Filomat 27:2 (2013), 371–380 DOI 10.2298/FIL1302371A
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Weighted Approximation by New Bernstein-Chlodowsky-Gadjiev Operators ALI˙ ARALa , TUNCER ACARa a Kırıkkale
University Faculty of Science and Arts, Department of Mathematics,Yahs¸ihan, KIRIKKALE, Turkey
Abstract. In the present paper, we introduce Bernstein-Chlodowsky-Gadjiev operators taking into consideration the polynomials introduced by Gadjiev and Ghorbanalizadeh [2]. The interval of convergence of the operators is a moved interval as polynomials given in [2] but grows as n → ∞ as in the classical Bernstein-Chlodowsky polynomials. Also their knots are shifted and depend on x. We firstly study weighted approximation properties of these operators and show that these operators are more efficient in weighted approximating to function having polynomial growth since these operators contain a factor bn tending to infinity. Secondly we calculate derivative of new Bernstein-ChlodowskyGadjiev operators and give a weighted approximation theorem in Lipchitz space for the derivatives of these operators.
1. Introduction Due to the polynomials have significant applications in a lot of area such as mathematics and physics, nowadays a variety of their generalizations have been studied increasingly. Recently, in [2] Gadjiev and Ghorbanalizadeh constructed a new generalization of Bernstein-Stancu type polynomials given by ( )n n ( )( )( )r ( )n−r n + β2 ∑ r + α1 n ( ) α2 n + α2 Sn,α,β f ; x = f x− −x , (1) n n + β1 r n + β2 n + β2 r=0 α2 2 where n+β ≤ x ≤ n+α n+β2 and αk , βk , k = 1, 2 are positive real numbers satisfying 0 ≤ α2 ≤ α1 ≤ β2 ≤ β1 . 2 They studied convergence properties of these operators, showed that the new polynomials are sequences of linear positive operators in the space of continuous functions and the interval of convergence of these polynomials is a moved interval and grows to [0, 1] . Following polynomials were introduced by I. Chlodowsky [1] in 1937 as a generalization of the Bernstein polynomials. The classical Bernstein-Chlodowsky operators are n ( ) ( ) ( )r ( ) ( ) ∑ r n x x n−r Cn f ; x = f bn 1− , n r bn bn r=0
2010 Mathematics Subject Classification. Primary 41A25; Secondary 41A36 Keywords. Bernstein-Chlodowsky-Gadjiev operators, weighted approximation, Lipschitz space Received: 10 October 2012; Accepted: 10 December 2012 Communicated by Dragana Cvetkovic Ilic The first author is supported by The Scientific and Technological Research Council of Turkey under Project No: 112T548. Email addresses:
[email protected] (ALI˙ ARAL),
[email protected] (TUNCER ACAR)
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where f is a function defined on [0, ∞) and bounded on every finite interval [0, b] ⊂ [0, ∞) and (bn )n≥1 is a positive increasing sequence with the properties bn → ∞ and
bn → 0 as n → ∞. n
(2)
In [8], weighted approximation properties of Bernstein-Chlodowsky operators were investigated and some generalization of Bernstein-Chlodowsky operators were given in [4] and [7]. In the present paper, using above ideas, we introduce a new construction of Bernstein-ChlodowskyGadjiev type operators as following:
Tn,α,β
(
( )n n ( )( )( )r ( )n−r n + β2 ∑ ) α2 x r + α1 n x n + α2 bn − − f;x = f α3 x + β3 n n + β1 r bn n + β2 n + β2 bn r=0
(3)
α2 2 where n+β bn ≤ x ≤ n+α n+β2 bn , bn satisfies (2) and αk , βk , k = 1, 2, 3 are positive real numbers satisfying α3 +β3 = 1 2 and 0 ≤ α2 ≤ α1 ≤ β2 ≤ β1 . If we chose
1. 2. 3. 4.
α1 α1 α2 α3
= α2 = α3 = β1 = β2 = 0 with bn = 1, then we obtain the classical Bernstein polynomials, = α2 = α3 = β1 = β2 = 0, then we obtain the classical Bernstein-Chlodowsky polynomials, = α3 = β2 = 0 with bn = 1, then we obtain the classical Bernstein-Stancu polynomials given in [9], = 0 with bn = 1, then we obtain the Bernstein-Stancu polynomials defined by (1).
The new Bernstein-Chlodowsky-Gadjiev operators, based on functions defined on [0, ∞), which are bounded on every [0, bn ] ⊂ [0, ∞) with (2), become an approximation process in approximating unbounded functions on the unbounded infinite interval [0, ∞). Also as known that, since an immediate analog of the Bohman-Korovkin theorem does not hold in the unbounded interval, some restrictions are needed. Now we give these restrictions and notations will be used throughout the paper. Let B2 [0, ∞) be the space of all functions f defined on the semi-axis [0, ∞) satisfying the inequality | f (x)| ≤ M f (1 + x2 ), where M f is a positive constant only depending on function f . Introduce C2 [0, ∞) = B2 [0, ∞) ∩ C[0, ∞) and
f (x) C∗2 [0, ∞) = = K < ∞ f ∈ C2 [0, ∞) : lim . f x→∞ 1 + x2
These spaces endowed with the norm
f (x)
f = sup . 2 2 x≥0 1 + x
(4)
As it follows from the Gadjiev papers [5] and [6], the Korovkin-type theorems for positive linear operators does not hold in the space C2 [0, ∞) but holds in the space of C∗2 [0, ∞) in the norm of B2 [0, ∞) and has the following forms: Theorem 1.1. If the sequence of positive linear operators Ln from C2 [0, ∞) to B2 [0, ∞) satisfies conditions
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lim ∥Ln (tν ; x) − xν ∥2 = 0, ν = 0, 1, 2.
n→∞
then for any function f ∈ C∗2 [0, ∞)
lim Ln f − f 2 = 0. n→∞
Theorem 1.2. For any sequence of linear positive operators (Ln )n≥1 satisfying the conditions of Theorem 1.1, there exists a function f ∗ ∈ C2 [0, ∞), for which
lim Ln f ∗ − f ∗ 2 , 0. n→∞
2. Weighted Approximation
( ) In this section we study approximation properties of Tn,α,β f using the Theorem 1.1.
Theorem 2.1. We have lim
n→∞
sup
( ) Tn,α,β f ; x − f (x) 1 + x2
α2 n+α2 n+β2 bn ≤x≤ n+β2 bn
=0
for any function f ∈ C∗2 [0, ∞). Proof. We use the method given in [8]. For simplification, we shall use following definition: ( )n n ( )( )r ( )n−r n + β2 ∑ ( r ) n x ( ) α2 n + α2 x ∗ Tn,α,β f ; x = f − − . n n r bn n + β2 n + β2 bn r=0
(5)
By the binomial expansion and (3) it is obvious that ∗ (1; x) Tn,α,β (1; x) = Tn,α,β ( )n n ( ) ( )r ( )n−r n + β2 ∑ n x α2 n + α2 x = − − n n + β2 bn r bn n + β2 r=0 )n ( )n ( n + β2 x α2 n + α2 x − + − = 1. = n bn n + β2 n + β2 bn
It follows that lim
n→∞
Tn,α,β (1; x) − 1
sup
α2 n+α2 n+β2 bn ≤x≤ n+β2 bn
From (5), we have ∗ Tn,α,β
1 + x2
= 0.
(6)
(7)
(
)n n ( ) ( )r ( )n−r n + β2 ∑ r n x α2 n + α2 x (t; x) = − − n n r bn n + β2 n + β2 bn r=0 ( )n n−1 ( )( )r+1 ( )n−r−1 n + β2 ∑ n − 1 x α2 n + α2 x = − − n r bn n + β2 n + β2 bn r=0 )n ( )n−1 ( ) ( n + β2 n x α2 − = n n + β2 bn n + β2 ( ) ( ) n + β2 x α2 = − n bn n + β2
(8)
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and ∗ Tn,α,β
( ) t2 ; x
)r ( )n−r ( )( r2 n x n + α2 α2 x = − − n + β2 bn n2 r bn n + β2 r=0 ( )n n ( )( )r ( )n−r n + β2 ∑ r (r − 1) n x α2 n + α2 x = − − n r bn n + β2 n + β2 bn n2 r=2 ( )n n ( )( )r ( )n−r n + β2 ∑ r n x α2 x n + α2 + − − n n + β2 bn n2 r bn n + β2 (
n + β2 n
)n ∑ n
r=1
)n )( )r+2 ( )n−r−2 n−2 ( n + β2 n − 1 ∑ n − 2 x α2 n + α2 x = − − n n r=0 bn n + β2 n + β2 bn r )n n−1 ( ( )( )r+1 ( )n−r−2 n + β2 1 ∑ n − 1 x α2 n + α2 x + − − n n r=0 r bn n + β2 n + β2 bn ( )2 ( )2 ( ) ( ) n + β2 1 x (n − 1) n + β2 x α2 α2 = − + − . n n bn n + β2 n n bn n + β2 (
(9)
By the definitions (3) and (5) we have (
Tn,α,β (t; x) = =
=
)n n ( )( )( )r ( )n−r n + β2 ∑ n x α2 n + α2 x r + α1 bn − − α3 x + β3 n n + β1 r bn n + β2 n + β2 bn r=0 ( )n ∑ ) ( ) ( ) ( n r n−r n + β2 α2 n + α2 x n x α3 x − − n n + β2 bn r bn n + β2 r=0 ( )n ( )( )r ( )n−r n ∑ n + β2 nβ3 r n x α2 n + α2 x + bn − − n n + β1 r=0 n r bn n + β2 n + β2 bn )n ) ( )n−r ( ( ) ( n r ∑ β3 α1 n + β2 α2 n + α2 x n x bn − − + n n + β1 r=0 r bn n + β2 n + β2 bn ∗ (1; x) + α3 xTn,α,β
nβ3 β3 α1 ∗ ∗ (t; x) + Tn,α,β (1; x) bn Tn,α,β bn . n + β1 n + β1
Taking into account (6) and (8) we get ) ( ) n + β2 α1 − α2 β3 x + β3 bn . Tn,α,β (t; x) = α3 x + n + β1 n + β1 (
Therefore lim
sup
Tn,α,β (t; x) − x
1 + x2 ( ) ) ( n + β2 α1 − α2 ≤ lim α3 + β3 − 1 + β3 bn n→∞ n + β1 n + β1
n→∞
n+α2 α2 n+β2 bn ≤x≤ n+β2 bn
= α3 + β3 − 1 = 0.
(10)
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Similarly ( ) Tn,α,β t2 ; x
(
)n n ( )2 ( ) ( )r ( )n−r n + β2 ∑ r + α1 n x α2 n + α2 x = α3 x + β3 bn − − n n + β1 r bn n + β2 n + β2 bn r=0 ( )n n ( ) ( )r ( )n−r n + β2 ∑ n x α2 n + α2 x = (α3 x)2 − − r bn n + β2 n n + β2 bn r=0 ( )n ∑ ) ( ) ( )r ( )n−r ( n n + β2 n x α2 n + α2 x r + α1 + bn − − 2xα3 β3 n n + β1 r bn n + β2 n + β2 bn r=0 ( )n n ( )2 ( ) ( )r ( )n−r n + β2 ∑ r + α1 n x α2 x n + α2 + β3 bn − − . n n + β1 r bn n + β2 n + β2 bn r=0
Again using (5) we can write ( ) Tn,α,β t2 ; x
) n ∗ (t; x) bn Tn,α,β = (α3 x) n + β1 ( ) ( )2 ( ) nβ3 α1 ∗ ∗ + 2xα3 β3 bn Tn,α,β (1; x) + bn Tn,α,β t2 ; x n + β1 n + β1 )2 ( 2 2β α1 n ∗ β3 α1 ∗ (1; x) (t; x) + + ( 3 )2 b2n Tn,α,β bn Tn,α,β n + β 1 n + β1 2
∗ Tn,α,β
( (1; x) + 2xα3 β3
and from (6), (8) and (9) we get ( )2 ( ) β 3 α1 α 1 2 2 Tn,α,β t ; x = (α3 x) + 2xα3 β3 bn + bn + n + β1 n + β1 ) ( ) ( n + β2 α2 x− bn + β3 n + β1 n + β2 ( )( ) ) ( β3 (n − 1) n + β2 β3 2β3 α1 α2 b n bn + x− + bn × 2xα3 + n + β1 n n + β1 n + β2 n + β1
(11)
Obviously
lim
sup
( ) Tn,α,β t2 ; x − x2
1 + x2 { 2 [ ( )( ( )) ] n + β2 β3 (n − 1) n + β2 x 2 α + β3 = lim sup 2α3 + −1 n→∞ α2 n + β1 n n + β1 1 + x2 3 n+α2 b ≤x≤ b n n n+β2 n+β2 [ ( )( ( ) ) β3 (n − 1) α2 bn β3 2α3 β3 α1 n + β2 2β3 α1 x bn + β3 bn − + bn + n + β1 n + β1 n n + β1 n + β1 1 + x2 n + β1 ( ( )( ( ))] )2 β3 (n − 1) n + β2 α2 b n 1 β3 α1 −β3 2α3 + + bn n + β1 n n + β1 1 + x2 n + β1 )( ( ( ))]} β3 2β3 α1 n−1 α2 bn bn − bn α2 − 1 −β3 n + β1 n + β1 n + β1 n
n→∞
n+α2 α2 n+β2 bn ≤x≤ n+β2 bn
= α23 + 2α3 β3 + β23 − 1 = 0.
(12)
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If we use the operators α2 bn f (x) if 0 ≤ x ≤ n+β 1 ( ) ( ) α2 T 2 if n+β2 bn ≤ x ≤ n+α Tn f ; x = n,α,β f ; x n+β2 bn n+α2 f (x) if n+β2 bn ≤ x < ∞ then we can write
( )
lim Tn f − f 2 = lim
n→∞
n→∞
sup
( ) Tn,α,β f ; x − f (x) 1 + x2
n+α2 α2 n+β2 bn ≤x≤ n+β2 bn
(13)
.
(14)
From (7), (10) and (12) it follows that lim ∥Tn (tν ; ) − xν ∥2 = 0.
n→∞
As an immediate application of Theorem 1.1, we have
( )
lim Tn f − f 2 = 0 n→∞
for f ∈ C∗2 [0, ∞). From (14), we have desired result. Theorem 2.2. We have 1 lim √ n→∞ bn
sup
α2 n+α2 n+β2 bn ≤x≤ n+β2 bn
( ) Tn,α,β f ; x − f (x) 1 + x2
=0
for any function f ∈ C∗2 [0, ∞). | f (x)| Proof. Since f ∈ C∗2 [0, ∞) we can write limx→∞ 1+x2 = K f . It is sufficient to study with the functions satisfying ( ( )) | f (x)| the condition limx→∞ 1+x2 = 0 for example φ (x) = f (x) − K f 1 + x2 . That is, there exist a sufficiently large | f (x)| number x0 > 0 such that 1+x2 < ε for x > x0 . Considering the operators defined in (13), we get ( ) ( ) ( ) Tn f ; x − f (x) Tn f ; x − f (x) Tn f ; x − f (x) 1 1 1 sup sup ≤ √ + √ sup √ 1 + x2 1 + x2 1 + x2 bn 0≤xx0 ( ) Tn 1 + t2 ; x
1
( ) 1
≤ √ Tn f − f C[0, x ] + √ f 2 sup 0 1 + x2 bn bn x>x0 f (x) 1 + √ sup . bn x>x0 1 + x2 The first term of above summation tends to zero as n → ∞ by the Korovkin’s theorem. Since x > x0 , the last term of above summation tends to zero as n → ∞. Also we can write from (11) ( ) ( )2 Tn 1 + t2 ; x α23 + 2α3 β3 α3 β 3 α1 β3 α1 1 1 ≤ +2 √ bn + √ bn √ sup √ 1 + x2 bn x>x0 bn bn n + β1 bn n + β1 2 2 β23 1 β3 (n − 1) 1 1 2β3 α1 bn + √ + √ bn +√ n bn n + β1 bn bn n + β1 which tend to zero as n → ∞. Since ( ) Tn f ; x − f (x) 1 1 lim √ sup = lim √ 2 n→∞ n→∞ 1+x bn 0≤x n
A. Aral, T. Acar / Filomat 27:2 (2013), 371–380
then we have P (x) =
378
( ) k r! (n + k − r)! bn+k s=0 s (r − s)! (n + s − r)! ( )r−s ( )n+s−r x α2 n + k + α2 x × − − . bn+k n + k + β2 n + k + β2 bn+k (
1
k )k ∑
(−1)k−s
Since
( ) ( ) n+k r! (n + k − r)! (n + k)! n = n! r (r − s)! (n + s − r)! r−s ( ) we can write the kth derivative of Tn+k,α,β f ; x )n+k ∑ )( ) ( ) ( n ∑ k ( ) ( (n + k)! 1 k n + k + β2 n r + s + α1 k−s k (−1) bn+k f n! bn+k n+k n + k + β1 r s r=0 s=0 ( )r ( )n−r x α2 x n + k + α2 × − − . bn+k n + k + β2 n + k + β2 bn+k Since k ∑
(−1)k−s
s=0
( ) ( ) ( ) k r + s + α1 r + α1 f bn+k = ∆kh f bn+k s n + k + β1 n + k + β1
where the operator ∆h is applied with step h = bn+k /n + k + β1 , we have desired result. Theorem 3.2. Let the function f be a (k − 1)-times continuous differentiable on [0, ∞) and its kth derivative belongs to LipM α, 0 < α ≤ 1 for some integer k ≥ 1. Then we have ( ) (k) T (k) (x) f ; x − f n+k,α,β lim sup = 0. n→∞ α 1 + xα n+k+α2 2 n+k+β2 bn+k ≤x≤ n+k+β2
bn+k
Proof. We know that ( ) (bn+k )k r + α1 k ∆h f bn+k = f (k) (ξr ) ( )k n + k + β1 n + k + β1
( ) ( ) where(r + α1 ) bn+k / n + k + β1 < ξr < (r + α1 + k) bn+k / n + k + β1 . If we take ξr =
r + α1 + θ r k bn+k , 0 < θr < 1 n + k + β1
we can write by Lemma 3.1 ( ) (k) Tn+k,α,β f ; x
=
)n+k ∑ )( ) ( n n + k + β2 n (k) r + α1 + θr k b f n+k ( )k n + k n + k + β r 1 n! n + k + β1 r=0 ( )r ( )n−r x n + k + α2 x α2 × − − . bn+k n + k + β2 n + k + β2 bn+k (n + k)!
We can easily verify that lim
n→∞
(n + k)! ( )k = 1. n! n + k + β1
(
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Thus we have
=
( ) (k) Tn+k,α,β f ; x − f (k) (x) ( )n+k ∑ ( ) )( ) n ( (n + k)! n n + k + β2 (k) r + α1 + θr k (k) f bn+k − f (x) ( )k n+k n + k + β1 r n! n + k + β1 r=0 ( )r ( )n−r } α2 x n + k + α2 x − × − bn+k n + k + β2 n + k + β2 bn+k ( )k n + k + β2 (n + k)! (k) + f (x) ( − 1 . )k n+k n! n + k + β1
Since f (k) ∈ LipM α we have ( ) T (k) f ; x − f (k) (x) n+k,α,β ( )k ( )n n α ( ) n + k + β2 (n + k)! n + k + β2 ∑ r + α1 + θr k n ≤ M ( )k n + k + β1 bn+k − x r n + k n + k n! n + k + β1 r=0 )r ( )n−r } ( α2 n + k + α2 x x − − × bn+k n + k + β2 n + k + β2 bn+k )k ( n + k + β (n + k)! 2 + f (k) (x) ( − 1 . )k n+k n! n + k + β1 Applying Holder’s inequality and use the inequality ¨ f (k) (x) ≤ f (k) (0) + Mxα ≤ M f (1 + xα ) we have
≤
( ) (k) T (k) (x) f ; x − f n+k,α,β )k ( )n n ( )2 ( ) ( n + k + β2 (n + k)! n n + k + β2 ∑ r + α1 + θr k M ( bn+k − x )k n+k n+k n + k + β1 r n! n + k + β1 r=0 ( )r ( )n−r }α/2 x α2 n + k + α2 x × − − bn+k n + k + β2 n + k + β2 bn+k ( )k n + k + β (n + k)! 2 +M f (1 + xα ) ( − 1 . )k n+k n! n + k + β1
It is obvious that ( ) (k) T (k) ≤ (x) f ; x − f n+k,α,β
M
(n + k)!
(
n + k + β2 n+k
)k
[
)] α2 ( T˜ n,α,β (t − x)2 ; x
( )k n! n + k + β1 ( )k n + k + β (n + k)! 2 +M f (1 + xα ) ( − 1 , )k n+k n! n + k + β1
where ( ) n ) n ( ( ) ( n+k+β )n ∑ ( r+α1 +k x T˜ n,α,β f ; x = n+k 2 f n+k+β1 bn+k bn+k − r r=0
)r α2 n+k+β2
(
n+k+α2 n+k+β2
−
x bn+k
)n−r
.
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( ) By calculating T˜ n,α,β (t − x)2 ; x , we get ( ) ( )k (k) T (k) (x) f ; x − f ] α2 n + k + β2 n+k,α,β (n + k)! 1 [ 2 ≤ M x γ + xσ + τ n n n ( ) α α k 1+x n+k 1+x n! n + k + β1 ( )k n + k + β2 (n + k)! +M f ( − 1 , )k n+k n! n + k + β1 where (( )2 ) ) (n + k + β ) ( ) ( n 1 2 n + k + β2 2 γn := −1 −n , n+k n+k+β n+k n+k+β 1
(
1
) [(
) ) ( ) ( n 2 n + k + β2 2n σn α2 − 2 (α1 + k) − 2 α2 n+k n+k n + k + β1 ) (n + k + β ) ] ( n 2 +2 (α1 + k + 1) , n + k n + k + β1 ( ( )2 )]2 [( ) ) ( bn+k bn+k n n α − (α1 + k) τn := + 2α2 . n+k 2 n + k + β1 n + k + β1 n+k [ ] α2 2 Taking supremum overall x ∈ n+k+β bn+k , n+k+α n+k+β2 bn+k and passing to limit with 2 n → ∞ respectively, we obtain ( ) (k) T (k) (x) f ; x − f n+k,α,β = 0, lim sup n→∞ α2 1 + xα n+α b ≤x≤ 2 b bn+k : = n + k + β1
n+β2 n
n+β2
n
which is desired. 4. Acknowledgement The authors are thankful to the referees for valuable suggestions, leading to an overall improvement in the paper. References [1] I. Chlodowsky, Sur le developpement des fonctions definies dans un intervalle infini en series de polynomes de M. S. Bernstein, Compositio Math. 4 (1937) 380–393. [2] A. D. Gadjiev and A. M. Ghorbanalizadeh, Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables, Appl. Math. Compt.216 (2010) 890–901. [3] M. S. Floater, On the convergence of derivatives of Bernstein approximation, J. Approx. Theory 134 (2005) 130–135. [4] A.D. Gadjiev, I. Efendiev, E. Ibikli, Generalized Bernstein-Chlodowsky polynomials, Rocky Mount. J. Math. V.28 No 4 (1998) 1267–1277 . [5] A.D. Gadzhiev, The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P.P. Korovkin, English translated, Sov. Math. Dokl. Vol 15 No 5 (1974). [6] A.D. Gadzhiev, P.P Korovkin type theorems, Mathem. Zametki Vol 20 No: 5 (1976) Engl. Transl., Math. Notes 20 995–998. [7] E. A. Gadjieva, E. Ibikli, On generalization of Bernstein-Chlodowsky polynomials, Hacet. Bull. Nat. Sci. Eng. 24 (1995) 31–40. [8] E. A. Gadjieva and E. Ibikli, Weigted approximation by Bernstein-Chlodowsky polynomials, Indian J. Pure Appl. Math. 30 (1997) 83–87. [9] D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rew. Roum. Math. Pure. Appl. 13 (1968) 1173–1194.
