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

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


Introduction
The Bernstein-Durrmeyer polynomials where p n,ν t n v 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 where is the kernel of the operators.
For m ∈ N 0 the set of nonnegative integers , the mth order moment for the operators M n is defined as The Iterative combination T n,k : where 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 papproximation 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 < a 1 < a 2 < a 3 < b 3 < b 2 < b 1 < 1 and I j a j , b j , j 1, 2, 3. Further, C is a constant not always the same.

Preliminaries and Auxiliary Results
In the sequel, we will require the following results.
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 Using Lemmas 2.1 and 2.3, we can prove the following.

2.4
Let f ∈ L p a, b , 1 p < ∞, and I 1 ⊂ a, b .Then, for sufficiently small η > 0 the Steklov mean f η,m of mth order corresponding to f is defined as follows: where Δ m h is the forward difference operator with step length h.
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 as 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.
where K j are certain constants independent of f.
The dual operator M n corresponding to the operator M n is defined as Then, the corresponding mth order moment is given by μ n,m u M n t − u m ; u Lemma 2.9.For the function μ n,m u , there holds the recurrence relation

2.10
Proof.In view of the relation u 1 − u p n,k u k − nu p n,k u , we get

2.11
Expanding t − t 2 as a polynomial in t − u and integrating by parts, we get 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.

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.

ISRN Mathematical Analysis
Theorem 3.1.If p > 1, f ∈ L p 0, 1 , f has derivatives of order 2k on I 1 with f 2k−1 ∈ AC I 1 , and f 2k ∈ L p I 1 , then for sufficiently large n

3.1
Moreover, if f ∈ L 1 0, 1 , f has derivatives up to the order 2k − 1 on I 1 with f 2k−2 ∈ AC I 1 , and f 2k−1 ∈ BV I 1 , then for sufficiently large n there holds where C is a certain constant independent of f and n.
Proof.Let p > 1, then for all u ∈ I 1 and t ∈ I 2 , we can write where ϕ u is the characteristic function of the interval I 1 and Therefore, operating by T n,k on both sides of 3.3 , we obtain three terms, say E 1 , E 2 , and E 3 corresponding to the three terms in the right-hand side of 3.3 .
In view of Lemmas 2.4 and 2.8, we get

3.5
Let h f 2k be the Hardy-Littleood majorant of f 2k on I 1 .Then, in order to estimate E 2 , it is sufficient to consider the estimate for J 1 Applying H ölder's inequality, Lemma 2.1, and then Fubini's theorem, we get

ISRN Mathematical Analysis 7
Now, in view of Lemmas 2.1 and 2.7, we have Consequently,

3.10
On an application of H ölder's inequality, Lemma 2.1, and Fubini's theorem, we get Cn −k f L p 0,1 .

3.11
Now in view of Lemmas 2.1 and 2.8, we have the inequality

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 ∈ I 2 and for all u ∈ I 1 , as

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 E 4 , E 5 , and E 6 , say corresponding to the three terms in the righthand side of 3.13 .

ISRN Mathematical Analysis
Now proceeding as in the case of the estimate of E 1 , we have

3.14
It can easily be shown that Consequently, by induction, we get Therefore, in order to get an estimate for E 5 , it is sufficient to consider the estimate for

3.17
For each n, there exists the integer r r n applying Fubini's theorem .

3.18
To estimate E 6 , for all u ∈ 0, 1 \ I 1 , t ∈ I 2 , we can choose a δ > 0 such that |u − t| δ.Therefore, we get the inequality

3.19
Since, W n t, u is symmetric in t and u, there follows

3.20
In view of Lemma 2.8, we obtain

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 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 supp g ⊂ I 1 and g 1 in I 2 .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

3.23
Let ϕ be the characteristic function of I 3 .Then,

3.30
In view of property c of Steklov's means, we get the inequality J 3 Cω 2k f, η, p, I 1 .

3.31
Choosing η 1/ √ n, the result follows from the estimates of J 1 -J 3 .
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 .
Cω 2k f, η, p, I 2 .3.26In the case p 1, 3.22 is obtained by boundedness of the operator M n .Now, as above in Theorem 3.1 for u ∈ I 2 \ I 3 , we can choose a δ > 0 such that |u − t| δ.Then, from Fubini's theorem and moment estimates 1 of dual operator, we getJ 5 δ −2k I 2 \I 3 W n t, u f − f η,2k u − t 2k du.