Some approximation results on Bleimann-Butzer-Hahn operators defined by (p,q)-integers
M. Mursaleen, Md. Nasiruzzaman, Asif Khan, Khursheed J. Ansari
aa r X i v : . [ m a t h . C A ] N ov SOME APPROXIMATION RESULTS ONBLEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q ) -INTEGERS M. MURSALEEN , MD. NASIRUZZAMAN , ASIF KHAN AND KHURSHEED J.ANSARI ∗ Abstract.
In this paper, we introduce a generalization of the Bleimann-Butzer-Hahn operators based on ( p, q )-integers and obtain Korovkin’s typeapproximation theorem for these operators. Furthermore, we compute conver-gence of these operators by using the modulus of continuity. Introduction and preliminaries
Bleimann, Butzer and Hahn (BBH) introduced the following operators in [2]as follows; L n ( f ; x ) = 1(1 + x ) n n X k =0 f (cid:18) kn − k + 1 (cid:19) (cid:20) nk (cid:21) x k , x ≥ q -type generalization of Bernstein polynomials was in-troduced by Lupa [7]. In 1997, Phillips [11] introduced another modificationof Bernstein polynomials. Also he obtained the rate of convergence and theVoronovskaja’s type asymptotic expansion for these polynomials.The BBH-type operators based on q -integers are defined as follows L qn ( f ; x ) = 1 ℓ n ( x ) n X k =0 f (cid:18) [ k ] q [ n − k + 1] q q k (cid:19) q k ( k − (cid:20) nk (cid:21) q x k (1.2)where ℓ n ( x ) = Q n − k =0 (1 + q s x ).Recently, Mursaleen et al [8] applied ( p, q )-calculus in approximation theory andintroduced first ( p, q )-analogue of Bernstein operators. They also introduced andstudied approximation properties of ( p, q )-analogue of Bernstein-Stancu opera-tors in [9].Let us recall certain notations on ( p, q )-calculus.The ( p, q ) integers [ n ] p,q are defined by[ n ] p,q = p n − q n p − q , n = 0 , , , · · · , < q < p ≤ . Date : Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author.2010
Mathematics Subject Classification.
Primary 41A10; Secondary 441A25, 41A36.
Key words and phrases. ( p, q )-integers; ( p, q )-Bernstein operators; ( p, q )-Bleimann-Butzer-Hahn operators; q -Bleimann-Butzer-Hahn operators; modulus of continuity. whereas q -integers are given by[ n ] q = 1 − q n − q , n = 0 , , , · · · , < q < . It is very clear that q -integers and ( p, q )-integers are different, that is we cannotobtain ( p, q ) integers just by replacing q by qp in the definition of q -integers but ifwe put p = 1 in definition of ( p, q ) integers then q -integers becomes a particularcase of ( p, q ) integers. Thus we can say that ( p, q )-calculus can be taken as ageneralization of q -calculus.Now by some simple calculation and induction on n, we have ( p, q )-binomialexpansion as follows( ax + by ) np,q := n X k =0 p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q a n − k b k x n − k y k , ( x + y ) np,q = ( x + y )( px + qy )( p x + q y ) · · · ( p n − x + q n − y ) , (1 − x ) np,q = (1 − x )( p − qx )( p − q x ) · · · ( p n − − q n − x )and the ( p, q )-binomial coefficients are defined by (cid:20) nk (cid:21) p,q = [ n ] p,q ![ k ] p,q ![ n − k ] p,q ! . Again it can be easily verified that ( p, q )-binomial expansion is different from q -binomial expansion and is not a replacement of q by qp .By some simple calculation, we have the following relation q k [ n − k + 1] p,q = [ n + 1] p,q − p n − k +1 [ k ] p,q . For details on q -calculus and ( p, q )-calculus, one can refer [15], [5, 12, 13],respectively.Now based on ( p, q )-integers, we construct ( p, q )-analogue of BBH-operators,and we call it as ( p, q )-Bleimann-Butzer-Hahn-Operators and investigate its Korovokin’s-type approximation properties, by using the test functions (cid:0) t t (cid:1) ν for ν = 0 , , C B ( R + ) be the set of all bounded and continuous functions on R + , then C B ( R + ) is linear normed space with k f k C B = sup x ≥ | f ( x ) | . Let ω denotes modulus of continuity satisfying the following condition:(1) ω is a non-negative increasing function on R + (2) ω ( δ + δ ) ≤ ω ( δ ) + ω ( δ )(3) lim δ → ω ( δ ) = 0. LEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q )-INTEGERS 3
Let H ω be the space of all real-valued functions f defined on the semiaxis R + satisfying the condition | f ( x ) − f ( y ) |≤ ω (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x x − y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , for any x, y ∈ R + . Theorem 1.1. [4]
Let { A n } be the sequence of positive linear operators from H ω into C B ( R + ) , satisfying the conditions lim n →∞ k A n (cid:18)(cid:18) t t (cid:19) ν ; x (cid:19) − (cid:18) x x (cid:19) ν k C B , for ν = 0 , , . Then for any function f ∈ H ω lim n →∞ k A n ( f ) − f k C B = 0 . We define ( p, q )-Bleimann-Butzer and Hahn-type operators based on ( p, q )-integers as follows: L p,qn ( f ; x ) = 1 ℓ p,qn ( x ) n X k =0 f (cid:18) p n − k +1 [ k ] p,q [ n − k + 1] p,q q k (cid:19) p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k (1.3)where, x ≥ , < q < p ≤ ℓ p,qn ( x ) = n − Y s =0 ( p s + q s x )and f is defined on semiaxis R + .And also by induction, we construct the Euler identity based on ( p, q )-analoguedefined as follows: n − Y s =0 ( p s + q s x ) = n X k =0 p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k (1.4)If we put p = 1, then we obtain q -BBH-operators. If we take f (cid:16) [ k ] p,q [ n − k +1] p,q (cid:17) in-stead of f (cid:16) p n − k +1 [ k ] p,q [ n − k +1] p,q q k (cid:17) in (1.3), then we obtain usual generalization of Bleimann,Butzer and Hahn operators based on ( p, q )-integers, then in this case it is impos-sible to obtain explicit expressions for the monomials t ν and (cid:0) t t (cid:1) ν for ν = 1 , (cid:0) t t (cid:1) ν for ν = 0 , ,
2. We em-phasize that these operators are more flexible than the classical BBH-operatorsand q -analogue of BBH-operators. That is depending on the selection of ( p, q )-integers, the rate of convergence of ( p, q )-BBH-operators is atleast as good as theclassical one. M. MURSALEEN, MD. NASIRUZZAMAN, ASIF KHAN, AND KHURSHEED J. ANSARI Main results
Lemma 2.1.
Let L p,qn ( f ; x ) be given by (1.3) , then for any x ≥ and < q
1, we have n X k =0 p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k = n − Y s =0 ( p s + q s x ) = ℓ p,qn ( x ) , which completes the proof.(2) Let t = p n − k +1 [ k ] p,q [ n − k +1] p,q q k , then tt +1 = [ k ] p,q p n +1 − k [ n +1] p,q L p,qn (cid:18) t t ; x (cid:19) = 1 ℓ p,qn ( x ) n X k =1 [ k ] p,q p n − k +1 [ n + 1] p,q p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k = 1 ℓ p,qn ( x ) n X k =1 [ n ] p,q p n − k +1 [ n + 1] p,q p ( n − k )( n − k − q k ( k − (cid:20) n − k − (cid:21) p,q x k = x (cid:18) ℓ p,qn ( x ) · [ n ] p,q [ n + 1] p,q p (cid:19) n − X k =0 p ( n − k )( n − k − q k ( k − (cid:20) n − k (cid:21) p,q ( qx ) k = p [ n ] p,q [ n + 1] p,q x x . (3) L p,qn (cid:16) t (1+ t ) ; x (cid:17) = ℓ p,qn ( x ) P nk =1 [ k ] p,q p n − k +1) [ n +1] p,q p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k .By some simple calculation, we have[ k ] p,q = p k − + q [ k − p,q , and [ k ] p,q = q [ k ] p,q [ k − p,q + p k − [ k ] p,q , using it we, get L p,qn (cid:18) t (1 + t ) ; x (cid:19) = 1 ℓ p,qn ( x ) n X k =2 q [ k ] p,q [ k − p,q p n − k +2 [ n + 1] p,q p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k + 1 ℓ p,qn ( x ) n X k =1 p k − [ k ] p,q p n − k +2 [ n + 1] p,q p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k LEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q )-INTEGERS 5 = 1 ℓ p,qn ( x ) q [ n ] p,q [ n − p,q [ n + 1] p,q n X k =2 p ( (2 n − k +2)+ ( n − k )( n − k − ) q k ( k − (cid:20) n − k − (cid:21) p,q x k + 1 ℓ p,qn ( x ) [ n ] p,q [ n + 1] p,q n X k =1 p ( ( k − n − k +2)+ ( n − k )( n − k − ) q k ( k − (cid:20) n − k − (cid:21) p,q x k = x ℓ p,qn ( x ) q [ n ] p,q [ n − p,q [ n + 1] p,q n − X k =0 p ( (2 n − k − ( n − k − n − k − ) q ( k +1)( k +2)2 (cid:20) n − k (cid:21) p,q x k + x ℓ p,qn ( x ) [ n ] p,q [ n + 1] p,q n − X k =0 p ( k +(2 n − k )+ ( n − k − n − k − ) q k ( k +1)2 (cid:20) n − k (cid:21) p,q x k = x ℓ p,qn ( x ) pq [ n ] p,q [ n − p,q [ n + 1] p,q n − X k =0 p ( n − k )( n − k − q k ( k − (cid:20) n − k (cid:21) p,q ( q x ) k + x ℓ p,qn ( x ) p n +1 [ n ] p,q [ n + 1] p,q n − X k =0 p ( n − k )( n − k − q k ( k − (cid:20) n − k (cid:21) p,q ( qx ) k = pq [ n ] p,q [ n − p,q [ n + 1] p,q x (1 + x )( p + qx ) + p n +1 [ n ] p,q [ n + 1] p,q (cid:18) x x (cid:19) . (cid:3) Korovkin type approximation properties.
In this section, we obtain the Korovkin’s type approximation properties forour operators defined by (1.3), with the help of Theorem 1.1.In order to obtain the convergence results for the operators L p,qn , we take q = q n , p = p n where q n ∈ (0 ,
1) and p n ∈ ( q n ,
1] satisfying,lim n p n = 1 , lim n q n = 1 (2.1) Theorem 2.2.
Let p = p n , q = q n satisfying (2.1) , for < q n < p n ≤ and if L p n ,q n n is defined by (1.3) , then for any function f ∈ H ω , lim n k L p n ,q n n ( f ; x ) − f k C B = 0 . Proof.
Using the Theorem 1.1, we see that it is sufficient to verify that followingthree conditions:lim n →∞ k L p n ,q n n (cid:18)(cid:18) t t (cid:19) ν ; x (cid:19) − (cid:18) x x (cid:19) ν k C B = 0 , ν = 0 , , ν = 0. Now it is easyto see that from (2) of Lemma 2.1 k L p n ,q n n (cid:18)(cid:18) t t (cid:19) ν ; x (cid:19) − (cid:18) x x (cid:19) ν k C B ≤ (cid:12)(cid:12)(cid:12)(cid:12) p n [ n ] p n ,q n [ n + 1] p n ,q n − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) p n q n (cid:19) (cid:18) − p nn n + 1] p n ,q n (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) . M. MURSALEEN, MD. NASIRUZZAMAN, ASIF KHAN, AND KHURSHEED J. ANSARI
Since q n [ n ] p n ,q n = [ n + 1] p n ,q n − p nn , [ n + 1] p n ,q n → ∞ as n → ∞ , the condition(2.2) holds for ν = 1. To verify this condition for ν = 2, consider (3) of Lemma2.1. Then, we see that k L p n ,q n n (cid:16)(cid:0) t t (cid:1) ; x (cid:17) − (cid:0) x x (cid:1) k C B = sup x ≥ n x (1+ x ) (cid:16) p n q n [ n ] pn,qn [ n − pn,qn [ n +1] pn,qn . xp n + q n x − (cid:17) + p n +1 n [ n ] pn,qn [ n +1] pn,qn · x x o .A small calculation leads to[ n ] p n ,q n [ n − p n ,q n [ n + 1] p n ,q n = 1 q n (cid:26) − p nn (cid:18) q n p n (cid:19) n + 1] p n ,q n + ( p nn ) (cid:18) q n p n (cid:19) n + 1] p n ,q n (cid:27) , and [ n ] p n ,q n [ n + 1] p n ,q n = 1 q n (cid:18) n + 1] p n ,q n − p nn n + 1] p n ,q n (cid:19) . Thus, we have k L p n ,q n n (cid:16)(cid:0) t t (cid:1) ; x (cid:17) − (cid:0) x x (cid:1) k C B ≤ p n q n n − p nn (cid:16) q n p n (cid:17) n +1] pn,qn + ( p nn ) (cid:16) q n p n (cid:17) n +1] pn,qn − o + p nn p n q n (cid:16) n +1] pn,qn − p nn n +1] pn,qn (cid:17) . This implies that the condition (2.2) holds for ν = 2 and the proof is completedby Theorem 1.1. (cid:3) Rate of Convergence.
In this section, we calculate the rate of convergence of operators (1.3) bymeans of modulus of continuity and Lipschitz type maximal functions.The modulus of continuity for f ∈ H ω is defined by e ω ( f ; δ ) = X | t t − x x |≤ δ,x,t ≥ | f ( t ) − f ( x ) | where e ω ( f ; δ ) satisfies the following conditions. For all f ∈ H ω ( R + )(1) lim δ → e ω ( f ; δ ) = 0(2) | f ( t ) − f ( x ) |≤ e ω ( f ; δ ) (cid:16) | t t − x x | δ + 1 (cid:17) Theorem 2.3.
Let p = p n , q = q n satisfy (2.1) , for < q n < p n ≤ and if L p n ,q n n is defined by (1.3) . Then for each x ≥ and for any function f ∈ H ω , wehave | L p n ,q n n ( f ; x ) − f |≤ e ω ( f ; p δ n ( x )) , where δ n ( x ) = x (1 + x ) (cid:18) p n q n [ n ] p n ,q n [ n − p n ,q n [ n + 1] p n ,q n xp n + q n x − p n [ n ] p n ,q n [ n + 1] p n ,q n + 1 (cid:19) + p n +1 n [ n ] p n ,q n [ n + 1] p n ,q n x x . LEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q )-INTEGERS 7
Proof. | L p n ,q n n ( f ; x ) − f | ≤ L p n ,q n n ( | f ( t ) − f ( x ) | ; x ) ≤ e ω ( f ; δ ) (cid:26) δ L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12) ; x (cid:19)(cid:27) . Now by using the Cauchy-Schwarz inequality, we have | L p n ,q n n ( f ; x ) − f | ≤ e ω ( f ; δ n ) δ n " L p n ,q n n (cid:18) t t − x x (cid:19) ; x ! ( L p n ,q n n (1; x )) ≤ e ω ( f ; δ n ) (cid:26) δ n h x (1+ x ) (cid:16) p n q n [ n ] pn,qn [ n − pn,qn [ n +1] pn,qn xp n + q n x − p n [ n ] pn,qn [ n +1] pn,qn + 1 (cid:17) + p n +1 n [ n ] pn,qn [ n +1] pn,qn x x i (cid:27) .This completes the proof. (cid:3) Now we will give an estimate concerning the rate of convergence by meansof Lipschitz type maximal functions. In [1], the Lipschitz type maximal functionspace on E ⊂ R + is defined as f W α,E = { f : sup(1 + x ) α e f α ( x ) ≤ M y ) α : x ≤ , and y ∈ E } (2.3)where f is bounded and continuous function on R + , M is a positive constant,0 < α ≤ f α as follows: f α ( x, t ) = X t> t = x | f ( t ) − f ( x ) || x − t | α . (2.4)We denote by d ( x, E ) the distance between x and E , that is d ( x, E ) = inf {| x − y | ; y ∈ E } . Theorem 2.4.
For all f ∈ f W α,E , we have | L p n ,q n n ( f ; x ) − f ( x ) |≤ M (cid:16) δ α n ( x ) + 2 ( d ( x, E )) α (cid:17) (2.5) where δ n ( x ) is defined as in Theorem 2.3.Proof. Let E denote the closure of the set E . Then there exits a x ∈ E suchthat | x − x | = d ( x, E ), where x ∈ R + . Thus we can write | f − f ( x ) |≤| f − f ( x ) | + | f ( x ) − f ( x ) | . Since L p n ,q n n is a positive linear operator, f ∈ f W α,E and by using the previousinequality, we have | L p n ,q n n ( f ; x ) − f ( x ) |≤| L p n ,q n n ( | f − f ( x ) | ; x )+ | f ( x ) − f ( x ) | L p n ,q n n (1; x ) ≤ M (cid:18) L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) + | x − x | α (1 + x ) α (1 + x ) α L p n ,q n n (1; x ) (cid:19) . M. MURSALEEN, MD. NASIRUZZAMAN, ASIF KHAN, AND KHURSHEED J. ANSARI
Since ( a + b ) α ≤ a α + b α , which consequently imply L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) ≤ L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) + L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x x − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) ≤ L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) + | x − x | α (1 + x ) α (1 + x ) α L p n ,q n n (1; x ) . By using the Hlder inequality with p = α and q = − α , we have L p n ,q n n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) t t − x x (cid:12)(cid:12)(cid:12)(cid:12) α ; x (cid:19) ≤ L p n ,q n n (cid:16)(cid:0) t t − x x (cid:1) ; x (cid:17) α ( L p n ,q n n (1; x )) − α + | x − x | α (1 + x ) α (1 + x ) α L p n ,q n n (1; x )= δ α n ( x ) + | x − x | α (1 + x ) α (1 + x ) α . This completes the proof. (cid:3)
Corollary 2.5.
If we take E = R + as a particular case of Theorem 2.4, then forall f ∈ f W α, R + , we have | L p n ,q n n ( f ; x ) − f ( x ) |≤ M δ α n ( x ) , where δ n ( x ) is defined in Theorem 2.3. Theorem 2.6. If x ∈ (0 , ∞ ) \ (cid:26) p n − k +1 [ k ] p,q [ n − k +1] p,q q k (cid:12)(cid:12)(cid:12)(cid:12) k = 0 , , , · · · , n (cid:27) , then L p,qn ( f ; x ) − f (cid:16) pxq (cid:17) = − x n +1 ℓ p,qn ( x ) h pxq ; p [ n ] p,q q n ; f i pq n ( n − − + xℓ p,qn ( x ) n − X k =0 (cid:20) pxq ; p n − k +1 [ k ] p,q [ n − k + 1] p,q q k ; f (cid:21) n − k ] p,q p ( n − k )( n − k +1)2 +1 q k ( k − − (cid:20) nk (cid:21) p,q x k . (2.6) Proof.
By using (1.3), we have L p,qn ( f ; x ) − f (cid:16) pxq (cid:17) = ℓ p,qn ( x ) P nk =0 h f (cid:16) p n − k +1 [ k ] p,q [ n − k +1] p,q q k (cid:17) − f (cid:16) pxq (cid:17)i p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k = − ℓ p,qn ( x ) P nk =0 (cid:16) pxq − p n − k +1 [ k ] p,q [ n − k +1] p,q q k (cid:17) h pxq ; p n − k +1 [ k ] p,q [ n − k +1] p,q q k ; f i p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k By using [ k ] p,q [ n − k +1] p,q (cid:20) nk (cid:21) p,q = (cid:20) nk − (cid:21) p,q , we have L p,qn ( f ; x ) − f (cid:18) pxq (cid:19) = − xℓ p,qn ( x ) n X k =0 (cid:20) pxq ; p n − k +1 [ k ] p,q [ n − k + 1] p,q q k ; f (cid:21) p ( n − k )( n − k − +1 q k ( k − − (cid:20) nk (cid:21) p,q x k . + 1 ℓ p,qn ( x ) n X k =1 (cid:20) pxq ; p n − k +1 [ k ] p,q [ n − k + 1] p,q q k ; f (cid:21) p ( n − k )( n − k − − ( k − n − q k ( k − − k (cid:20) nk − (cid:21) p,q x k LEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q )-INTEGERS 9 = − xℓ p,qn ( x ) n X k =0 (cid:20) pxq ; p n − k +1 [ k ] p,q [ n − k + 1] p,q q k ; f (cid:21) p ( n − k )( n − k − +1 q k ( k − − (cid:20) nk (cid:21) p,q x k . + xℓ p,qn ( x ) n − X k =0 (cid:20) pxq ; p n − k [ k + 1] p,q [ n − k ] p,q q k +1 ; f (cid:21) p ( n − k − n − k − − ( k − n ) q k ( k +1)2 − ( k +1) (cid:20) nk (cid:21) p,q x k = − x n +1 ℓ p,qn ( x ) h pxq ; p [ n ] p,q q n ; f i pq n ( n − − + xℓ p,qn ( x ) P n − k =0 nh pxq ; p n − k [ k +1] p,q [ n − k ] p,q q k +1 ; f i − h pxq ; p n − k +1 [ k ] p,q [ n − k +1] p,q q k ; f io p ( n − k )( n − k − +1 q k ( k − − (cid:20) nk (cid:21) p,q x k . Now by using the result h pxq ; p n − k [ k +1] p,q [ n − k ] p,q q k +1 ; f i − h pxq ; p n − k +1 [ k ] p,q [ n − k +1] p,q q k ; f i = (cid:18) p n − k [ k + 1] p,q [ n − k ] p,q q k +1 − p n − k +1 [ k ] p,q [ n − k + 1] p,q q k (cid:19) f (cid:20) pxq ; p n − k +1 [ k ] p,q [ n − k + 1] p,q q k ; p n − k [ k + 1] p,q [ n − k ] p,q q k +1 ; f (cid:21) and p n − k [ k + 1] p,q [ n − k ] p,q q k +1 − p n − k +1 [ k ] p,q [ n − k + 1] p,q q k = [ n + 1] p,q , we have L p,qn ( f ; x ) − f (cid:16) pxq (cid:17) = − x n +1 ℓ p,qn ( x ) h pxq ; p [ n ] p,q q n ; f i pq n ( n − − + xℓ p,qn ( x ) P n − k =0 nh pxq ; p n − k +1 [ k ] p,q [ n − k +1] p,q q k ; f i p n − k [ n +1] p,q [ n − k ] p,q [ n − k +1] p,q q k +1 o p ( n − k )( n − k − +1 q k ( k − − (cid:20) nk (cid:21) p,q x k . This completes the proof. (cid:3)
Some Generalization of L p,qn . In this section, we present some generalization of the operators L p,qn based on( p, q )-integers similar to work done in [3, 1].We consider a sequence of linear positive operators based on ( p, q )-integersas follows: L ( p,q ) ,γn ( f ; x ) = 1 ℓ p,qn ( x ) n X k =0 f (cid:18) p n − k +1 [ k ] p,q + γb n,k (cid:19) p ( n − k )( n − k − q k ( k − (cid:20) nk (cid:21) p,q x k , ( γ ∈ R )(2.7)where b n,k satisfies the following conditions: p n − k +1 [ k ] p,q + b n,k = c n and [ n ] p,q c n → , for n → ∞ . It is easy to check that if b n,k = q k [ n − k + 1] p,q + β for any n, k and 0 < q < p ≤ c n = [ n +1] p,q + β . If we choose p = 1, then operators reduce to generalizationof q -BBH opeartors defined in [1], and which turn out to be D.D. Stancu-typegeneralization of Bleimann, Butzer, and Hahn operators based on q -integers [14].If we choose γ = 0 , q = 1 as in [1] for p = 1, then the operators become thespecial case of the Balzs-type generalization of the q -BBH operators [1] given in[3]. Theorem 2.7.
Let p = p n , q = q n satisfying (2.1) , for < q n < p n ≤ and if L ( p n ,q n ) ,γn is defined by (2.7) , then for any function f ∈ f W α, [0 , ∞ ) , we have lim n k L ( p n ,q n ) ,γn ( f ; x ) − f ( x ) k C B ≤ M × max (cid:26)(cid:16) [ n ] pn,qn c n + γ (cid:17) α (cid:16) γ [ n ] pn,qn (cid:17) α , (cid:12)(cid:12)(cid:12)(cid:12) − [ n +1] pn,qn c n + γ (cid:12)(cid:12)(cid:12)(cid:12) α (cid:16) p n [ n ] pn,qn [ n +1] pn,qn (cid:17) α , − p n [ n ] pn,qn [ n +1] pn,qn + q n [ n ] pn,qn [ n − pn,qn [ n +1] pn,qn (cid:27) . Proof.
Using (1.3) and (2.7), we have | L ( p,q ) ,γn ( f ; x ) − f ( x ) |≤ ℓ pn,qnn ( x ) P nk =0 (cid:12)(cid:12)(cid:12)(cid:12) f (cid:16) p n − k +1 n [ k ] pn,qn + γb n,k (cid:17) − f (cid:16) p n − k +1 n [ k ] pn,qn γ + b n,k (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) p ( n − k )( n − k − n q k ( k − n (cid:20) nk (cid:21) p n ,q n x k + 1 ℓ p n ,q n n ( x ) n X k =0 (cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) p n − k +1 n [ k ] p n ,q n γ + b n,k (cid:19) − f (cid:18) p n − k +1 n [ k ] p n ,q n [ n − k + 1] p n ,q n q kn (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) p ( n − k )( n − k − n q k ( k − n (cid:20) nk (cid:21) p n ,q n x k + | L p n ,q n n ( f ; x ) − f ( x ) | . Since f ∈ f W α, [0 , ∞ ) and by using the Corollary 2.5, we can write | L ( p,q ) ,γn ( f ; x ) − f ( x ) |≤ Mℓ pn,qnn ( x ) P nk =0 (cid:12)(cid:12)(cid:12)(cid:12) p n − k +1 n [ k ] pn,qn + γp n − k +1 n [ k ] pn,qn + γ + b n,k − p n − k +1 n [ k ] pn,qn γ + p n − k +1 n [ k ] pn,qn + b n,k (cid:12)(cid:12)(cid:12)(cid:12) α p ( n − k )( n − k − n q k ( k − n (cid:20) nk (cid:21) p n ,q n x k + Mℓ pn,qnn ( x ) P nk =0 (cid:12)(cid:12)(cid:12)(cid:12) p n − k +1 n [ k ] pn,qn p n − k +1 n [ k ] pn,qn + γ + b n,k − p n − k +1 n [ k ] pn,qn p n − k +1 n [ k ] pn,qn +[ n − k +1] pn,qn q kn (cid:12)(cid:12)(cid:12)(cid:12) × p ( n − k )( n − k − n q k ( k − n (cid:20) nk (cid:21) p n ,q n x k + M δ α n ( x ) . This implies that | L ( p,q ) ,γn ( f ; x ) − f ( x ) |≤ M (cid:16) [ n ] pn,qn c n + γ (cid:17) α (cid:16) γ [ n ] pn,qn (cid:17) α + Mℓ p n ,q n n ( x ) (cid:12)(cid:12)(cid:12)(cid:12) − [ n + 1] p n ,q n c n + γ (cid:12)(cid:12)(cid:12)(cid:12) α n X k =0 (cid:18) p n − k +1 n [ k ] p n ,q n [ n + 1] p n ,q n (cid:19) α p ( n − k )( n − k − n q k ( k − n (cid:20) nk (cid:21) p n ,q n x k + M δ α n ( x )= M (cid:18) [ n ] p n ,q n c n + γ (cid:19) α (cid:18) γ [ n ] p n ,q n (cid:19) α + M (cid:12)(cid:12)(cid:12)(cid:12) − [ n + 1] p n ,q n c n + γ (cid:12)(cid:12)(cid:12)(cid:12) α L p n ,q n n (cid:18)(cid:18) t t (cid:19) α ; x (cid:19) + M δ α n ( x ) . LEIMANN-BUTZER-HAHN OPERATORS DEFINED BY ( p, q )-INTEGERS 11
Using the Hlder inequality for p = α , q = − α , we get | L ( p,q ) ,γn ( f ; x ) − f ( x ) |≤ M (cid:16) [ n ] pn,qn c n + γ (cid:17) α (cid:16) γ [ n ] pn,qn (cid:17) α + M (cid:12)(cid:12)(cid:12)(cid:12) − [ n +1] pn,qn c n + γ (cid:12)(cid:12)(cid:12)(cid:12) α L p n ,q n n (cid:0) t t ; x (cid:1) α ( L p n ,q n n (1; x )) − α + M δ α n ( x ) ≤ M (cid:16) [ n ] pn,qn c n + γ (cid:17) α (cid:16) γ [ n ] pn,qn (cid:17) α + M (cid:12)(cid:12)(cid:12)(cid:12) − [ n +1] pn,qn c n + γ (cid:12)(cid:12)(cid:12)(cid:12) α (cid:16) p n [ n ] pn,qn [ n +1] pn,qn x x (cid:17) α + M δ α n ( x ) . This completes the proof. (cid:3)
Acknowledgement.
Acknowledgements could be placed at the end of thetext but precede the references.
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