Filomat 27:4 (2013), 721–730 DOI 10.2298/FIL1304721M
Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat
Approximation Theorems for q-Bernstein-Kantorovich Operators N. I. Mahmudova , P. Sabancigila a Eastern
Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey
Abstract. In the present paper we introduce a q-analogue of the Bernstein-Kantorovich operators and investigate their approximation properties. We study local and global approximation properties and Voronovskaja type theorem for the q-Bernstein-Kantorovich operators in case 0 < q < 1.
1. Introduction In the last two decades interesting generalizations of Bernstein polynomials were proposed by Lupas¸ [15] and by Phillips [20]. Generalizations of the Bernstein polynomials based on the q-integers attracted a lot of interest and was studied widely by a number of authors. A survey of the obtained results and references on the subject can be found in [19]. Recently some new generalizations of well known positive linear operators, based on q-integers were introduced and studied by several authors, see [23], [5], [6], [8], [21], [22], [16]. The classical Kantorovich operator B∗n , n = 1, 2, ... is defined by (cf. [14]) B∗n
(
∑ ) f ; x := (n + 1) n
(
k=0
)
n k
∫ x (1 − x)n−k
k+1/n+1
k
f (t) dt k/n+1
) ) ∫ 1 ( n ( ∑ k+t n n−k k f = x (1 − x) dt, k n+1 0
f : [0, 1] → R.
(1)
k=0
These operators have been extensively considered in the mathematical literature. Also, a number of generalizations have been introduced by different authors (see, for instance [24], [25], [26]). In this paper, inspired by (1), we introduce a q-type generalization of Bernstein-Kantorovich polynomial operators as follows. ( ) ∑ ( ) B∗n,q f, x := pn,k q; x n
k=0
∫
(
1
f 0
) [k] + qk t dq t, [n + 1]
where f ∈ C [0, 1] , 0 < q < 1. 2010 Mathematics Subject Classification. Primary 41A46 ; Secondary 33D99, 41A25) Keywords. q-integers, positive operator, q-Bernstein-Kantorovich operator Received: 26 November 2011; Accepted: 8 December 2012 Communicated by Gradimir Milovanovic Email addresses:
[email protected] (N. I. Mahmudov),
[email protected] (P. Sabancigil)
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The paper is organized as follows. In Section 2, we give standard notations that will be used throughout the paper, introduce q-Bernstein-Kantorovich operators and evaluate the moments of B∗n,q . In Section 3 we study local and global convergence properties of the q-Bernstein-Kantorovich operators and prove Voronovskaja-type asymptotic formula. In the final section we give statistical approximation result for the q-Bernstein-Kantorovich operators.
2. q-Bernstein-Kantorovich operators Let q > 0. For any n ∈ N ∪ {0}, the q-integer [n] = [n]q is defined by [n] := 1 + q + ... + qn−1 , [0] := 0; and the q-factorial [n]! = [n]q ! by [n]! := [1] [2] ... [n] ,
[0]! := 1.
For integers 0 ≤ k ≤ n, the q-binomial coefficient is defined by [
]
n k
:=
[n]! . [k]! [n − k]!
The q-analogue of integration in the interval [0, A] (see [13]) is defined by ∫
( ) ∑ ( n) n f (t) dq t := A 1 − q f Aq q , 0 < q < 1. ∞
A
0
n=0
Let 0 < q < 1. Based on the q-integration we propose the Kantorovich type q-Bernstein polynomial as follows.
B∗n,q
(
) ∑ ( ) pn,k q; x f, x = n
k=0
) ∫1 ( [k] + qk t f dq t, 0 ≤ x ≤ 1, n ∈ N [n + 1] 0
where pn,k
(
) q; x :=
[
n k
] x (1 − k
x)qn−k
, (1 −
x)nq
:=
n−1 ∏ (
) 1 − qs x .
s=0
It can be seen that for q → 1− the q-Bernstein-Kantorovich operator becomes the classical BernsteinKantorovich operator. Lemma 2.1. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have B∗n,q (tm , x) =
) ) m−j ( m ( ∑ ∑ ( ) )i [n] j m− j ( n m q − 1 Bn,q t j+i , x . [ ] m i j [n + 1] m − j + 1 i=0 j=0
(2)
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Proof. The recurrence formula can be derived by direct computation. B∗n,q
(t , x) = m
n ∑
pn,k
(
)∑ q; x m
k=0
=
n ∑
( ) pn,k q; x
k=0
=
m ( ∑ j=0
m j
m ( ∑ m = j j=0
m ( ∑ m = j j=0
=
m ( ∑ m j j=0
j=0 0 m ( ∑ j=0
)
∫1 (
m j
m j )
)
[k] j qk(m−j) tm−j dq t [n + 1]m
qk(m−j) [k] j [ ] [n + 1]m m − j + 1
n ( ∑ )m−j [k] j ( ) [n] j qk − 1 + 1 p q; x ] [ m j n,k [n + 1] m − j + 1 k=0 [n] ) ( ) m− j n ∑ ∑ )i [k] j ( ) [n] j m− j ( k q pn,k q; x − 1 [ ] m j i [n + 1] m − j + 1 k=0 i=0 [n] ) ) m−j ( n ∑ )i ∑ [k] j+i ( ) [n] j m− j ( n q −1 p q; x [ ] j+i n,k i [n + 1]m m − j + 1 i=0 k=0 [n] ) ) m−j ( ∑ ( ) )i [n] j m− j ( n q − 1 Bn,q t j+i , x . [ ] m i [n + 1] m − j + 1 i=0
Lemma 2.2. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have 2q [n] 1 1 B∗n,q (1, x) = 1, B∗n,q (t, x) = x+ , [2] [n + 1] [2] [n + 1] ( ) ( ) q q + 2 q [n] [n − 1] 4q + 7q2 + q3 [n] 1 1 2 B∗n,q t2 , x = x + x+ . 2 2 [3] [2] [3] [3] [n + 1]2 [n + 1] [n + 1] Proof. Taking into account (2), by direct computation, we obtain explicit formulas for B∗n,q (t, x) and ( ) B∗n,q t2 , x as follows. ( ) ( ) [n] 1 Bn,q (1, x) + qn − 1 Bn,q (t, x) + Bn,q (t, x) [n + 1] [2] [n + 1] ( n ) q −1 2q [n] [n] 1 1 = + x+ = x+ [2] [n + 1] [n + 1] [2] [n + 1] [2] [n + 1] [2] [n + 1]
B∗n,q (t, x) =
and ( ) B∗n,q t2 , x =
(
1 [3] [n + 1]
2
( )) ( ) ( )2 Bn,q (1, x) + 2 qn − 1 Bn,q (t, x) + qn − 1 Bn,q t2 , x
( ) 2 B t , x n,q [2] [n + 1]2 [n + 1]2 ) ( ) ( )2 ( [n]2 2 [n] qn − 1 qn − 1 1 1 + + + x2 = 1 − [n] [3] [n + 1]2 [n + 1]2 [2] [n + 1]2 [3] [n + 1]2 ( ) ( n )2 ( ) 2 [n] qn − 1 q −1 2 qn − 1 [n]2 2 [n] x + + + + + [n] [n + 1]2 [2] [n] [n + 1]2 [3] [n] [n + 1]2 [2] [n + 1]2 [3] [n + 1]2 +
=
2 [n]
( ( )) ( ) Bn,q (t, x) + qn − 1 Bn,q t2 , x +
[n]2
2q + 3q2 + q3 q [n] [n − 1] 2 4q + 7q2 + q3 [n] 1 x + x+ . 2 2 [2] [3] [2] [3] [n + 1] [n + 1] [3] [n + 1]2
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Remark 2.3. It is observed from the above lemma that for q = 1, we get the moments of the Bernstein-Kantorovich operators. Lemma 2.4. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have ( ) ( ) ( ) ( ) 4 1 1 C 2 4 ∗ ∗ Bn,q (t − x) , x ≤ x (1 − x) + , x (1 − x) + , Bn,q (t − x) , x ≤ [n] [n] [n]2 [n]2 where C is a positive absolute constant. Proof. Note that estimation of the moments for the q-Bernstein operators is given in [17]. The proof is based on the estimations of the second and fourth order central moments of the q-Bersntein polynomials. ( ) ( ) C 1 x (1 − x) , Bn,q (t − x)4 , x ≤ Bn,q (t − x)2 , x = x (1 − x) . [n] [n]2 Indeed
( ) B∗n,q (t − x)2 , x =
n ∑
( ) pn,k q; x
k=0
∫1 ( 0
)2 )2 ∫1 ( k n ∑ [k] + qk t qt qn [k] ( ) [k] − x dq t = pn,k q; x − + − x dq t [n + 1] [n + 1] [n] [n + 1] [n] k=0
0
)2 ∫1 ( n ∑ ( ) [k] dq t + 2 pn,k q; x − x dq t [n] k=0 k=0 0 0 ( ) 4 2 4 1 4 + + x (1 − x) ≤ x (1 − x) + . ≤ [n] [n] [3] [n + 1]2 [n + 1]2 [n]
n ∑ ( ) ≤ 2 pn,k q; x
∫1 (
qk t qn [k] − [n + 1] [n] [n + 1]
)2
A similar calculus reveals: ( ) B∗n,q (t − x)4 , x =
n ∑
( ) pn,k q; x
k=0
∫1 ( 0
)4 )4 ∫1 ( k n ∑ [k] + qk t qt qn [k] ( ) [k] pn,k q; x − x dq t = − + − x dq t [n + 1] [n + 1] [n] [n + 1] [n] 0
)4 ∫1 ( n ∑ ( ) [k] dq t + 4 pn,k q; x − x dq t [n] k=0 k=0 0 0 ( ) 32 32 4 C 1 ≤ + + Cx (1 − x) ≤ x (1 − x) + . [n]2 [n]2 [5] [n + 1]4 [n + 1]4 [n]2
n ∑ ( ) ≤ 4 pn,k q; x
∫1 (
k=0
qk t qn [k] − [n + 1] [n] [n + 1]
)4
Lemma 2.5. Assume that 0 < qn < 1, qn → 1 and qnn → a as n → ∞. Then we have 1+a 1 x+ , 2 2 ) ( 1 2 2 2 2 ∗ lim [n]qn Bn,qn (t − x) ; x = − x − ax + x. n→∞ 3 3 lim [n]qn B∗n,qn (t − x; x) = −
n→∞
( ) Proof. To prove the lemma we use formulas for B∗n,qn (t; x) and B∗n,qn t2 ; x given in Lemma 2.2. ( { } ) [n]qn 2qn [n]qn 1 ∗ −1 x+ lim [n]qn Bn,qn (t − x; x) = lim [n]qn n→∞ n→∞ [2]qn [n + 1]qn [2]qn [n + 1]qn { } [n]qn 1 + qn+1 [n]qn 1+a 1 1 n = lim − =− x+ x+ . n→∞ [n + 1]qn [2]qn [2]qn [n + 1]qn 2 2
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( ) lim [n]qn B∗n,q (t − x)2 , x n→∞ ( ( ) ) = lim [n]qn B∗n,q t2 , x − x2 − 2xB∗n,q (t − x, x) n→∞ ( ) 2 3 qn qn + 2 [n]2qn − [n]qn x2 + lim [n] 4qn + 7qn + qn [n]qn x − 1 = lim [n]qn q n n→∞ n→∞ [3]qn [2]qn [3]qn [n + 1]2q [n + 1]2qn n − lim [n]qn 2xB∗n,qn (t − x, x) n→∞
) ( ) 4qn + 7q2n + q3n ( )( x = lim qn 1 − qnn 2qn + q2n + 2 x2 − lim 4qn + 3q2n + 2q3n x2 + lim n→∞ n→∞ n→∞ [2]qn [3]qn − lim [n]qn 2xB∗n,qn (t − x, x) n→∞
5 (1 − a) x2 − 3x2 + 2x + (1 + a) x2 − x 3 1 2 = − x2 − ax2 + x. 3 3 =
3. Local and global approximation We begin by considering the following K-functional: {
} ( )
K2 f, δ2 := inf f − 1 + δ2 1′′ : 1 ∈ C2 [0, 1] , δ ≥ 0, where { } C2 [0, 1] := 1 : 1, 1′ , 1′′ ∈ C [0, 1] . Then, in view of a known result [7], there exists an absolute constant C0 > 0 such that ( ) ( ) K2 f, δ2 ≤ C0 ω2 f, δ where
( ) ω2 f, δ := sup sup f (x − h) − 2 f (x) + f (x + h) 0 0 such that ) ( √ 1 + qn+1 x − 1 ( ) (x) δ n + ω f, B∗n,q f ; x − f (x) ≤ Cω2 f, [2] [n + 1] , [n] where f ∈ C [0, 1], δn (x) = φ2 (x) +
1 , 0 ≤ x ≤ 1 and 0 < q < 1. [n]
Proof. Let ( ) ( ) e B∗n,q f ; x = B∗n,q f ; x + f (x) − f (an x + bn ) , where f ∈ C [0, 1], an =
2q [n] 1+q [n+1]
and bn = ∫
t
1 (t) = 1 (x) + 1′ (x) (t − x) + x
1 1 1+q [n+1] .
Using the Taylor formula
(t − s) 1′′ (s) ds,
1 ∈ C2 [0, 1] ,
(3)
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we have ( ) e B∗n,q 1; x = 1 (x) + B∗n,q
(∫
t x
) ∫ (t − s) 1 (s) ds; x − ′′
an x+bn
(an x + bn − s) 1′′ (s) ds,
1 ∈ C2 [0, 1] .
x
Hence ) ∫ a x+b ( ∫ t n n ′′ ′′ |t − s| 1 (s) ds ; x + |an x + bn − s| 1 (s) ds x x ( )
2 2 ≤ 1′′ B∗n,q (t − x) ; x + 1′′ (an x + bn − x) ( ) } {
4 1 4 2 2 ′′ x (1 − x) + + x + ≤ 1 [n] [n] [n]2 [n]2
10
= δn (x) 1′′ . [n]
( ) e B∗n,q 1; x − 1 (x) ≤ B∗n,q
(4)
Using (4) and the uniform boundedness of e B∗n,q we get ( ) ∗ ( ) ( ) B∗n,q f ; x − f (x) ≤ e Bn,q 1; x − 1 (x) + f (x) − 1 (x) + f (an x + bn ) − f (x) B∗n,q f − 1; x + e
10
( ) ≤ 4 f − 1 + δn (x) 1′′ + ω f, |(an − 1) x + bn | . [n] Taking the infimum on the right hand side over all 1 ∈ C2 [0, 1], we obtain ( ) ( ) ( ) δ (x) B∗n,q f ; x − f (x) ≤ 10K2 f ; n + ω f, |(an − 1) x + bn | , [n] which together with (3) gives the proof of the theorem. Corollary 3.2. Assume that qn ∈ (0, 1) , qn → 1 as n → ∞. For any f ∈ C2 [0, 1] we have
( ) lim B∗n,qn f − f = 0. n→∞
We next present the direct global approximation theorem for the operators B∗n,q . In order to state the theorem we need the weighted K-functional of second order for f ∈ C[0, 1] defined by {
( )
( )} K2,ϕ f, δ2 := inf f − 1 + δ2 ϕ2 1′′ : 1 ∈ W 2 φ ,
δ ≥ 0, φ2 (x) = x (1 − x)
where } ( ) { W 2 φ := 1 ∈ C [0, 1] : 1′ ∈ AC [0, 1] , φ2 1′′ ∈ C [0, 1] , and 1′ ∈ AC [0, 1] means that 1 is differentiable and 1′ is absolutely continuous in [0, 1]. Moreover, the Ditzian-Totik modulus of second order is given by ( ) ) ( ) φ( sup f x − φ (x) h − 2 f (x) + f x + φ (x) h . ω2 f, δ := sup 0 0 such that ( ) ) (
( ) 1 1 → −
B∗n,q f − f
≤ Cωφ2 f, √ + ω ψ f, , [n] [n] where f ∈ C[0, 1], 0 < q < 1, φ2 (x) = x (1 − x), ψ (x) = 2x + 1. Proof. Let ( ) ( ) e B∗n,q f ; x = B∗n,q f ; x + f (x) − f (an x + bn ) , where f ∈ C [0, 1], an =
2q [n] 1+q [n+1]
and bn = ∫
t
1 (t) = 1 (x) + 1′ (x) (t − x) +
1 1 1+q [n+1] .
Using the Taylor formula ( ) 1 ∈ W2 φ ,
(t − s) 1′′ (s) ds,
x
we have ( ) e B∗n,q 1; x = 1 (x) + B∗n,q
(∫
Hence ( ) e B∗n,q 1; x − 1 (x) ≤ B∗n,q
t
) ∫ (t − s) 1′′ (s) ds; x −
x
an x+bn
(an x + bn − s) 1′′ (s) ds,
( ) 1 ∈ W2 φ .
x
) ∫ a x+b ( ∫ t n n ′′ ′′ |t − s| 1 (s) ds ; x + |an x + bn − s| 1 (s) ds . x x
(5)
Because the function δ2n is concave on [0, 1], we have for u = t + τ(x − t), τ ∈ [0, 1], the estimate τ |x − t| |x − t| |t − s| τ |x − t| ( )≤ 2 = 2 ≤ . 2 (t) 2 (x) 2 (t) δ2n (s) δn (s) δn (x) δn + τ δn − δn Hence, by (5), we find ) ( ∫ t
∫ an x+bn ( ) − s| |t ∗ 2 ′′ ∗ 2 ′′ 2 e Bn,q 1; x − 1 (x) ≤ δn 1 Bn,q ds ; x + δn 1 |an x + bn − s| /δn (s) ds x δ2n (s) x
( ) )
δ2n 1′′ ( ≤ 2 B∗n,q (t − x)2 ; x + (an x + bn − x)2 δ (x)
n
{ ( ) }
δ2n 1′′ 4 1 4 2 2 ≤ 2 x (1 − x) + + x + [n] δ (x) [n] [n]2 [n]2
n
{ ( )}
δ2n 1′′ 10 1 10
2 ′′
δ 1 . ≤ 2 x (1 − x) + = [n] [n] n δn (x) [n] Since
δ2n 1′′
≤
φ2 1′′
+
1
′′
1 [n + 1]
we have ( )
10
( ) 1
′′
∗ 2 ′′ e Bn,q 1; x − 1 (x) ≤
φ 1 +
1 . [n] [n]
(6)
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Using (6) and the uniform boundedness of e B∗n,q we get ( ) ( ) ∗ ( ) B∗n,q f ; x − f (x) ≤ e B∗n,q f − 1; x + e Bn,q 1; x − 1 (x) + f (x) − 1 (x) + f (an x + bn ) − f (x) ) (
10
1
′′
2 ′′
φ 1 +
1 + f (an x + bn ) − f (x) . ≤ 4 f − 1 + [n] [n] ( ) Taking the infimum on the right hand side over all 1 ∈ W 2 φ , we obtain ) ( ( ) 1 ∗ Bn,q f ; x − f (x) ≤ 10K2,φ f ; + f (an x + bn ) − f (x) . [n] On the other hand ( ) f (an x + bn ) − f (x) = f x + ψ (x) ((an − 1) x + bn ) − f (x) ( )) ( 1 + qn+1 1 x+ − f (x) ≤ sup f x + ψ (t) − ψ (x) [2] [n + 1] [2] [n + 1] ψ (x) ) ( n+1 1+q 1 − x+ ≤→ ω ψ f ; − ψ (x) [2] [n + 1] [2] [n + 1] ψ (x) ∗ ) ( Bn,q (t; x) − x → 2x + 1 → − ≤ − ≤ ω ψ f ; . ω f ; ψ ψ (x) [2] [n] ψ (x)
728
(7)
(8)
( √ ) ) ( φ 1 1 and the Ditzian-Totik modulus ω2 f, [n] we Hence, by (7) and (8), using the equivalence of K2,φ f, [n] get the desired estimate. Next we prove Voronovskaja type result for q-Bernstein-Kantorovich operators. Theorem 3.4. Assume that qn ∈ (0, 1) , qn → 1 and qnn → a as n → ∞. For any f ∈ C2 [0, 1] the following equality holds ( ) ( ) ( ) ( ) 1+a 1 1 1 2 lim [n]qn B∗n,qn f ; x − f (x) = f ′ (x) − + f ′′ (x) − x2 − ax2 + x x+ n→∞ 2 2 2 3 3 uniformly on [0, 1] . Proof. Let f ∈ C2 [0, 1] and x ∈ [0, 1] be fixed. By the Taylor formula we may write 1 ′′ f (x) (t − x)2 + r (t; x) (t − x)2 , (9) 2 where r (t; x) is the Peano form of the remainder, r (·; x) ∈ C [0, 1] and limt→x r (t; x) = 0. Applying B∗n,qn to (9) we obtain ( ) ( ) [n]qn B∗n,qn f ; x − f (x) = f ′ (x) [n]qn B∗n,qn (t − x; x) ) ( ) ( 1 + f ′′ (x) [n]qn B∗n,qn (t − x)2 ; x + [n]qn B∗n,qn r (t; x) (t − x)2 ; x . 2 By the Cauchy-Schwartz inequality, we have √ ) ( ) √ ( (10) B∗n,qn r (t; x) (t − x)2 ; x ≤ B∗n,qn (r2 (t; x) ; x) B∗n,qn (t − x)4 ; x . f (t) = f (x) + f ′ (x) (t − x) +
Observe that r2 (x; x) = 0 and r2 (·; x) ∈ C [0, 1] . Then it follows from Corollary 3.2 that ) ( lim B∗n,qn r2 (t; x) ; x = r2 (x; x) = 0 n→∞
uniformly with respect to x ∈ [0, 1] . Now from (10), (11) and Lemma 2.5 we get immediately ( ) lim [n]qn B∗n,qn r (t; x) (t − x)2 ; x = 0. n→∞
The proof is completed.
(11)
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4. Statistical approximation At this moment, we recall the concept of statistical convergence. The density of a subset K of N is given n ∑ by δ(K) := limn→∞ n1 χK (k) , whenever the limit exists, where χK is the characteristic function of K. A k=1
sequence x = {xn }n∈N is said to be statistically convergent to L if for any ε > 0, δ {n ∈ N : |xn − L| ≥ ε} = 0 and it is denoted{by} st − lim x = L (see[10]). Assume that qn n∈N be sequence from (0, 1] such that st − lim qn = 1. n
{ } Observe that for any sequence qn n∈N ⊂ (0, 1], satisfying (12) and for fixed x ∈ [0, 1] , we have ) ( x − 1 1 + qn+1 n δn (x) = 0, st − lim = stA − lim n [n]q n [2]q [n + 1]q n n n
(12)
(13)
which yields √ δn (x) = 0, st − lim ω2 f, n [n]qn and
) ( 1 + qn+1 x − 1 n = 0 st − lim ω f, [2]qn [n + 1]qn n
(14)
(15)
respectively. So, Theorem 3.1 gives the following statistical approximation theorem. { } Theorem 4.1. Assume that, qn n∈N is a sequence satisfying (12). Then, for all f ∈ C [0, 1] and fixed x ∈ [0, 1] , we have st − lim B∗n,q ( f ; x) − f (x) = 0. n
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