Journal of Applied Mathematics

Volume 2012 (2012), Article ID 546784, 28 pages

http://dx.doi.org/10.1155/2012/546784

## Approximation by Lupas-Type Operators and Szász-Mirakyan-Type Operators

^{1}Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea^{2}Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 28 July 2011; Accepted 5 January 2012

Academic Editor: Yuantong Gu

Copyright © 2012 Hee Sun Jung and Ryozi Sakai. 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.

#### Abstract

Lupas-type operators and Szász-Mirakyan-type operators are the modifications of Bernstein polynomials to infinite intervals. In this paper, we investigate the convergence of Lupas-type operators and Szász-Mirakyan-type operators on .

#### 1. Introduction and Main Results

For , Bernstein operator is defined as follows: Let and then we define Derriennic [1] gave a modified operator of such as and obtained the result that for , Lupas investigated a family of linear positive operators which mapped the class of all bounded and continuous functions on into such that Moreover, Sahai and Prasad [2] modified Lupas operators as follows: Let be integrable on and let be a positive integer. Then we define where In this paper, we assume that is a positive integer. Then they obtained the following;

Theorem 1.1 (see [2], Theorem 1). *If is integrable on and admits its th and th derivatives, which are bounded at a point , and ( is a positive integer ) as , then
*

*Theorem 1.1 holds only for bounded , so it does not mean the norm convergence on . In this paper, we improve Theorem 1.1 with respect to the norm convergence on .*

*Let and let be a positive weight, that is, for . For a function on , we define the norm by
For convenience, for nonnegative integers , , and , we let
Then we have the following results:*

*Theorem 1.2. Let . Let and be nonnegative integers and . Let satisfy
Then we have uniformly for and ,
In particular, if , then we have uniformly for ,
*

*Remark 1.3. *(a) We see that for nonnegative integers , , and ,

(b) The following weight is useful.

*Let
*

*Theorem 1.4. Let and be nonnegative integers and . Let satisfy
Then we have uniformly for and ,
*

*Let us define the weighted modulus of smoothness by
where
*

*Theorem 1.5. Let and be nonnegative integers and . Let . Then we have uniformly for and ,
*

*The Szász-Mirakyan operators are also generalizations of Bernstein polynomials on infinite intervals. They are defined by:
where
*

*In [3], the class of Szász-Mirakyan operators was defined as follows:
where and
*

*Theorem 1.6 (see [3]). Let and be fixed numbers. Then there exists . depending only on and such that, for every uniformly continuous and bounded function on , the following inequalities hold;(a)(b)
where . (c) for every fixed , we have for every continuous with , , bounded on ,*

*Now, we modify the Szász-Mirakyan operators as follows: let be integrable on , then we define
where is a nonnegative integer. Then we have the following results:*

*Theorem 1.7. Let , and be nonnegative integers. Let satisfies
Then one has uniformly for and ,
In particular, let . If one supposes , then one has uniformly for and ,
*

*Remark 1.8. *(a) We note that for nonnegative integers and ,

(b) The following weight is useful.
where is defined in Remark 1.3.

*Theorem 1.9. Let , , and be nonnegative integers. Let satisfies
Then one has uniformly for and ,
*

*Theorem 1.10. Let , , and be nonnegative integers. Then one has for ,
*

*2. Proofs of Results*

*2. Proofs of Results**First, we will prove results for Lupas-type operators such as Theorems 1.2, 1.4, and 1.5. To prove theorems, we need some lemmas.*

*Lemma 2.1. Let and be nonnegative integers and . Let
Then(i) , (ii)
(iii)for ,
where ; (iv)for ,
where is a polynomial of degree such that the coefficients are bounded independently of and they are positive for .*

*Proof. *(i), (ii), and (iii) have been proved in [2, Lemma 1]. So we may show only the part of (2.4). For , (2.4) holds. Let us assume (2.4) for . We note
So, we have by the assumption of induction,
Here, if is even, then
and if is odd, then
Hence, we have
and here we see that is a polynomial of degree such that the coefficients of are bounded independently of . Moreover, we see from (2.6) that the coefficients of are positive for .

*Lemma 2.2 (see [2, Lemma 2]). Let be a nonnegative integer and . Then one has for :
*

*Let
Then we have
where is defined by (1.10).*

*Proof of Theorem 1.2. *Let . By the second inequality in (1.11),
Let ,
First, we see by (2.13) and Lemma 2.1,
Next, we estimate . By the first inequality in (1.11),
Here, using
and the notation:
we have
Then, we obtain
Here, we used the following that for ,
because
And we know that
Thus, we obtain
Therefore, we have uniformly on ,
Here, if we let , then we have
that is, (1.12) is proved. So, we also have a norm convergence (1.13).

*Proof of Theorem 1.4. *We know that for ,
where . Then we obtain from (2.10) and (2.27),
and from (2.28),
Using , we have
Therefore, we have
Since we know that for ,
we have

*Lemma 2.3. Let and be nonnegative integers and . Let satisfies
Then one has uniformly for , and ,
*

*Proof. *Using , we have
The assumption (2.35) means
Then we can obtain by (2.10),
Consequently, since is uniformly bounded on , we have the result.

*The Steklov function for is defined as follows:*

* Then for the Steklov function with respect to , we have the following properties.*

*Lemma 2.4 (cf.[4]). Let and be a positive and nonincreasing function on . Then (i) ;
(ii)
(iii)
(iv)
*

*Proof. *(i) For , we have the Steklov functions and as follows. We note
Then, we can see from (2.44),
Similarly to (2.44), we know
Therefore, we have from (2.46),
Therefore, (i) is proved.

(ii) We easily see from (2.44) that

(iii) From (2.46), we have

(iv) From (2.47), we have

*Proof of Theorem 1.5. *We know that for ,
Then, we have
From (2.51) and (2.41) of Lemma 2.4,
Here, we suppose and then we know that
From Theorem 1.4, (2.51), (2.42), and (2.43) of Lemma 2.4, we have
Therefore, we have
If we let , then
because .

*From now on, we will prove Theorems 1.7, 1.9, and 1.10, which are the results for the Szász-Mirakyan operators, analogously to the case of Lupas-type operators.*

*Lemma 2.5. Let be a nonnegative integer. Then one has for ,
*

*Proof. *We know that
Therefore, we have

*Lemma 2.6. Let , , and be nonnegative integers.
Then one has(i) and ; (ii)For
(iii)
where is a polynomial of degree such that the coefficients of are bounded independently of .*

*Proof. *Let . Then (i)

(ii) Using , we obtain
Here, we see
Then substituting (2.66) for (2.65), we consider the following;
Then, we have
Here the last equation follows by parts of integration. Furthermore, we have
Therefore, we have

(iii) It is proved by the same method as the proof of Lemma 2.1 (iv).

*Proof of Theorem 1.7. *Let . By the second inequality in (1.30),
Let , ,
First, we see that by (2.71) and Lemma 2.6(i),
Next, to estimate , we split it into two parts:
First, we estimate
Then, using the following facts:
we have
Then, using (2.18) and Lemma 2.6, we have