Approximation by Lupas-Type Operators and Sz ´asz-Mirakyan-Type Operators

Lupas-type operators and Sz´asz-Mirakyan-type operators are the modiﬁcations of Bernstein polynomials to inﬁnite intervals. In this paper, we investigate the convergence of Lupas-type operators and Sz´asz-Mirakyan-type operators on (cid:2) 0 , ∞ (cid:3) .


Introduction and Main Results
For f ∈ C 0, 1 , Bernstein operator B n f x is defined as follows: Let In this paper, we assume that n is a positive integer. Then they obtained the following; Theorem 1.1 holds only for bounded x K, so it does not mean the norm convergence on 0, ∞ . In this paper, we improve Theorem 1.1 with respect to the norm convergence on 0, ∞ . Let 0 < p ∞ and let w be a positive weight, that is, w x 0 for x ∈ R. For a function g on 0, ∞ , we define the norm by g L p 0,∞ :

1.9
For convenience, for nonnegative integers n 2, r, and n − r − 2 0, we let A n,r : Journal of Applied Mathematics 3 Then we have the following results: Let α and r be nonnegative integers and n − r − 2 0. Let f ∈ C r 1 0, ∞ satisfy Then we have uniformly for f and n, In particular, if x 1 α 2 w x L p 0,∞ < ∞, then we have uniformly for n, a We see that for nonnegative integers n 2, r, and n − r − 2 0, The following weight is useful.

1.15
Let ψ x : Theorem 1.4. Let r and β be nonnegative integers and n − r − 2 0. Let f ∈ C r 2 0, ∞ satisfy Then we have uniformly for f and n, Theorem 1.5. Let β and r be nonnegative integers and n − r − 2 0. Let f ∈ C r 0, ∞ . Then we have uniformly for f and n,

1.21
The Szász-Mirakyan operators are also generalizations of Bernstein polynomials on infinite intervals. They are defined by: In 3 , the class of Szász-Mirakyan operators S n;r,q f; x was defined as follows: where q > 0 and
c for every fixed x ∈ 0, ∞ , we have for every continuous f with f j x e −qx , j 0, 1, 2, bounded on 0, ∞ , Now, we modify the Szász-Mirakyan operators as follows: let f be integrable on 0, ∞ , then we define where β is a nonnegative integer. Then we have the following results: Theorem 1.7. Let α, β and r be nonnegative integers. Let f ∈ C r 1 0, ∞ satisfies Then one has uniformly for f and n, 1.31 In particular, let 0 < p ∞. If one supposes e βx x 1 α 2 w x L p 0,∞ < ∞, then one has uniformly for f and n, The following weight is useful.
where w λ x is defined in Remark 1.3. Theorem 1.9. Let β, γ, and r be nonnegative integers. Let f ∈ C r 2 0, ∞ satisfies Then one has uniformly for f and n,

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 m and r be nonnegative integers and n > m r 1. Let
Then i T n,0,r x 1,

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where n > m r 2; iv for m 0, where q n,m,r x is a polynomial of degree m such that the coefficients are bounded independently of n and they are positive for n > m r 1.
Proof. i , ii , and iii have been proved in 2, Lemma 1 . So we may show only the part of 2.4 . For m 1, 2, 2.4 holds. Let us assume 2.4 for m 2 . We note T n,m,r x O 1 n m 1 /2 q m,r x , q m,r x ∈ P m−1 .
2.5 So, we have by the assumption of induction,

2.6
Here, if m is even, then and if m is odd, then Hence, we have and here we see that q n,m 1,r x is a polynomial of degree m 1 such that the coefficients of q n,m 1,r x are bounded independently of n. Moreover, we see from 2.6 that the coefficients of q n,m 1,r x are positive for n > m r 2.
Lemma 2.2 see 2, Lemma 2 . Let r be a nonnegative integer and n − r − 2 0. Then one has for f ∈ C r 0, ∞ : Let

2.15
Next, we estimate B. By the first inequality in 1.11 ,

2.16
Here, using and the notation:

2.19
Then, we obtain

2.20
Here, we used the following that for i 1,

2.22
And we know that

2.23
Thus, we obtain 2.24 Journal of Applied Mathematics 11 Therefore, we have uniformly on n, Here, if we let γ 1/3, 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 f ∈ C r 2 0, ∞ ,

12
Journal of Applied Mathematics Therefore, we have

2.35
Then one has uniformly for n, f and x ∈ 0, ∞ ,

2.36
Journal of Applied Mathematics 13 Proof. Using 1 y 2β C y − x 2β 1 x 2β , we have The assumption 2.35 means f r y C 1 y 2β .

2.38
Then we can obtain by 2.10 ,
The Steklov function f h x for f ∈ C 0, ∞ is defined as follows: Then for the Steklov function f h x with respect to f ∈ C 0, ∞ , we have the following properties.

Lemma 2.4 cf. 4 .
Let f x ∈ C 0, ∞ and η x be a positive and nonincreasing function on Proof. i For f ∈ C 0, ∞ , we have the Steklov functions f h x and f h x as follows. We note Then, we can see from 2.44 ,

2.45
Similarly to 2.44 , we know x h x f u h − f u du .

2.46
Therefore, we have from 2.46 ,

2.47
Therefore, i is proved.
ii We easily see from 2.44 that

2.49
iv From 2.47 , we have

2.50
Proof of Theorem 1.5. We know that for f x ∈ C r 0, ∞ , Then, we have

2.56
If we let h 1/ √ n, then
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.

2.64
ii Using xS

2.65
Here, we see

2.68
Here the last equation follows by parts of integration. Furthermore, we have

2.79
Then, using 2.18 and Lemma 2.6, we have