International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 2 (2017), 206-215 http://www.etamaths.com DUNKL GENERALIZATION OF q-PARAMETRIC SZÁSZ-MIRAKJAN OPERATORS M. MURSALEEN∗, MD. NASIRUZZAMAN AND A.A.H. AL-ABIED Abstract. In this paper, we construct q-parametric Szász-Mirakjan operators generated by the q- Dunkl generalization of the exponential function. We obtain Korovkin’s type approximation theorem and compute convergence of these operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. 1. Introduction In 1912, Bernstein [5] introduced the following sequence of operators Bn : C[0, 1] → C[0, 1] defined by Bn(f; x) = n∑ k=0 ( n k ) xk(1 −x)n−kf ( k n ) , (n ∈ N) x ∈ [0, 1], f ∈ C[0, 1]. (1.1) In 1950, Szász [27] introduced the operators Sn(f; x) = e −nx ∞∑ k=0 (nx)k k! f ( k n ) , x ≥ 0, f ∈ C[0,∞). (1.2) For the last two decades, the application of q-calculus emerged as a new area in the field of approx- imation theory. The first q-analogue of Bernstein polynomials was introduced by Lupaş [15] and later Phillips [23] considered another q-analogue of the Bernstein polynomials. Later on, many authors in- troduced q-generalization of various operators and investigated several approximation properties. For instance, [1], [2], [3], [8], [9], [10], [12], [16]– [22], [24]. The q-integer [n]q, the q-factorial [n]q! and the q-binomial coefficient are defined by (see [13]) [n]q := { 1−qn 1−q , if q ∈ R + \{1} n, if q = 1, for n ∈ N and [0]q = 0, [n]q! := { [n]q[n− 1]q · · · [1]q, n ≥ 1, 1, n = 0,[ n k ] q := [n]q! [k]q![n−k]q! , respectively. The q-analogue of (1 + x)n is the polynomial (1 + x)nq := { (1 + x)(1 + qx) · · ·(1 + qn−1x) n = 1, 2, 3, · · · 1 n = 0. A q-analogue of the common Pochhammer symbol also called a q-shifted factorial is defined as (x; q)0 = 1, (x; q)n = n−1∏ j=0 (1 −qjx), (x; q)∞ = ∞∏ j=0 (1 −qjx). Received 1st September, 2016; accepted 8th November, 2016; published 1st March, 2017. 2010 Mathematics Subject Classification. 41A25, 41A36, 33C45. Key words and phrases. q-integers; Szász operator; Szász-Mirakjan operator; Dunkl analogue; modulus of continuity; Lipschitz class. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 206 DUNKL GENERALIZATION OF q-PARAMETRIC SZÁSZ-MIRAKJAN OPERATORS 207 The Gauss binomial formula is given by (x + a)nq = n∑ k=0 [ n k ] q qk(k−1)/2akxn−k. There are two q-analogue of the exponential function ez, defined as (see also [14]) For | z |< 1 1−q and | q |< 1, e(z) = ∞∑ k=0 zk k! = 1 1 − ((1 −q)z)∞q , (1.3) and for | q |< 1, E(z) = ∞∏ j=0 ( 1 + (1 −q)qjz )∞ q = ∞∑ k=0 q k(k−1) 2 zk k! = (1 + (1 −q)z)∞q , (1.4) where (1 −x)∞q = ∏∞ j=0(1 −q jx). The q−analogue of Bernstein operators [23] is defined as follows: Bn,q(f; x) = n∑ k=0 [ n k ] q xk n−k−1∏ s=0 (1 −qsx) f ( [k]q [m]q ) , x ∈ [0, 1],n ∈ N. (1.5) In [4] q-Szász-Mirakjan operators were defined as follows: Sn,q(f; x) := E ( − [n]qx bn ) ∞∑ k=0 f ( [k]qbn [n]q ) [n]kqx k [k]q!bkn , (1.6) where 0 ≤ x < bn (1−q)[n]q , f ∈ C[0,∞) and {bn} is a sequence of positive numbers such that limn→∞ bn = ∞. Sucu [26] defined a Dunkl analogue of Szász operators via a generalization of the exponential function given by [25] as follows: S∗n(f; x) := 1 eµ(nx) ∞∑ k=0 (nx)k γµ(k) f ( k + 2µθk n ) (n ∈ N), (1.7) where x ≥ 0,f ∈ C[0,∞),µ ≥ 0 and eµ(x) = ∞∑ n=0 xn γµ(n) . Also here γµ(2k) = 22kk!Γ ( k + µ + 1 2 ) Γ ( µ + 1 2 ) , and γµ(2k + 1) = 22k+1k!Γ ( k + µ + 3 2 ) Γ ( µ + 1 2 ) . A recursion formula for γµ is given by γµ(k + 1) = (k + 1 + 2µθk+1)γµ(k), k = 0, 1, 2, · · · , where θk = { 0 if k ∈ 2N 1 if k ∈ 2N + 1 In [6], Cheikh et al. studied the q-Dunkl classical q-Hermite type polynomials and presented the definitions of q-Dunkl analogues of exponential functions, recursion relations and notaions for µ > −1 2 and 0 < q < 1, respectively. eµ,q(x) = ∞∑ n=0 xn γµ,q(n) , x ∈ R, (1.8) 208 MURSALEEN, NASIRUZZAMAN AND AL-ABIED Eµ,q(x) = ∞∑ n=0 q n(n−1) 2 xn γµ,q(n) , x ∈ R, (1.9) γµ,q(n + 1) = ( 1 −q2µθn+1+n+1 1 −q ) γµ,q(n), n ∈ N, (1.10) θn = { 0 if n ∈ 2N, 1 if n ∈ 2N + 1. An explicit formula for γµ,q(n) is given by γµ,q(n) = (q2µ+1,q2)[ n+1 2 ](q 2,q2)[ n 2 ] (1 −q)n γµ,q(n), n ∈ N. Some of the special cases of γµ,q(n) are listed as: γµ,q(0) = 1, γµ,q(1) = 1 −q2µ+1 1 −q , γµ,q(2) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q ) , γµ,q(3) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q )( 1 −q2µ+3 1 −q ) , γµ,q(4) = ( 1 −q2µ+1 1 −q )( 1 −q2 1 −q )( 1 −q2µ+3 1 −q )( 1 −q4 1 −q ) . In [11], Içöz and Çekim gave a Dunkl generalization of Szász operators via q-calculs as: Dn,q(f; x) = 1 eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) f ( 1 −q2µθk+k 1 −qn ) (n ∈ N), (1.11) where µ > 1 2 , x ≥ 0, 0 < q < 1 and f ∈ C[0,∞). In this paper, we define a Dunkl generalization of q-parametric Szász-Mirakjan operators: For any x ∈ [0,∞), f ∈ C[0,∞), 0 < q < 1, and µ > 1 2 , we define D∗n,q(f; x) = 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 f ( 1 −q2µθk+k qk−2(1 −qn) ) (n ∈ N). (1.12) 2. Main results Lemma 2.1. Let D∗n,q(. ; .) be the operators given by (1.12). Then we have the following identities: (1) D∗n,q(e0; x) = 1 (2) D∗n,q(e1; x) = qx (3) qx2 + q 2(1+µ) [n]q [1 − 2µ]qx ≤ D∗n,q(e2; x) ≤ qx2 + q2(1+µ) [n]q [1 + 2µ]qx, where ej(t) = t j, j = 0, 1, 2, · · · . Proof. (1) D∗n,q(1; x) = 1 Eµ,q([n]qx) ∑∞ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 = 1 (from (1.9)). DUNKL GENERALIZATION OF q-PARAMETRIC SZÁSZ-MIRAKJAN OPERATORS 209 (2) D∗n,q(e1; x) = 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) ) = q [n]q 1 Eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q (k−1)(k−2) 2 = q [n]q 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−1) 2 = qx Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 = qx. (3) D∗n,q(e2; x) = 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) )2 = 1 [n]2q 1 Eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q k2−5k+8 2 ( 1 −q2µθk+k (1 −q) ) = q2 [n]2q 1 Eµ,q([n]qx) ∞∑ k=1 ([n]qx) k γµ,q(k − 1) q (k−1)(k−4) 2 ( 1 −q2µθk+k (1 −q) ) , hence D∗n,q(e2; x) = q2 [n]2q 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−3) 2 ( 1 −q2µθk+1+k+1 (1 −q) ) . (2.1) A simple calculation yields [2µθk+1 + k + 1]q = [2µθk + k]q + q 2µθk+k[2µ(−1)k + 1]q. (2.2) Replacing k by 2k, then (2.2) implies that [2µθ2k+1 + 2k + 1]q = ( 1 −q2µθ2k+2k 1 −q ) + q2µθ2k+2k[1 + 2µ]q, (2.3) and by replacing k by 2k + 1, we have [2µθ2k+2 + 2k + 2]q = ( 1 −q2µθ2k+1+2k+1 1 −q ) + q2µθ2k+1+2k+1[1 − 2µ]q. (2.4) Now by separating (2.1), to even and odd terms and using (2.3) and (2.4) D∗n,q(e2; x) = q2 [n]2q 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k+1 γµ,q(k) q k(k−3) 2 ( 1 −q2µθk+k (1 −q) )∣∣∣∣ for k=2k,2k+1 + q2 [n]2q 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) 2k+1 γµ,q(2k) qk(2k−3)q2µθ2k+2k[1 + 2µ]q + q2 [n]2q 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) 2k+2 γµ,q(2k + 1) q(k−1)(2k+1)q2µθ2k+1+2k+1[1 − 2µ]q. Since [1 − 2µ]q ≤ [1 + 2µ]q, (2.5) 210 MURSALEEN, NASIRUZZAMAN AND AL-ABIED using the inequality (2.5), we have D∗n,q(e2; x) ≤ qx 2 + q2 [n]q x Eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 (q[n]qx) 2k γµ,q(2k) qk(2k−3) + q2(µ+1) [n]q x Eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 (q[n]qx) 2k+1 γµ,q(2k + 1) q(k−1)(2k+1) ≤ qx2 + q2(µ+1) [n]q x Eµ,q([n]qx) [1 + 2µ]q ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ≤ qx2 + q2(µ+1) [n]q [1 + 2µ]qx. Similarly, we can show that D∗n,q(e2; x) ≥ qx 2 + q2(µ+1) [n]q [1 − 2µ]qx. � Lemma 2.2. Let the operators D∗n,q(. ; .) be given by (1.12). Then we have the following identities: (1) D∗n,q(e1 − 1; x) = qx− 1 (2) D∗n,q(e1 −x; x) = (q − 1)x (3) (1 −q)x2 + q 2(1+µ) [n]q [1 − 2µ]qx ≤ D∗n,q((e1 −x)2; x) ≤ (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx. Next, we obtain the Korovkin’s type approximation properties for our operators defined by (1.12). In order to obtain the convergence results for the operators D∗n,q(., .), we write q = qn where qn ∈ (0, 1) such that, lim n qn → 1 (2.6) Theorem 2.1. Let q = qn satisfy (2.6), for 0 < qn < 1 and if D ∗ n,qn (. ; .) be the operators given by (1.12). Then for any function f ∈ X[0,∞) ∩H, D∗n,qn(f; x) = f(x) uniformly on each compact subset of [0,∞). Proof. The proof is based on the well known Korovkin’s theorem regarding the convergence of a sequence of linear positive operators, so it is enough to prove the conditions D∗n,qn((ej; x) = x j, j = 0, 1, 2, {as n →∞} uniformly on [0, 1]. Clearly from (2.6) and 1 [n]qn → 0, (n →∞), we have lim n→∞ D∗n,qn(e1; x) = x, limn→∞ D∗n,qn(e2; x) = x 2. Which completes the proof. � Let CB(R+) be the set of all bounded and continuous functions on R+ = [0,∞), which is linear normed space with ‖ f ‖CB = sup x≥0 | f(x) | . We write H := {f : x ∈ [0,∞), f(x) 1 + x2 is convergent as x →∞}. DUNKL GENERALIZATION OF q-PARAMETRIC SZÁSZ-MIRAKJAN OPERATORS 211 We recall the weighted spaces defined as follows: Pρ(R+) = {f :| f(x) |≤ Mfρ(x)} , Qρ(R+) = { f : f ∈ Pρ(R+) ∩C[0,∞) } , Qkρ(R +) = { f : f ∈ Qρ(R+) and lim x→∞ f(x) ρ(x) = k(k is a constant) } , where ρ(x) = 1 + x2 is a weight function and Mf is a constant depending only on f. Qρ(R+) is a normed space with the norm ‖ f ‖ρ= supx≥0 |f(x)| ρ(x) . Theorem 2.2. Let q = qn satisfy (2.6), for 0 < qn < 1 and if D ∗ n,qn (. ; .) be the operators given by (1.12). Then for any function f ∈ Qkρ(R+) we have lim n→∞ ‖ D∗n,qn(f; x) −f ‖ρ= 0. Proof. From Lemma 2.1, the first condition of (1) is fulfilled for τ = 0. Now for τ = 1, 2 it is easy to see from (2), (3) of Lemma 2.1 by using (2.6) that ‖ D∗n,qn (e1) τ ; x) −xτ ‖ρ= 0. This complete the proof. � 3. Rate of convergence Next, we calculate the rate of convergence of operators (1.12) by means of modulus of continuity and Lipschitz type maximal functions. Let f ∈ C[0,∞]. The modulus of continuity of f denoted by ω(f,δ) gives the maximum oscillation of f in any interval of length not exceeding δ > 0 and it is given by the relation ω(f,δ) = sup |y−x|≤δ | f(y) −f(x) |, x,y ∈ [0,∞). (3.1) It is known that limδ→0+ ω(f,δ) = 0 for f ∈ C[0,∞) and for any δ > 0 one has | f(y) −f(x) |≤ ( | y −x | δ + 1 ) ω(f,δ). (3.2) Theorem 3.1. Let q = qn satisfy (2.6) for x ≥ 0, 0 < qn < 1 and if D∗n,qn(. ; .) be the operators defined by (1.12). Then for any function f ∈ C̃[0,∞), we have | D∗n,qn(f; x) −f(x) |≤ { 1 + √ (1 −qn)[n]qnx2 + q 2(1+µ) n [1 + 2µ]qnx } ω ( f; 1√ [n]qn ) , where C̃[0,∞) is the space of uniformly continuous functions on R+ and ω(f,δ) is the modulus of continuity of the function f ∈ C̃[0,∞) defined in (3.2). 212 MURSALEEN, NASIRUZZAMAN AND AL-ABIED Proof. We prove it by using the result (3.2),(3.3) and Cauchy-Schwarz inequality: | D∗n,q(f; x) −f(x) | ≤ 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣f ( 1 −q2µθk+k qk−2(1 −qn) ) −f(x) ∣∣∣∣ ≤ 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 { 1 + 1 δ ∣∣∣∣ ( 1 −q2µθk+k qk−2(1 −qn) ) −x ∣∣∣∣ } ω(f; δ) = { 1 + 1 δ ( 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣ 1 −q2µθk+kqk−2(1 −qn) −x ∣∣∣∣ )} ω(f; δ) ≤  1 + 1δ ( 1 Eµ,q([n]qx) ∞∑ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ( 1 −q2µθk+k qk−2(1 −qn) −x )2)12 ( D∗n,q(e0; x) )1 2  ω(f; δ) = { 1 + 1 δ ( D∗n,q(e1 −x) 2; x )1 2 } ω(f; δ) ≤ { 1 + 1 δ √ (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx } ω(f; δ), if we choose δ = δn = √ 1 [n]q , then we get our result. � Now we give the rate of convergence of the operators D∗n,q(f; x) defined in (1.12) in terms of the elements of the usual Lipschitz class LipM (ν). Let f ∈ C[0,∞), M > 0 and 0 < ν ≤ 1. We recall that f belongs to the class LipM (ν) if LipM (ν) = {f :| f(ζ1) −f(ζ2) |≤ M | ζ1 − ζ2 |ν (ζ1,ζ2 ∈ [0,∞))} (3.3) is satisfied. Theorem 3.2. Let D∗n,q(. ; .) be the operator defined in (1.12). Then for each f ∈ LipM (ν), (M > 0, 0 < ν ≤ 1) satisfying (3.3), we have | D∗n,q(f; x) −f(x) |≤ M (λn(x)) ν 2 where λn(x) = D ∗ n,q ( (e1 −x)2; x ) . Proof. We prove it by using the result (3.3) and Hölder inequality: | D∗n,q(f; x) −f(x) |≤| D ∗ n,q(f(e1) −f(x); x) |≤ D ∗ n,q (| f(e1) −f(x) |; x) ≤| MD ∗ n,q (| e1 −x | ν; x) . Therefore | D∗n,q(f; x) −f(x) | ≤ M 1 Eµ,q([n]qx) ∑∞ k=0 ([n]qx) k γµ,q(k) q k(k−1) 2 ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣ν ≤ M 1 Eµ,q([n]qx) ∑∞ k=0 ( ([n]qx) kq k(k−1) 2 γµ,q(k) )2−ν 2 ( ([n]qx) kq k(k−1) 2 γµ,q(k) )ν 2 ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣ν ≤ M ( 1 (Eµ,q([n]qx)) ∑∞ k=0 ([n]qx) kq k(k−1) 2 γµ,q(k) )2−ν 2 ( 1 (Eµ,q([n]qx)) ∑∞ k=0 ([n]qx) kq k(k−1) 2 γµ,q(k) ∣∣∣∣ 1−q2µθk+kqk−2(1−qn) −x ∣∣∣∣2 )ν 2 ≤ M ( D∗n,q(e1 −x)2; x )ν 2 . This complete the proof. � We denote CB[0,∞) for the space of all bounded and continuous functions on R+ = [0,∞), and C2B(R +) = {g ∈ CB(R+) : g′,g′′ ∈ CB(R+)}, (3.4) DUNKL GENERALIZATION OF q-PARAMETRIC SZÁSZ-MIRAKJAN OPERATORS 213 with the norm ‖ g ‖C2 B (R+)=‖ g ‖CB(R+) + ‖ g ′ ‖CB(R+) + ‖ g ′′ ‖CB(R+), (3.5) also ‖ g ‖CB(R+)= sup x∈R+ | g(x) | . (3.6) Theorem 3.3. Let D∗n,q(. ; .) be the operators defined by (1.12). Then for any g ∈ C2B(R +), we have | D∗n,q(f; x) −f(x) |≤ ( (1 −q)x + λn(x) 2 ) ‖ g ‖C2 B (R+) where λn(x) is given as in Theorem 3.2. Proof. Let g ∈ C2B(R +). Then by using the generalized mean value theorem in the Taylor series expansion we have g(e1) = g(x) + g ′(x)(e1 −x) + g′′(ψ) (e1 −x)2 2 , ψ ∈ (x,e1). By applying linearity property on D∗n,q, we have D∗n,q(g,x) −g(x) = g ′(x)D∗n,q ((e1 −x); x) + g′′(ψ) 2 D∗n,q ( (e1 −x)2; x ) , which implies that, | D∗n,q(g; x) −g(x) |≤ (1 −q)x ‖ g ′ ‖CB(R+) + ( (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx ) ‖ g′′ ‖CB(R+) 2 From (3.5) we have ‖ g′ ‖CB[0,∞)≤‖ g ‖C2B[0,∞). | D∗n,q(g; x) −g(x) |≤ (1 −q)x ‖ g ‖C2B(R+) + ( (1 −q)x2 + q2(1+µ) [n]q [1 + 2µ]qx ) ‖ g ‖C2 B (R+) 2 . From 3 of Lemma 2.2, we get the required result. � The Peetre’s K-functional is defined by K2(f,δ) = inf C2 B (R+) {( ‖ f −g ‖CB(R+) +δ ‖ g ′′ ‖C2 B (R+) ) : g ∈W2 } , (3.7) where W2 = { g ∈ CB(R+) : g′,g′′ ∈ CB(R+) } . (3.8) Then there exits a positive constant C > 0 such that K2(f,δ) ≤ Cω2(f,δ 1 2 ), δ > 0, where the second order modulus of continuity is given by ω2(f,δ 1 2 ) = sup 0 0 in [7] we use the relation K2(f; δ) ≤ C{ω2(f; √ δ) + min(1,δ) ‖ f ‖}. This complete the proof. � Acknowledgement. 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