iterative combination of Bernstein–Durrmeyer polynomials

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