On a Kantorovich variant of (p,q)-Szasz-Mirakjan operators
aa r X i v : . [ m a t h . C A ] D ec On a Kantorovich variant of ( p, q ) -Sz´asz-Mirakjanoperators M. Mursaleen ∗ , Khursheed J. Ansari and Abylkassymova Elmira Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India chair ”The theory and methods of teaching informatics”, Science-Pedagogical Faculty, M.Auezov South Kazakhstan State University, Tauke Khan Avenue 5, Shymkent 160012,[email protected]; [email protected]; smanchik d armen @ mail.ru Abstract
In the present paper we propose a Kantorovich variant of ( p, q )-analogue of Sz´asz-Mirakjan operators. Weestablish the moments of the operators with the help of a recurrence relation that we have derived and then provethe basic convergence theorem. Next, the local approximation as well as weighted approximation properties of thesenew operators in terms of modulus of continuity are studied.
Keywords and phrases : ( p, q )-Sz´asz-Mirakjan operators; ( p, q )-Kantorovich-Sz´asz-Mirakjan operators; mod-ulus of continuity; weighted modulus of continuity; K -functional. AMS Subject Classifications (2010) : 41A10, 41A25, 41A36
1. Introduction and Notations
Approximation theory has been an established field of mathematics in the past century. Theapproximation of functions by positive linear operators is an important research topic in gen-eral mathematics and it also provides powerful tools to application areas such as computer-aidedgeometric design, numerical analysis, and solution of differential equations.During the last two decades, the applications of q -calculus emerged as a new area in thefield of approximation theory. The rapid development of q -calculus has led to the discovery ofvarious generalizations of Bernstein polynomials involving q -integers. Several researchers intro-duced and studied many positive linear operators based on q -integers, q -Bernstein basis, q -Betabasis, q -derivative and q -integrals etc. Using q -integers, Lupa¸s [10] introduced the first q -Bernsteinoperators [4] and investigated its approximating and shape-preserving properties. Another q -analogue of the Bernstein polynomials is due to Phillips [16]. Since then several generalizations ofwell-known positive linear operators based on q -integers have been introduced and studied theirapproximation properties. Aral [2] and Aral and Gupta [3] proposed and studied some q -analogueof Sz´asz-Mirakjan operators [18], defined on positive real axis. Also Mahmudov [11] introduceda q -parametric Sz´asz-Mirakjan operators and studied their convergence properties. Recently, ap-proximation properties for Kings type q-BernsteinKantorovich operators have been studied in [15].Very recently, Mursaleen et al applied ( p, q )-calculus in approximation theory and introducedthe ( p, q )-analogue of Bernstein operators [12, 13] and ( p, q )-Bernstein-Stancu operators [14] andinvestigated their approximation properties. Also Acar [1] has introduced ( p, q ) parametric gen-eralization of Sz´asz-Mirakjan operators. In the present work we have proposed a Kantorovichvariant of Sz´asz-Mirakjan operators and establish the moments of the operators with the help of arecurrence relation that we have derived and then prove the basic convergence theorem. Next, thelocal approximation as well as weighted approximation properties of these new operators in termsof modulus of continuity are studied. ∗ Corresponding author p, q )-integer was introduced in order to generalize or unify several forms of q -oscillatoralgebras well known in the earlier physics literature related to the representation theory of singleparameter quantum algebras [5]. Let us recall certain notations of ( 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 ≤ . The ( p, q )-facorial and ( p, q )-Binomial coefficients are defined by[ n ] p,q ! := (cid:26) [ n ] p,q [ n − p,q . . . [1] p,q , n ∈ N ;1 , n = 0and (cid:20) nk (cid:21) p,q := [ n ] p,q ![ k ] p,q ![ n − k ] p,q ! , respectively. Further, the ( p, q )-binomial expansions are given as( ax + by ) np,q := n X k =0 p ( n − k ) q ( k ) a n − k b k x n − k y k and ( x − y ) np,q := ( x − y )( px − qy )( p x − q y ) · · · ( p n − x − q n − y ) . Let m and n be two non-negative integers. Then the following assertion is valid( x − y ) m + np,q := ( x − y ) mp,q ( p m x − q m y ) np,q . Also, the ( p, q )-derivative of a function f , denoted by D p,q f , is defined by( D p,q f )( x ) := f ( px ) − f ( qx )( p − q ) x , x = 0 , ( D p,q f )(0) := f ′ (0)provided that f is differentiable at 0.The ( p, q )-derivative fulfils the following product rules D p,q ( f ( x ) g ( x )) := f ( px ) D p,q g ( x ) + g ( qx ) D p,q f ( x ) ,D p,q ( f ( x ) g ( x )) := f ( px ) D p,q g ( x ) + g ( qx ) D p,q f ( x ) . Moreover, D p,q (cid:18) f ( x ) g ( x ) (cid:19) := g ( qx ) D p,q f ( x ) − f ( qx ) D p,q g ( x ) g ( px ) g ( qx ) ,D p,q (cid:18) f ( x ) g ( x ) (cid:19) := g ( px ) D p,q f ( x ) − f ( px ) D p,q g ( x ) g ( px ) g ( qx ) . We consider the ( p, q )-exponential functions in the following forms: e p,q ( x ) = ∞ X n =0 p n ( n − / x n [ n ] p,q ! ,E p,q ( x ) = ∞ X n =0 q n ( n − / x n [ n ] p,q ! , e p,q ( x ) E p,q ( − x ) = 1. The definite integrals of the function f are definedby Z a f ( x ) d p,q x = ( q − p ) a ∞ X k =0 p k q k +1 f (cid:18) p k q k +1 a (cid:19) , when (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) < , and Z a f ( x ) d p,q x = ( p − q ) a ∞ X k =0 q k p k +1 f (cid:18) q k p k +1 a (cid:19) , when (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) > . Details on ( p, q )-calculus can be found in [5, 8, 17]. For p = 1, all the notions of ( p, q )-calculusare reduced to q -calculus.
2. Operators and estimation of moments
Now we set the ( p, q )-Sz´asz-Mirakjan basis function as s n ( p, q ; x ) =: E p,q (cid:0) − [ n ] p,q x (cid:1) ∞ X k =0 q k ( k − (cid:0) [ n ] p,q x (cid:1) k [ k ] p,q ! . For q ∈ (0 , p ), p ∈ ( q,
1] and x ∈ [0 , ∞ ), s n ( p, q ; x ) ≥
0. We can easily check that ∞ X k =0 s n ( p, q ; x ) =: E p,q (cid:0) − [ n ] p,q x (cid:1) ∞ X k =0 q k ( k − (cid:0) [ n ] p,q x (cid:1) k [ k ] p,q ! = 1 . For 0 < q < p ≤ p, q )-Sz´asz-Mirakjan operators are defined as S n ( f, p, q ; x ) = [ n ] p,q ∞ X k =0 p − k q k s n,k ( p, q ; x ) f (cid:18) [ k ] p,q q k − [ n ] p,q (cid:19) , x ∈ [0 , ∞ ) . (1)From the definition of the ( p, q )-Sz´asz-Mirakjan operators we derive the following formulas. Lemma 1.
Let 0 < q < p ≤
1. We have(i) S n (1 , p, q ; x ) = 1;(ii) S n ( t, p, q ; x ) = x ;(iii) S n ( t , p, q ; x ) = px q + x [ n ] p,q ;(iv) S n ( t , p, q ; x ) = p q x + p +2 pqq [ n ] p,q x + q [ n ] p,q x ;(v) S n ( t , p, q ; x ) = p q x + p ( p +2 q +3 q ) q [ n ] p,q x + p ( p +3 pq +3 q ) q [ n ] p,q x + q [ n ] p,q x .Now we propose our Kantorovich variant of ( p, q )-Sz´asz-Mirakjan operators (1) as follows:For f ∈ C [0 , ∞ ), 0 < q < p ≤ n , K n ( f, p, q ; x ) = [ n ] p,q ∞ X k =0 p − k q k s n,k ( p, q ; x ) Z [ k +1] p,qqk [ n ] p,q [ k ] p,qqk − n ] p,q f ( t ) d p,q t (2)3e will derive the recurrence formula for K n ( t m , p, q ; x ) and calculate K n ( t m , p, q ; x ) for m = 0 , , Lemma 2.
For the operators K n we have K n ( t m , p, q ; x ) = 1[ m + 1] p,q m X j =0 j X i =0 p i q i [ n ] j − ip,q (cid:18) ji (cid:19) S n ( t m + i − j , p, q ; x ) . (3) Proof.
Using the expansion a m +1 − b m +1 = ( a − b )( a m + a m − b + · · · + ab m − + b m ) we have Z [ k +1] p,qqk [ n ] p,q [ k ] p,qqk − n ] p,q t m d p,q t = 1[ m + 1] p,q ((cid:18) [ k + 1] p,q q k [ n ] p,q (cid:19) m +1 − (cid:18) [ k ] p,q q k − [ n ] p,q (cid:19) m +1 ) . Using [ k + 1] p,q = p k + q [ k ] p,q and also [ k + 1] p,q = q k + p [ k ] p,q , we have Z [ k +1] p,qqk [ n ] p,q [ k ] p,qqk − n ] p,q t m d p,q t = 1[ m + 1] p,q p k q k [ n ] p,q m X j =0 (cid:18) [ k + 1] p,q q k [ n ] p,q (cid:19) j (cid:18) [ k ] p,q q k − [ n ] p,q (cid:19) m − j = 1[ m + 1] p,q p k q k [ n ] p,q m X j =0 (cid:18) q k + p [ k ] p,q q k [ n ] p,q (cid:19) j (cid:18) [ k ] p,q q k − [ n ] p,q (cid:19) m − j = 1[ m + 1] p,q p k q k [ n ] p,q m X j =0 j X i =0 (cid:18) ji (cid:19) p i [ k ] ip,q q k ( j − i ) q kj [ n ] jp,q [ k ] m − jp,q q ( k − m − j ) [ n ] m − jp,q = 1[ m + 1] p,q p k q k [ n ] p,q m X j =0 j X i =0 (cid:18) ji (cid:19) p i [ k ] m + i − jp,q q ki [ n ] mp,q q ( k − m − j ) . Writing this in the definition of K n ( t m , p, q ; x ), we get K n ( t m , p, q ; x ) = [ n ] p,q ∞ X k =0 p − k q k s n,k ( p, q ; x ) Z [ k +1] p,qqk [ n ] p,q [ k ] p,qqk − n ] p,q t m d p,q t = 1[ m + 1] p,q m X j =0 ∞ X k =0 s n,k ( p, q ; x ) j X i =0 p i q i [ n ] j − ip,q (cid:18) ji (cid:19) [ k ] m + i − jp,q q ( k − m + i − j ) [ n ] m + i − jp,q = 1[ m + 1] p,q m X j =0 j X i =0 p i q i [ n ] j − ip,q (cid:18) ji (cid:19) ∞ X k =0 [ k ] m + i − jp,q q ( k − m + i − j ) [ n ] m + i − jp,q s n,k ( p, q ; x )= 1[ m + 1] p,q m X j =0 j X i =0 p i q i [ n ] j − ip,q (cid:18) ji (cid:19) S n ( t m + i − j , p, q ; x ) . Using the recurrence formula (3) we may easily calculate K n ( t m , p, q ; x ) for m = 0 , , Lemma 3.
We have(i) K n (1 , p, q ; x ) = 1;(ii) K n ( t, p, q ; x ) = q x + p,q [ n ] p,q ; 4iii) K n ( t , p, q ; x ) = pq x + (cid:16) p +[2] p,q q [3] p,q [ n ] p,q + q [ n ] p,q (cid:17) x + p,q [ n ] p,q ;(iv) K n ( t , p, q ; x ) = p q x + (cid:16) p +2 pqq [ n ] p,q + p (3 p +2 pq + q ) q [4] p,q [ n ] p,q (cid:17) x + (cid:16) q [ n ] p,q + p +2 pq + q q [4] p,q [ n ] p,q + p + qq [4] p,q [ n ] p,q (cid:17) x + p,q [ n ] p,q ;(v) K n ( t , p, q ; x ) = p q x + (cid:16) p ( p +2 q +3 q ) q [ n ] p,q + p (4 p +3 p q +2 pq + q ) q [5] p,q [ n ] p,q (cid:17) x + (cid:16) p ( p +3 pq +3 q ) q [ n ] p,q + ( p +2 pq )(4 p +3 p q +2 pq + q ) q [5] p,q [ n ] p,q + p (6 p +3 pq + q ) q [5] p,q [ n ] p,q (cid:17) x + (cid:16) q [ n ] p,q + p +3 p q +2 pq + q q [5] p,q [ n ] p,q + p +3 pq + q q [5] p,q [ n ] p,q + p + qq [5] p,q [ n ] p,q (cid:17) x + p,q [ n ] p,q ;(iv) K n (cid:0) ( t − x ) , p, q ; x (cid:1) = − qq x + p,q [ n ] p,q ;(v) K n (cid:0) ( t − x ) , p, q ; x (cid:1) = (cid:16) pq − q +1 (cid:17) x + (cid:16) p +[2] p,q q [3] p,q [ n ] p,q + q [ n ] p,q − p,q [ n ] p,q (cid:17) x + p,q [ n ] p,q ; (4)(vi) K n (cid:0) ( t − x ) , p, q ; x (cid:1) = x (cid:16) p q − p q + pq − q + 1 (cid:17) + x [ n ] p,q (cid:16) p ( p +2 q +3 q ) q + p (4 p +3 p q +2 pq + q ) q [5] p,q − p +2 pq ) q − p (3 p +2 pq + q ) q [4] p,q + p + q ) q [3] p,q + q − p,q (cid:17) + x [ n ] p,q (cid:16) p ( p +3 pq +3 q ) q + ( p +2 pq )(4 p +3 p q +2 pq + q ) q [5] p,q + p (6 p +3 pq + q ) q [5] p,q − p +2 pq + q ) q − q + p,q (cid:17) + x [ n ] p,q (cid:16) q + p +3 p q +2 pq + q q [5] p,q + p +3 pq + q q [5] p,q + p + qq [5] p,q (cid:17) . Proof.
Obviously, with the help of Lemma 1, we can get K n ( t, p, q ; x ) = 1[2] p,q (cid:26)(cid:16) pq (cid:17) S n ( t, p, q ; x ) + 1[ n ] p,q S n (1 , p, q ; x ) (cid:27) = 1 q x + 1[2] p,q [ n ] p,q ,K n ( t , p, q ; x ) = 1[3] p,q (cid:26)(cid:16) pq + p q (cid:17) S n ( t , p, q ; x ) + (cid:16) n ] p,q + 2 pq [ n ] p,q (cid:17) S n ( t, p, q ; x ) + 1[ n ] p,q S n (1 , p, q ; x ) (cid:27) = 1 q S n ( t , p, q ; x ) + p + [2] p,q q [3] p,q [ n ] p,q S n ( t, p, q ; x ) + 1[3] p,q [ n ] p,q S n (1 , p, q ; x )= pq x + (cid:16) p + [2] p,q q [3] p,q [ n ] p,q + 1 q [ n ] p,q (cid:17) x + 1[3] p,q [ n ] p,q . Using the linearity of the operators, we can have K n (cid:0) ( t − x ) , p, q ; x (cid:1) = K n ( t , p, q ; x ) − xK n ( t, p, q ; x ) + x K n (1 , p, q ; x )= pq x + (cid:16) p + [2] p,q q [3] p,q [ n ] p,q + 1 q [ n ] p,q (cid:17) x + 1[3] p,q [ n ] p,q − x (cid:16) q x + 1[2] p,q [ n ] p,q (cid:17) + x = (cid:16) pq − q + 1 (cid:17) x + (cid:16) p + [2] p,q q [3] p,q [ n ] p,q + 1 q [ n ] p,q − p,q [ n ] p,q (cid:17) x + 1[3] p,q [ n ] p,q . Remark
For q ∈ (0 ,
1) and p ∈ ( q,
1] it is obvious that lim n →∞ [ n ] p,q = p − q . In order to reach toconvergence results of the operator K n we take sequences q n ∈ (0 ,
1) and p n ∈ ( q n ,
1] such thatlim n →∞ p n = 1 lim n →∞ q n = 1. So we get that lim n →∞ [ n ] p n ,q n = ∞ .5hus the above remark provides an example that such a sequence can always be constructed.If we choose for a > b > q n = nn + a < nn + b = p n such that 0 < q n < p n ≤
1, it can be easilyseen that lim n →∞ p n = 1 , lim n →∞ q n = 1 and lim n →∞ p nn = e − b , lim n →∞ q nn = e − a . Hence we guarantee thatlim n →∞ [ n ] p n ,q n = ∞ .
3. Direct approximation result
In this section we study the Korovkin’s approximation property of the Kantorovich variant of( p, q )-Sz´asz operators.
Theorem 4.
Let 0 < q n < p n ≤ A >
0. Then for each f ∈ C m [0 , ∞ ) := (cid:8) f ∈ C [0 , ∞ ) : | f ( x ) | ≤ M f (1 + x m ) , for some M f > f, m > (cid:9) where C m [0 , ∞ ) be endowed withthe norm k f k m = sup x ∈ [0 , ∞ ) | f ( x ) | x m , the sequence of operators K n ( f, p n , q n ; x ) converges to f uniformlyon [0 , A ] if and only if lim n →∞ p n = 1 and lim n →∞ q n = 1. Proof.
First, we assume that lim n →∞ p n = 1 and lim n →∞ q n = 1. Now, we have to show that K n ( f, p n , q n ; x ) converges to f uniformly on [0 , A ].From Lemma 3, we see that K n (1 , p n , q n ; x ) → , K n ( t, p n , q n ; x ) → x, K n ( t , p n , q n ; x ) → x , uniformly on [0 , A ] as n → ∞ .Therefore, the well-known property of the Korovkin theorem implies that K n ( f, p n , q n ; x ) con-verges to f uniformly on [0 , A ] provided f ∈ C m [0 , ∞ ).We show the converse part by contradiction. Assume that p n and q n do not converge to 1. Thenthey must contain subsequences p n k ∈ (0 , q n k ∈ (0 , p n k → a ∈ [0 ,
1) and q n k → b ∈ [0 ,
1) as k → ∞ , respectively.Thus, 1[ n k ] p nk ,q nk = p n k − q n k ( p n k ) n k − ( q n k ) n k → k → ∞ and we get K n ( t, p n k , q n k ; x ) − x = 1 q n k x + 1[2] p nk ,q nk [ n ] p nk ,q nk − x → xb − x = 0 . This leads to a contradiction. Thus p n → q n → n → ∞ . Theorem 5.
Let f ∈ C [0 , ∞ ) , q = q n ∈ (0 ,
1) and p = p n ∈ ( q,
1] such that p n → , q n → n → ∞ and ω a +1 ( f, δ ) be the modulus of continuity on the finite interval [0 , a + 1] ⊂ [0 , ∞ ), where a >
0. Then | K n ( f, p, q ; x ) − f ( x ) | ≤ M f (1 + a ) δ n ( x ) + 2 ω a +1 ( f, δ n ( x ))where δ n ( x ) = q K n (cid:0) ( t − x ) , p, q ; x (cid:1) , given by (4).6 roof. For x ∈ [0 , a ] and t > a + 1, since t − x >
1, we have | f ( t ) − f ( x ) | ≤ M f (2+ x + t ) ≤ M f (cid:0) x +2( t − x ) (cid:1) ≤ M f (4+3 x )( t − x ) ≤ M f (1+ a )( t − x ) . (5)For x ∈ [0 , a ] and t ≤ a + 1, we have | f ( t ) − f ( x ) | ≤ ω a +1 ( f, | t − x | ) ≤ (cid:18) | t − x | δ (cid:19) ω a +1 ( f, δ ) (6)with δ > | f ( t ) − f ( x ) | ≤ M f (1 + a )( t − x ) + (cid:18) | t − x | δ (cid:19) ω a +1 ( f, δ ) , (7)for x ∈ [0 , a ] and t ≥
0. Thus, by applying the Cauchy-Schwarz’s inequality, we have | K n ( f, p, q ; x ) − f ( x ) | ≤ K n (cid:0) | f ( t ) − f ( x ) | , p, q ; x (cid:1) ≤ M f (1 + a ) K n (cid:0) ( t − x ) , p, q ; x (cid:1) + (cid:18) δ q K n (cid:0) ( t − x ) , p, q ; x (cid:1)(cid:19) ω a +1 ( f, δ ) ≤ M f (1 + a ) δ n ( x ) + 2 ω a +1 ( f, δ n ( x ))on choosing δ := δ n ( x ). This completes the proof of the theorem.
4. Local approximation
In this section we establish local approximation theorem for the Kantorovich variant of ( p, q )-Sz´asz operators. Let C B [0 , ∞ ) be the space of all real-valued continuous bounded functions f on[0 , ∞ ), endowed with the norm k f k = sup x ∈ [0 , ∞ ) | f ( x ) | . The Peetre’s K-functional is defined by K ( f, δ ) = inf g ∈ C [0 , ∞ ) (cid:8) k f − g k + δ k g ′′ k (cid:9) , where C B [0 , ∞ ) := (cid:8) g ∈ C B [0 , ∞ ) : g ′ , g ′′ ∈ C B [0 , ∞ ) (cid:9) . By [4, p.177, Theorem 2.4], there existsan absolute constant M > K ( f, δ ) ≤ M ω ( f, √ δ ) , (8)where δ > ω ( f, δ ) = sup
Let f ∈ C B [0 , ∞ ) and 0 < q < p ≤
1. Then, for every x ∈ [0 , ∞ ), we have | K n ( f, p, q ; x ) − f ( x ) | ≤ M ω (cid:16) f, p δ n ( x ) (cid:17) + ω (cid:16) f, p,q [ n ] p,q + 1 − qq x (cid:17) , where M is an absolute constant and 7 n ( x ) = K n (cid:0) ( t − x ) , p, q ; x (cid:1) + (cid:16) p,q [ n ] p,q + 1 − qq x (cid:17) . Proof.
For x ∈ [0 , ∞ ), we consider the auxiliary operators K ∗ n defined by K ∗ n ( f, p, q ; x ) = K n ( f, p, q ; x ) − f (cid:16) p,q [ n ] p,q + 1 q x (cid:17) + f ( x ) . (10)From Lemma 3, we observe that the operators K ∗ n ( f, p, q ; x ) are linear and reproduce the linearfunctions. Hence K ∗ n (cid:0) ( t − x ) , p, q ; x (cid:1) = K n (cid:0) ( t − x ) , p, q ; x (cid:1) − (cid:16) p,q [ n ] p,q + 1 q x − x (cid:17) = K n ( t, p, q ; x ) − xK n (1 , p, q ; x ) − (cid:16) p,q [ n ] p,q + 1 q x (cid:17) + x = 0 . (11)Let x ∈ [0 , ∞ ) and g ∈ C B [0 , ∞ ). Using the Taylor’s formula g ( t ) = g ( x ) + g ′ ( x )( t − x ) + Z tx ( t − u ) g ′′ ( u ) du. Applying K ∗ n to both sides of the above equation and using (11), we have K ∗ n ( g, p, q ; x ) − g ( x ) = K ∗ n (cid:0) ( t − x ) g ′ ( x ) , p, q ; x (cid:1) + K ∗ n (cid:18) Z tx ( t − u ) g ′′ ( u ) du, p, q ; x (cid:19) = g ′ ( x ) K ∗ n (cid:0) ( t − x ) , p, q ; x (cid:1) + K ( p,q ) n (cid:18) Z tx ( t − u ) g ′′ ( u ) du, p, q ; x (cid:19) − Z p,q [ n ] p,q + q xx (cid:16) p,q [ n ] p,q + 1 q x − u (cid:17) g ′′ ( u ) du = K n (cid:18) Z tx ( t − u ) g ′′ ( u ) du, p, q ; x (cid:19) − Z p,q [ n ] p,q + q xx (cid:16) p,q [ n ] p,q + 1 q x − u (cid:17) g ′′ ( u ) du. On the other hand, since (cid:12)(cid:12)(cid:12)(cid:12) Z tx ( t − u ) g ′′ ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z tx | t − u || g ′′ ( u ) | du ≤ k g ′′ k Z tx | t − u | du ≤ ( t − x ) k g ′′ k and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z p,q [ n ] p,q + q xx (cid:16) p,q [ n ] p,q + 1 q x − u (cid:17) g ′′ ( u ) du (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) p,q [ n ] p,q + 1 q x − u (cid:17) k g ′′ k
8e conclude that (cid:12)(cid:12)(cid:12) K ∗ n ( g, p, q ; x ) − g ( x ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) K n (cid:18) Z tx ( t − u ) g ′′ ( u ) du, p, q ; x (cid:19) − Z p,q [ n ] p,q + q xx (cid:16) p,q [ n ] p,q + 1 q x − u (cid:17) g ′′ ( u ) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ k g ′′ k K n (cid:0) ( t − x ) , p, q ; x (cid:1) + (cid:16) p,q [ n ] p,q + 1 q x − x (cid:17) k g ′′ k = δ n ( x ) k g ′′ k . (12)Now, taking into account boundedness of K ∗ n by (10), we have (cid:12)(cid:12) K ∗ n ( f, p, q ; x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) K n ( f, p, q ; x ) (cid:12)(cid:12) + 2 k f k ≤ k f k (13)Using (12) and (13) in (10), we obtain (cid:12)(cid:12) K n ( f, p, q ; x ) − f ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) K ∗ n ( f, p, q ; x ) − f ( x ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − f (cid:16) p,q [ n ] p,q + 1 q x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12) K ∗ n ( f − g, p, q ; x ) − ( f − g )( x ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − f (cid:16) p,q [ n ] p,q + 1 q x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) K ∗ n ( g, p, q ; x ) − g ( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) K ∗ n ( f − g, p, q ; x ) (cid:12)(cid:12) + (cid:12)(cid:12) ( f − g )( x ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) f ( x ) − f (cid:16) p,q [ n ] p,q + 1 q x (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) K ∗ n ( g, p, q ; x ) − g ( x ) (cid:12)(cid:12) ≤ k f − g k + ω (cid:16) f, p,q [ n ] p,q + 1 − qq x (cid:17) + δ n ( x ) k g ′′ k . Hence, taking the infimum on the right-hand side over all g ∈ C B [0 , ∞ ), we have the followingresult (cid:12)(cid:12) K n ( f, p, q ; x ) − f ( x ) (cid:12)(cid:12) ≤ K (cid:0) f, δ n ( x ) (cid:1) + ω (cid:16) f, p,q [ n ] p,q + 1 − qq x (cid:17) . In view of the property of K -functional (8), we get (cid:12)(cid:12) K n ( f, p, q ; x ) − f ( x ) (cid:12)(cid:12) ≤ M ω (cid:16) f, p δ n ( x ) (cid:17) + ω (cid:16) f, p,q [ n ] p,q + 1 − qq x (cid:17) . This completes the proof of the theorem.
5. Weighted approximation
Let f ∈ C ∗ [0 , ∞ ) := (cid:8) f ∈ C [0 , ∞ ) : lim x →∞ | f ( x ) | x < ∞ (cid:9) . Throughout the section, we assume that( p n ) and ( q n ) are sequences such that 0 < q n < p n ≤ p n → q n → n → ∞ . Theorem 7.
For each f ∈ C ∗ [0 , ∞ ), we havelim n →∞ k K n ( f, p n , q n ; x ) − f ( x ) k = 0 . roof. Using the Korovkin type theorem on weighted approximation in [7] we see that it issufficient to verify the following three conditionslim n →∞ k K n ( t i , p n , q n ; x ) − x i k = 0 , i = 0 , , . (14)Since K n (1 , p n , q n ; x ) = 1, (14) holds true for m = 0.By Lemma 3, we have k K n ( t, p n , q n ; x ) − x k = sup x ∈ [0 , ∞ ) | K n ( t, p n , q n ; x ) − x | x = sup x ∈ [0 , ∞ )
11 + x (cid:12)(cid:12)(cid:12)(cid:12) q n x + 1[2] p,q [ n ] p,q − x (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:16) q n − (cid:17) sup x ∈ [0 , ∞ ) x x + 1[2] p,q [ n ] p,q sup x ∈ [0 , ∞ )
11 + x ≤ q n − p,q [ n ] p,q . which implies that the condition in (14) holds for i = 1 as n → ∞ .Similarly we can write k K n ( t , p n , q n ; x ) − x k = sup x ∈ [0 , ∞ ) | K n ( t , p n , q n ; x ) − x | x ≤ (cid:16) p n q n − (cid:17) sup x ∈ [0 , ∞ ) x x + (cid:16) p n + q n q n [3] p n ,q n [ n ] p n ,q n + 1 q n [ n ] p n ,q n (cid:17) sup x ∈ [0 , ∞ ) x x + 1[3] p n ,q n [ n ] p n ,q n sup x ∈ [0 , ∞ )
11 + x ≤ p n q n − p n + q n q n [3] p n ,q n [ n ] p n ,q n + 1 q n [ n ] p n ,q n + 1[3] p n ,q n [ n ] p n ,q n , which implies that lim n →∞ k K n ( t , p n , q n ; x ) − x k = 0 , (14) holds for i = 2. Thus the proof is completed.We give the following theorem to approximate all functions in C ∗ [0 , ∞ ). This type of resultsare given in [6] for classical Sz´asz operators. Theorem 8.
For each f ∈ C ∗ [0 , ∞ ) and α >
0, we havelim n →∞ sup x ∈ [0 , ∞ ) (cid:12)(cid:12) K n ( f, p n , q n ; x ) − f ( x ) (cid:12)(cid:12) (1 + x ) α = 0 . roof. Let x ∈ [0 , ∞ ) be arbitrary but fixed. Thensup x ∈ [0 , ∞ ) (cid:12)(cid:12) K n ( f, p n , q n ; x ) − f ( x ) (cid:12)(cid:12) (1 + x ) α = sup x ≤ x (cid:12)(cid:12) K n ( f, p n , q n ; x ) − f ( x ) (cid:12)(cid:12) (1 + x ) α + sup x>x (cid:12)(cid:12) K n ( f, p n , q n ; x ) − f ( x ) (cid:12)(cid:12) (1 + x ) α ≤ k K n ( f ) − f k C [0 ,x ] + k f k sup x>x (cid:12)(cid:12) K n (cid:0) t , p, q ; x (cid:1)(cid:12)(cid:12) (1 + x ) α + sup x>x | f ( x ) | (1 + x ) α . (15)Since | f ( x ) | ≤ k f k (1 + x ), we have sup x>x | f ( x ) | (1+ x ) α ≤ k f k (1+ x ) α .Let ε > x to be so large that k f k (1 + x ) α < ε . (16)In view of Theorem 4, we obtain k f k lim n →∞ (cid:12)(cid:12) K n (cid:0) t , p, q ; x (cid:1)(cid:12)(cid:12) (1 + x ) α = 1 + x (1 + x ) α k f k = k f k (1 + x ) α ≤ k f k (1 + x ) α < ε . (17)Using Theorem 5, we can see that the first term of the inequality (15), implies that k K n ( f ) − f k C [0 ,x ] < ε , as n → ∞ . (18)Combining (16)-(18), we get that desired result.For f ∈ C ∗ [0 , ∞ ), the weighted modulus of continuity is defined asΩ ( f, δ ) = sup x ≥ , 11 + x ≤ C , for sufficiently large n (20)where C is a positive constant. From (20), there exists a positive constant K such that K n ( ϕ x , p, q ; x ) ≤ K (1 + x ), for sufficiently large n .Proceeding similarly x K n (1 + t , p, q ; x ) ≤ C , for sufficiently large n , where C is a positiveconstant.So there exists a positive constant K such that K n ( ϕ x , p, q ; x ) ≤ K (1 + x ), where x ∈ [0 , ∞ ) n is large enough. Also we get K n ( ψ x , p, q ; x ) = (cid:16) pq − q + 1 (cid:17) x + (cid:16) p + [2] p,q q [3] p,q [ n ] p,q + 1 q [ n ] p,q − p,q [ n ] p,q (cid:17) x + 1[3] p,q [ n ] p,q = α n x + β n x + γ n . Hence form (19), we have | K n ( f, p, q ; x ) − f ( x ) | ≤ (1 + x ) (cid:16) K + 1 δ n K p α n x + β n x + γ n (cid:17) Ω ( f, δ n ) . If we take δ n = max { α n , β n , γ n } , then we get | K n ( f, p, q ; x ) − f ( x ) | ≤ (1 + x ) (cid:16) K + K √ x + x + 1 (cid:17) Ω ( f, δ n ) ≤ K (1 + x λ )Ω ( f, δ n ) , for sufficiently large n and x ∈ [0 , ∞ ) . Hence the proof is completed. 12 eferences [1] T. Acar, ( p, q )-generalization of Sz´asz-Mirakjan operators, arXiv:submit/1263016[math.CA].[2] A. Aral, A generalization of Sz´asz-Mirakjan operators based on q -integers, Math. Comput.Model. 47(9-10), (2008) 1052-1062.[3] Ali Aral and V. Gupta, The q -derivative and applications to q -Sz´asz-Mirakjan operators,Calcolo 43(3), (2006) 151-170.[4] S.N. Bernstein, D´emostration du th´eor`eme de Weierstrass fond´ee sur le calcul de probabilit´es,Comm. Soc. Math. Kharkow (2), 13 (1912-1913) 1-2.[5] R. Chakrabarti and R. Jagannathan, A ( p, q )-oscillator realization of two parameter quantumalgebras, J. Phys. A: Math. Gen., 24 (1991) 711-718.[6] O. Doˇgru and E.A. Gadjieva, Agirlikli uzaylarda Sz´asz tipinde operat¨orler dizisinin s¨ureklifonksiyonlara yaklasimi’,II, Kizilirmak Uluslararasi Fen Bilimleri Kongresi Bildiri Kitabi, 29-37, Kirikkale (1998) (Konusmaci: O. Dogru) (T¨urk¸ce olarak sunulmus ve yayinlanmistir).[7] A.D. Gadjieva, A problem on the convergence of a sequence of positive linear operators onunbounded sets, and theorems that are analogous to P.P. Korovkin’s theorem, Dokl. Akad.Nauk SSSR 218 (1974), 1001-1004 (in Russian); Sov. Math. Dokl. 15 (1974), 1433-1436 (inEnglish).[8] R. Jagannathan and K.S. Rao, Two-parameter quantum algebras, twin-basic numbers, andassociated generalized hypergeometric series, Proceedings of the International Conference onNumber Theory and Mathematical Physics, December 2005, 20-21.[9] A.J. L´opez-Moreno, Weighted simultaneous approximation with Baskakov type operators,Acta Math. Hungar. 104(1-2), (2004) 143-151.[10] A. Lupa¸s, A q -analogue of the Bernstein operator, Seminar on Numerical and StatisticalCalculus, University of Cluj-Napoca, 9(1987), 85-92.[11] N.I. Mahmudov, On q -parametric Sz´asz-Mirakjan operators, Mediterr. J. Math. 7(3) (2010),297-311.[12] M. Mursaleen, K.J. Ansari and Asif Khan, On ( p, q )-analogue of Bernstein operators, Appl.Math. Comput., doi: 10.1016/j.amc.2015.04.090.[13] M. Mursaleen, K.J. Ansari and Asif Khan, On ( p, q )-analogue of Bernstein operators(Revised),arXiv:1503.07404v2 [math.CA] 20 Nov 2015.[14] M. Mursaleen, K.J. Ansari and Asif Khan, Some approximation results by ( p, q )-analogue ofBernstein-Stancu operators, Appl. Math. Comput. 264 (2015) 392-402.[15] M. Mursaleen, Faisal Khan and Asif Khan, Approximation properties for Kings type modified q -BernsteinKantorovich operators, Math. Meth. Appl. Sci. 2015, doi: 10.1002/mma.3454.[16] G.M. Phillips, Bernstein polynomials based on the q -integers, The heritage of P.L.Chebyshev,Ann. Numer. Math., 4 (1997) 511-518. 1317] P.N. Sadjang, On the fundamental theorem of ( p, q )-calculus and some ( p, qp, q