International Journal of Analysis and Applications ISSN 2291-8639 Volume 12, Number 1 (2016), 30-37 http://www.etamaths.com APPROXIMATION THEOREMS FOR q− ANALOUGE OF A LINEAR POSITIVE OPERATOR BY A. LUPAS KARUNESH KUMAR SINGH1, ASHA RAM GAIROLA2 AND DEEPMALA3,∗ Abstract. The purpose of the present paper is to introduce q− analouge of a sequence of linear and positive operators which was introduced by A. Lupas [1]. First, we estimate moments of the operators and then prove a basic convergence theorem. Next, a local direct approximation theorem is established. Further, we study the rate of convergence and point-wise estimate using the Lipschitz type maximal function. 1. Introduction At the International Dortmund Meeting held in Written (Germany, March, 1995), A. Lupas [1] introduced the following Linear positive operators: (1) Ln(f; x) = (1 −a)nx ∞∑ k=0 (nx)k k! akf ( k n ) ,x ≥ 0. with f : [0,∞] → R. If we impose that Lne1 = e1 we find that a = 1/2. Therefore operator (1) becomes Ln(f; x) = 2 −nx ∞∑ k=0 (nx)k 2kk! f ( k n ) ,x ≥ 0, where (α)0 = 1, (α)k = α(α + 1)...(α + k − 1),k ≥ 1. The q− analouge of the above operators is defined as: Ln,q(f; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! f ( [k]q [n]q ) ,x ≥ 0, We denote CB[0,∞) the space of real valued bounded continuous function f on the interval [0,∞), the norm on the space is defined as ‖f‖ = sup 0≤x<∞ |f(x)|. Let W2 = {g ∈ CB[0,∞) : g′,g′′ ∈ CB[0,∞)}. The Peetre’s K− functional is defined as K2(f,δ) = inf g∈W2 {‖f −g‖ + δ‖g′′‖}, where δ > 0. For f ∈ CB[0,∞) a usual modulus of continuity is given by ω(f,δ) = sup 0 0 such that K2(f,δ) ≤ Cω2(f, √ δ). In recent years, many results about the generalization of linear positive operators have been obtained by several mathematicians ([6]-[17]). 2. Moment estimates Lemma 1. The following relations hold: Ln,q(1; x) = 1,Ln,q(t; x) = x and Ln,q(t 2; x) = qx2 + 1 + q [n] x. Proof. We have Ln,q(1; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! = 1 Now, Ln,q(t; x) = 2 −[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! [k]q [n]q = 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k − 1]q![n]q = 2−[n]qx−1 [n]q ∞∑ k=1 [n]qx([n]qx + 1)k−1 2k−1[k − 1]q! = 2−[n]qx−1x ∞∑ k=1 ([n]qx + 1)k−1 2k−1[k − 1]q! = 2−[n]qx−1x ∞∑ k=0 ([n]qx + 1)k 2k[k]q! = x. Next, Ln,q(t 2; x) = 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! [k]2q [n]2q = 2−[n]qx ∞∑ k=0 [n]qx([n]qx + 1)k−1 2k[k]q[k − 1]q! [k]2q [n]2q = 2−[n]qx−1x ∞∑ k=1 ([n]qx + 1)k−1 2k−1[k − 1]q! [k]q [n]q = 2−[n]qx−1x [n]q ∞∑ k=1 ([n]qx + 1)k−1[k]q 2k−1[k − 1]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k[k + 1]q 2k[k]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k(1 + q[k]q) 2k[k]q! = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)k 2k[k]q! + 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)kq[k]q 2k[k]q! = I1 + I2, say. 32 SINGH, GAIROLA AND DEEPMALA We find that I1 = x [n]q . Now, I2 = 2−[n]qx−1x [n]q ∞∑ k=0 ([n]qx + 1)kq[k]q 2k[k]q! = 2−[n]qx−2qx [n]q ∞∑ k=1 ([n]qx + 1)([n]qx + 2)k−1 2k−1[k − 1]q! = 2−[n]qx−2qx([n]qx + 1) [n]q ∞∑ k=1 ([n]qx + 2)k−1 2k−1[k − 1]q! = 2−[n]qx−2qx([n]qx + 1) [n]q ∞∑ k=0 ([n]qx + 2)k 2k[k]q! = qx([n]qx + 1) [n]q . Hence, on combining I1 and I2, we get Ln,q(t 2; x) = (1 + q)x [n]q + qx2. � Let us define mth order moment by ψn,m(q; x) = Ln,q((t−x)m; x). Lemma 2. Let 0 < q < 1, then for x ∈ [0,∞) we have ψn,1(q; x) = 0 and ψn,2(q; x) = x([2] − (1 −q)[n]qx) [n]q . Proof. We have ψn,1(q; x) = Ln,q(t−x; x) = 0. Now, ψn,2(q; x) = Ln,q((t−x)2; x) = Ln,q(t 2 + x2 − 2tx; x) = (1 + q)x [n]q + (q − 1)x2. � 3. Basic Pointwise Convergence The operators Ln,q do not satisfy the conditions of the Bohman-Korovkin theorem in case 0 < q < 1. To make this theorem applicable, we can choose a sequence (qn) in place of the number q such that qn → 1 and qnn → 0 as n →∞. With this modification we obtain the following Korovkin type result: Theorem 1. Let f ∈ CB[0,∞) and qn be a real sequence in (0, 1) such that qn → 1 and qnn → 0 as n →∞. Then, for each x ∈ [0,∞) we have lim n→∞ Ln,qn(f; x) = f(x). Proof. The proof is based on the well known Korovkin theorem regarding the convergence of a sequence of linear positive operators. So, it is enough to prove the conditions lim n→∞ Ln,qn(t m; x) = xm,m = 0, 1, 2. Now, using Lemma 1 we obtain lim n→∞ Ln,qn(1; x) = 1, lim n→∞ Ln,qn(t; x) = x LINEAR POSITIVE OPERATORS BY A. LUPAS 33 and lim n→∞ Ln,qn(t; x) = lim n→∞ qnx 2 + 1 + qn [n]qn x = x2. This completes the proof. � 4. Direct results Theorem 2. Let f ∈ CB[0,∞) and q ∈ (0, 1). Then, for each x ∈ [0,∞) and n ∈ N there exists an absolute constant C > 0 such that |Ln,q(f; x) −f(x)|6 Cω2 ( f, √ x([2] − (1 −q)[n]qx) [n]q ) . Proof. Let g ∈ W2 and x,t ∈ [0,∞). Using Taylor’s expansion we can write g(t) = g(x) + g′(x)(t−x) + t∫ x (t−v)g′′(v)dv. On application of Lemma 2 we obtain Ln,q ( g(t); x) −g(x) ) = Ln,q   t∫ x (t−v)g′′(v)dv; x   . Now, we have ∣∣∣∣ t∫ x (t−v)g′′(v)dv ∣∣∣∣ ≤ (t−x)2‖g′′‖. Therefore |Ln,q(g(t); x) −g(x)| ≤ Ln,q ( (t−x)2; x ) ‖g′′‖ = x([2] − (1 −q)[n]qx) [n]q ‖g′′‖. By Lemma 1, we have |Ln,q(f; x)| ≤ 2−[n]qx ∞∑ k=0 ([n]qx)k 2k[k]q! ∣∣∣∣f ( [k]q [n]q )∣∣∣∣ ≤‖f‖. Thus |Ln,q(f; x) −f(x)| ≤ |Ln,q(f −g; x) − (f −g)(x)| + |Ln,q(g; x) −g(x)| ≤ 2‖f −g‖ + x([2]−(1−q)[n]qx) [n]q ‖g′′‖. At last, taking the infimum over all g ∈ W2 and on application of the inequality K2(f,δ) ≤ Cω2(f,δ1/2),δ > 0, we get the required result. This completes the proof of the theorem. � 5. Pointwise Estimates In this section, we obtain some pointwise estimates of the rate of convergence of the q− Baskakov- Durrmeyer operators. First, we discuss the relationship between the local smoothness of f and the local approximation. Theorem 3. Let 0 < α ≤ 1 and E be any bounded subset of the interval [0,∞). If f ∈ CB[0,∞) ∩ LipM (α) then we have |Ln,q(f; x) −f(x)| ≤ M{ψ α 2 n,2(q; x) + 2(d(x,E)) α},x ∈ [0,∞), where M is a constant depending on α and f, d(x,E) is the distance between x and E defined as d(x,E) = inf{|t−x|; t ∈ E} and ψn,2(q; x) = Ln,q((t−x)2; x) . 34 SINGH, GAIROLA AND DEEPMALA Proof. From the property of infimum, it follows that there exists a point t0 ∈ Ē such that d(x,E) = |t0 −x|. In view of the triangle inequality we have |f(t) −f(x)| ≤ |f(t) −f(t0)| + |f(t0) −f(x)|. Using the definition of LipM (α), we get |Ln,q(f; x) −f(x)| ≤ Ln,q(|f(t) −f(t0)|; x) + Ln,q(|f(x) −f(t0)|; x) ≤ M{Ln,q(|t− t0|α; x) + |x− t0|α} ≤ M{Ln,q(|t−x|α; x) + 2|x− t0|α}. Choosing p1 = 2 α and p2 = 2 2−α, we get 1 p1 + 1 p2 = 1. Then, Hölder’s inequality yields |Ln,q(f; x) −f(x)| ≤ M{(Ln,q(|t−x|αp1 ; x))1/p1 [Ln,q(1p2 ; x)]1/p2 + 2(d(x,E))α} ≤ M{(Ln,q((t−x)2; x))α/2 + 2(d(x,E))α} = M{ψα/2n,2 (q; x) + 2(d(x,E)) α}. This completes the proof of the theorem. � Next, we obtain a local direct estimate of operators Ln,q using the Lipschitz-type maximal function of order α introduced by Lenze [2] as (2) ω̃α(f,x) = sup t 6=x,t∈[0,∞) |f(t) −f(x)| |t−x|α , x ∈ [0,∞) andα ∈ (0, 1]. Theorem 4. Let 0 < α ≤ 1 and f ∈ CB[0,∞), then for all x ∈ [0,∞) we have |Ln,q(f; x) −f(x)| ≤ ω̃α(f,x)ψ α/2 n,2 (q; x). Proof. In view of (2), we get |f(t) −f(x)| ≤ ω̃α(f,x)|t−x|α and hence |Ln,q(f; x) −f(x)| ≤ Ln,q(|f(t) −f(x)|; x) ≤ ω̃α(f,x)Ln,q(|t−x|α; x). Now, using the Hölder’s inequality with p = 2 α and 1 q = 1 − 1 p , we obtain |Ln,q(f; x) −f(x)| ≤ ω̃α(f,x)(Ln,q(|t−x|2; x))α/2 = ω̃α(f,x)ψ α/2 n,2 (x). Thus, the proof is completed. � 6. Weighted Approximation In this section, we discuss about the weighted approximation theorem for the operators Ln,q(f). Let C∗ x2 [0,∞) be the subspace of all functions f ∈ Cx2 [0,∞) for which limx→∞ |f(x)| 1+x2 is finite. Theorem 5. Let qn be a sequence in (0, 1) such that qn → 1 and qnn → 0, as n → ∞. For each C∗ x2 [0,∞), we have (3) lim n→∞ ‖Ln,qn(f) −f‖x2 = 0. LINEAR POSITIVE OPERATORS BY A. LUPAS 35 Proof. In order to proof (3) it is sufficient to show that ([5]) (4) lim n→∞ ‖Ln,qn(t ν; x) −xν‖x2 = 0, ν = 0, 1, 2. Since, Ln,qn(1; x) = 1, (4) holds true for ν = 0. Now, by Lemma 1, we have ‖Ln,qn(t; x) −x‖x2 = sup x∈[0,∞) |Ln,qn(t; x) −x| 1 + x2 → 0, as n →∞. Therefore, (4) is true for ν = 1. Again, by Lemma 1, we may write ‖Ln,qn(t 2; x) −x2‖x2 = sup x∈[0,∞) |Ln,qn(t2; x) −x2| 1 + x2 = sup x∈[0,∞) ∣∣∣(1+qn)x+qn[n]qnx2[n]qn −x2 ∣∣∣ 1 + x2 ≤ 1 + qn [n]qn sup x∈[0,∞) x 1 + x2 + (qn − 1) sup x∈[0,∞) x2 1 + x2 = 1 + qn [n]qn + (qn − 1). Hence, (4) follows for ν = 2. This completes the proof of the theorem. � Theorem 6. Let f ∈ Cx2 [0,∞),q = qn ∈ (0, 1) such that qn → 1 and qnn → 0 as n → ∞ and ωa+1be its modulus of continuity on the finite interval [0,a + 1] ⊂ [0,∞), a > 0. Then, for every n ≥ 1 ‖Ln,q(f) −f‖C[0,a] ≤ 12Mf(1+a 2)a [n]q + 2ωa+1 ( f, √ 2a [n]q ) . Proof. For x ∈ [0,a] and t > a + 1. Since t−x > 1, we have |f(t) −f(x)| ≤ Mf (2 + x2 + t2) ≤ Mf (2 + 3x2 + 2(t−x)2) ≤ 3Mf (1 + x2 + (t−x)2) ≤ 6Mf (1 + x2)(t−x)2 ≤ 6Mf (1 + a2)(t−x)2.(5) For x ∈ [0,a] and t ≤ a + 1, we have |f(t) −f(x)| ≤ ωa+1(f, |t−x|) ≤ ( 1 + |t−x| δ ) ωa+1(f,δ),(6) where δ > 0. From (5) and (6), we can write |f(t) −f(x)| ≤ 6Mf (1 + a2)(t−x)2 + ( 1 + |t−x| δ ) ωa+1(f,δ)(7) For x ∈ [0,a] and t ≥ 0 and applying Schwarz inequality, we obtain |Ln,q(f; x) −f(x)| ≤ Ln,q(|f(t) −f(x)|; x) ≤ 6Mf (1 + a2)Ln,q((t−x)2; x) + ωa+1(f,δ) ( 1 + 1 δ Ln,q((t−x)2; x) 1 2 ) . 36 SINGH, GAIROLA AND DEEPMALA Hence, using Lemma 2, for every q ∈ (0, 1) and x ∈ [0,a] |Ln,q(f; x) −f(x)| ≤ 6Mf (1 + a2) x([2] − (1 −q)[n]qx) [n]q + Cωa+1(f,δ) ( 1 + 1 δ √ x([2] − (1 −q)[n]qx) [n]q ≤ 12Mf (1 + a 2)a [n]q + ωa+1(f,δ) ( 1 + 1 δ √ 2a [n]q ) . Taking δ = √ 2a [n]q , we get the required result. This completes the proof of Theorem. � Now, we prove a theorem to approximate all functions in Cx2 [0,∞). Such type of results are given in [4] for locally integrable functions. Theorem 7. Let q = qn ∈ (0, 1) such that qn → 1 and qnn → 0, as n → ∞. For each f ∈ C∗x2 [0,∞), and α > 1, we have lim n→∞ sup x∈[0,∞) |Ln,qn(f; x) −f(x)| (1 + x2)α = 0. Proof. For any fixed x0 > 0, sup x∈[0,∞) |Ln,qn(f; x) −f(x)| (1 + x2)α ≤ sup x≤x0 |Ln,qn(f; x) −f(x)| (1 + x2)α + sup x>x0 |Ln,qn(f; x) −f(x)| (1 + x2)α ≤ ‖Ln,qn(f) −f‖C[0,x0] + ‖f‖x2 sup x≥x0 |Ln,qn(1 + t2; x)| (1 + x2)α (8) + sup x≥x0 |f(x)| (1 + x2)α . Since, |f(x)| ≤ Mf (1 + x2), we have sup x≥x0 |f(x)| (1 + x2)α ≤ sup x≥x0 Mf (1 + x2)α−1 ≤ Mf (1 + x20) α−1 . Let � > 0 be arbitrary. We can choose x0 to be large that (9) Mf (1 + x20) α−1 < � 3 and in view of Lemma 1, we obtain ‖f‖x2 lim n→∞ |Ln,qn(1 + t2; x)| (1 + x2)α = 1 + x2 (1 + x2)α ‖f‖x2 = ‖f‖x2 (1 + x2)α−1 ≤ ‖f‖x2 (1 + x20) α−1 < � 3 .(10) Using Theorem 6 we can see that the first term of the inequality (8) implies that (11) ‖Ln,qn(f; .) −f‖C[0,x0] < � 3 , as n →∞. Combining (8)-(11), we get the desired result. LINEAR POSITIVE OPERATORS BY A. LUPAS 37 � Acknowledgement The research work of the third author Deepmala is supported by the Science and Engineering Research Board (SERB), Government of India under SERB NPDF scheme, File Number: PDF/2015/000799. References [1] A. Lupas, The approximation by some positive linear operators. In: proceedings of the International Dortmund meeting on Approximation Theory(M.W. Müller et al., eds.), akademie Verlag, Berlin, 1995, 201-229. [2] B. Lenze, On Lipschitz-type maximal functions and their smoothness spaces, Nederl. Akad. Wetensch. Indag. 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Lett., 4 (3) (2016), 1-8. 1Government Polytechnic, Rampur, Uttar Pradesh, India 2Department of Mathematics, Doon University, Dehradun, Uttarakhand-248 001, India 3SQC & OR Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata-700 108, West Bengal, India ∗Corresponding author: dmrai23@gmail.com, deepmaladm23@gmail.com