Direct estimates for certain integral type Operators EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 4, 2018, 958-975 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Direct estimates for certain integral type Operators Alok Kumar1, Dipti Tapiawala2,3, Lakshmi Narayan Mishra4,5,∗ 1 Department of Computer Science, Dev Sanskriti Vishwavidyalaya, Haridwar 249 411, Uttarakhand, India 2 Department of Mathematics, C U Shah university, Surendranagar-Ahmedabad High way Nr Kotharity village, Wadhwan City 363 030, Surendranagar, Gujarat, India 3 AS and H Department (Mathematics), Sardar Vallabhbhai Patel Institute of Technology, Vasad 388 306, Anand, Gujarat, India 4 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT) University, Vellore 632 014, Tamil Nadu, India 5 L. 1627 Awadh Puri Colony, Beniganj, Phase IIIrd, Opposite Industrial Training Institute (I.T.I.), Ayodhya main road, Faizabad 224 001, Uttar Pradesh, India Abstract. In this note, we study approximation properties of a family of linear positive operators and establish asymptotic formula, rate of convergence, local approximation theorem, global ap- proximation theorem, weighted approximation theorem and better approximation for this family of linear positive operators. 2010 Mathematics Subject Classifications: 41A25, 26A15, 40A35 Key Words and Phrases: Global approximation, Voronovskaja type theorem, K-functional, modulus of continuity, weighted approximation 1. Introduction In the year 2008, Mihes.an [38] constructed an important generalization of the well- known Szász operators depending on α ∈ R as G(α)n (f ;x) = ∞∑ k=0 m (α) n,k(x)f ( k n ) , x ∈ [0,∞) (1) where m (α) n,k(x) = (α)k k! . (nxα )k (1 + nx α )α+k , ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3305 Email addresses: alokkpma@gmail.com (A. Kumar), tapiawalad@yahoo.com (D. Tapiawala), lakshminarayanmishra04@gmail.com, l n mishra@yahoo.co.in (L. N. Mishra) http://www.ejpam.com 958 c© 2018 EJPAM All rights reserved. A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 959 and (α)k = α(α + 1)...(α + k − 1), (α)0 = 1, is the rising factorial and α + nx > 0. The operator G(α)n preserve the linear polynomials, and for special values of α, one can obtain some well-known operators. Recently, Kajla [15] introduced a new sequence of summation-integral type operators and established some approximation properties e.g. weighted approximation, asymptotic formula and error estimation in terms of modulus of smoothness. Very recently, Gupta and Agrawal [11] proposed the integral modification of the operators (1) by taking weights of Beta basis functions as follows: M (α) n (f ;x) = ∞∑ k=1 m (α) n,k(x) ∫ ∞ 0 bn,k(t)f(t)dt+ ( α α+ nx )α f(0), (2) where bn,k(t) = 1 B(n+ 1, k) . tk−1 (1 + t)k+n+1 , and B(m,n) being the Beta function defined as B(m,n) = Γ(m)Γ(n) Γ(m+ n) , m, n > 0. They obtain different approximation properties for these operators. For the different val- ues of α, we get different special cases. Some of the special cases are discussed in [11]. In [42], Stancu introduced and investigated a new parameter-dependent linear positive operators of Bernstein type associated to a function f ∈ C[0, 1]. The new construction of his operators shows that the new sequence of Bernstein polynomials present a better approach with the suitable selection of the parameters. In the recent years, Stancu type generalization of the certain operators introduced by several researchers and obtained different type of approximation properties of many oper- ators, we refer some of the important papers in this direction as [1], [16], [23], [24], [25], [33] etc. Inspired by the above work, We introduce the Stancu type generalization of the operators (2): M (β,γ) n,α (f ;x) = ∞∑ k=1 m (α) n,k(x) ∫ ∞ 0 bn,k(t)f ( nt+ β n+ γ ) dt+ ( α α+ nx )α f ( β n+ γ ) . (3) In this present work, our focus is to study the approximation properties of the operators (3) in terms of first and second order modulus of continuity. We estimate the rate of convergence of these operators in terms of modulus of continuity. Furthermore, we inves- tigate weighted approximation theorems. Lastly we study King type modification of the operators (3). A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 960 2. Moment estimates In the sequel, we shall need the following auxiliary results which will be necessary to prove our main results. Lemma 1. [11] For the operators M (α) n (f ;x), we have (i) M (α) n (1;x) = 1, (ii) M (α) n (t;x) = x, (iii) M (α) n (t2;x) = x[nx(α+ 1) + 2α] α(n− 1) . Lemma 2. For the operators M (β,γ) n,α (f ;x), we have (i) M (β,γ) n,α (1;x) = 1, (ii) M (β,γ) n,α (t;x) = nx+ β n+ γ , (iii) M (β,γ) n,α (t2;x) = { n3(α+ 1) α(n− 1)(n+ γ)2 } x2 + { 2n2 + 2nβ(n− 1) (n− 1)(n+ γ)2 } x+ β2 (n+ γ)2 . Proof. For x ∈ [0,∞), in view of Lemma 1, we have M (β,γ) n,α (1;x) = 1. The first order moment is given by M (β,γ) n,α (t;x) = n n+ γ M (α) n (t;x) + β n+ γ = nx+ β n+ γ . The second order moment is given by M (β,γ) n,α (t2;x) = ( n n+ γ )2 M (α) n (t2;x) + 2nβ (n+ γ)2 M (α) n (t;x) + ( β n+ γ )2 = { n3(α+ 1) α(n− 1)(n+ γ)2 } x2 + { 2n2 + 2nβ(n− 1) (n− 1)(n+ γ)2 } x+ β2 (n+ γ)2 . Lemma 3. For f ∈ CB[0,∞) (space of all real valued bounded functions on [0,∞) endowed with norm ‖ f ‖CB [0,∞)= sup x∈[0,∞) |f(x)|), ‖M (β,γ) n,α (f) ‖≤‖ f ‖ . A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 961 Proof. In view of (3) and Lemma 2, we get ‖M (β,γ) n,α (f)‖ ≤ ‖f‖M (β,γ) n,α (1;x) = ‖f‖. Remark 1. For every x ∈ [0,∞), we have M (β,γ) n,α ((t− x);x) = β − γx n+ γ and M (β,γ) n,α ( (t− x)2;x ) = { n2(n+ α) + αγ2(n− 1) α(n− 1)(n+ γ)2 } x2 + { 2n2 + 2βγ(1− n) (n− 1)(n+ γ)2 } x+ β2 (n+ γ)2 = ξ(β,γ)n,α (x). 3. Main results Throughout this paper, we assume that α = α(n) → ∞, as n → ∞ and lim n→∞ n α(n) = l(∈ R). Let ei(t) = ti, i = 0, 1, 2. Theorem 1. Let f ∈ C[0,∞). Then lim n→∞ M (β,γ) n,α (f ;x) = f(x), uniformly in each compact subset of [0,∞). Proof. In view of Lemma 2, we get lim n→∞ M (β,γ) n,α (ei;x) = xi, i = 0, 1, 2, uniformly in each compact subset of [0,∞). Applying Bohman-Korovkin theorem, it follows that lim n→∞ M (β,γ) n,α (f ;x) = f(x), uniformly in each compact subset of [0,∞). 3.1. Voronovskaja type theorem In this section we prove Voronvoskaja type asymptotic theorem for the operators M (β,γ) n,α . Theorem 2. Let f be a bounded and integrable function on [0,∞), second derivative of f exists at a fixed point x ∈ [0,∞), then lim n→∞ n ( M (β,γ) n,α (f ;x)− f(x) ) = (β − γx)f ′(x) + 1 2 ( 2x+ (l + 1)x2 ) f ′′(x). Proof. Let x ∈ [0,∞) be fixed. Using Taylor’s expansion formula of function f , it follows f(t) = f(x) + (t− x)f ′(x) + 1 2 f ′′(x)(t− x)2 + r(t, x)(t− x)2, (4) A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 962 where r(t, x) is a bounded function and lim t→x r(t, x) = 0. Applying M (β,γ) n,α (f ;x) on both sides of (4), we get n ( M (β,γ) n,α (f ;x)− f(x) ) = nf ′(x)M (β,γ) n,α ((t− x);x) + 1 2 nf ′′(x)M (β,γ) n,α ( (t− x)2;x ) +nM (β,γ) n,α ( (t− x)2r(t, x);x ) . In view of Remark 1, we have lim n→∞ nM (β,γ) n,α ((t− x);x) = β − γx (5) and lim n→∞ nM (β,γ) n,α ( (t− x)2;x ) = 2x+ (l + 1)x2. (6) Now, we shall show that lim n→∞ nM (β,γ) n,α ( r(t, x)(t− x)2;x ) = 0. By using Cauchy-Schwarz inequality, we have M (β,γ) n,α ( r(t, x)(t− x)2;x ) ≤ ( M (β,γ) n,α (r2(t, x);x) )1/2 ( M (β,γ) n,α ((t− x)4;x) )1/2 . (7) We observe that r2(x, x) = 0 and r2(., x) ∈ CB[0,∞). Then, it follows that lim n→∞ M (β,γ) n,α (r2(t, x);x) = r2(x, x) = 0. (8) Now, from (7) and (8) we obtain lim n→∞ nM (β,γ) n,α ( r(t, x)(t− x)2;x ) = 0. (9) From (5), (6) and (9), we get the required result. 3.2. Local approximation For CB[0,∞), let us consider the following K-functional: K2(f, δ) = inf g∈W 2 {‖ f − g ‖ +δ ‖ g′′ ‖}, where δ > 0 and W 2 = {g ∈ CB[0,∞) : g′, g′′ ∈ CB[0,∞)}. By, p. 177, Theorem 2.4 in [2], there exists an absolute constant M > 0 such that K2(f, δ) ≤Mω2(f, √ δ), (10) A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 963 where ω2(f, √ δ) = sup 0 0 be fixed. We define the following Lipschitz-type space (see [39]): Lip (a1,a2) M (r) = ( f ∈ C[0,∞) : |f(t)− f(x)| ≤M |t− x|r (t+ a1x2 + a2x)r/2 ; x, t ∈ [0,∞) ) , where M is any positive constant and 0 < r ≤ 1. Theorem 4. Let f ∈ Lip(a1,a2)M (r). Then, for all x > 0, we have |M (β,γ) n,α (f ;x)− f(x)| ≤M ( ξ (β,γ) n,α (x) a1x2 + a2x )r/2 . Proof. First we prove the theorem for r = 1. Then, for f ∈ Lip(a1,a2)M (1), and x > 0, we have |M (β,γ) n,α (f ;x)− f(x)| ≤ M (β,γ) n,α (|f(t)− f(x)|;x) ≤ MM (β,γ) n,α ( |t− x| (t+ a1x2 + a2x)1/2 ;x ) A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 965 ≤ M (a1x2 + a2x)1/2 M (β,γ) n,α (|t− x|;x). Applying Cauchy-Schwarz inequality, we get |M (β,γ) n,α (f ;x)− f(x)| ≤ M (a1x2 + a2x)1/2 ( M (β,γ) n,α ((t− x)2;x) )1/2 ≤ M ( ξ (β,γ) n,α (x) a1x2 + a2x )1/2 . Thus the result holds for r = 1. Now, we prove that the result is true for 0 < r < 1. Then, for f ∈ Lip(a1,a2)M (r), and x > 0, we get |M (β,γ) n,α (f ;x)− f(x)| ≤ M (a1x2 + a2x)r/2 M (β,γ) n,α (|t− x|r;x). Taking p = 1 r and q = p p−1 , applying the Hölders inequality, we have |M (β,γ) n,α (f ;x)− f(x)| ≤ M (a1x2 + a2x)r/2 ( M (β,γ) n,α (|t− x|;x) )r . Finally by Cauchy-Schwarz inequality, we get |M (β,γ) n,α (f ;x)− f(x)| ≤ M ( ξ (β,γ) n,α (x) a1x2 + a2x )r/2 . Thus, the proof is completed. 3.3. Global approximation In this section, the first and the second order Ditzian-Totik moduli of smoothness are defined as ω̄φ(f, δ) = sup 0<|h|≤δ sup x+hφ(x)∈[0,∞) | f(x+ hφ(x))− f(x) | and ω2,φ(f, √ δ) = sup 0<|h|≤ √ δ sup x±hφ(x)∈[0,∞) | f(x+ hφ(x))− 2f(x) + f(x− hφ(x)) |, respectively and the corresponding K-functional is K2,φ(f, δ) = inf g∈W 2(φ) {‖ f − g ‖ +δ ‖ φ2g′′ ‖}, A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 966 where δ > 0 and W 2(φ) = {g ∈ CB[0,∞) : g′ ∈ AC[0,∞), φ2g′′ ∈ CB[0,∞)} and g′ ∈ AC[0,∞) means that g′ is absolutely continuous on [0,∞). It is well known that (see [3]) K2,φ(f, δ) ∼ ω2,φ(f, √ δ) which means that there exist an absolute constant M > 0 such that M−1ω2,φ(f, √ δ) ≤ K2,φ(f, δ) ≤Mω2,φ(f, √ δ). (15) In the following we will consider φ(x) = 1 + x2. Theorem 5. Let f ∈ CB[0,∞) and x ∈ [0,∞). Then, there exist an absolute constant M > 0 such that |M (β,γ) n,α (f ;x)− f(x) |≤ 4K2,φ ( f, M 2n ) + ω̄φ ( f, √ M n ) , for n sufficiently large. Proof. Let g ∈W 2(φ). Applying Taylor’s expansion, we may write g(t) = g(x) + (t− x)g′(x) + ∫ t x (t− v)g′′(v)dv. Applying M (β,γ) n,α on both sides of the above equation, we get |M (β,γ) n,α (g;x)− g(x)| ≤ M (β,γ) n,α (∣∣∣∣ ∫ t x |t− v||g′′(v)|dv ∣∣∣∣;x)+ ∣∣∣∣ ∫ nx+β n+γ x ∣∣∣∣nx+ β n+ γ − v ∣∣∣∣|g′′(v)|dv ∣∣∣∣ ≤ ‖ φ2g′′ ‖ φ2(x) ( ξ(β,γ)n,α (x) + ( β − γx n+ γ )2) . (16) In view of Remark 1, it follows that there exist a positive constant M > 0 such that ξ (β,γ) n,α (x) φ2(x) ≤ M n , 1 φ2(x) ( β − γx n+ γ )2 ≤ M n2 . Thus, |M (β,γ) n,α (g;x)− g(x)| ≤M ‖ φ2g′′ ‖ ( 1 n + 1 n2 ) ≤ 2M n ‖ φ2g′′ ‖ . Now, |M (β,γ) n,α (f ;x)− f(x)| ≤ |M (β,γ) n,α (f − g;x)|+ |(f − g)(x)|+ |M (β,γ) n,α (g;x)− g(x)|+ ∣∣∣∣f (nx+ β n+ γ ) − f(x) ∣∣∣∣ A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 967 ≤ 4 ‖ f − g ‖ + 2M n ‖ φ2g′′ ‖ + ∣∣∣∣f (nx+ β n+ γ ) − f(x) ∣∣∣∣. Also, we obtain∣∣∣∣f (nx+ β n+ γ ) − f(x) ∣∣∣∣ = ∣∣∣∣f ( x+ φ(x) nx+β n+γ − x φ(x) ) − f(x) ∣∣∣∣ ≤ sup ∣∣∣∣f ( x+ φ(x) β−γx n+γ φ(x) ) − f(x) ∣∣∣∣ ≤ ω̄φ ( f, √ M n ) . Using the above equations, we get |M (β,γ) n,α (f ;x)− f(x) | ≤ 4 ( ‖ f − g ‖ + M 2n ‖ φ2g′′ ‖ ) + ω̄φ ( f, √ M n ) . Now, applying (15), the theorem is completed. 3.4. Rate of convergence Let ωa(f, δ) denote the usual modulus of continuity of f on the closed interval [0, a], a > 0, and defined as ωa(f, δ) = sup |t−x|≤δ sup x,t∈[0,a] |f(t)− f(x)|. We observe that for a function f ∈ CB[0,∞), the modulus of continuity ωa(f, δ) tends to zero. Now, we give a rate of convergence theorem for the operators M (β,γ) n,α . Theorem 6. Let f ∈ CB[0,∞) and ωa+1(f, δ) be its modulus of continuity on the finite interval [0, a+ 1] ⊂ [0,∞), where a > 0. Then, we have |M (β,γ) n,α (f ;x)− f(x)| ≤ 6Mf (1 + a2)ξ(β,γ)n,α (a) + 2ωa+1 ( f, √ ξ (β,γ) n,α (a) ) , where ξ (β,γ) n,α (a) is defined in Remark 1 and Mf is a constant depending only on f . Proof. For x ∈ [0, a] and t > a+ 1. Since t− x > 1, we have |f(t)− f(x)| ≤ Mf (2 + x2 + t2) ≤ Mf (t− x)2(2 + 3x2 + 2(t− x)2) ≤ 6Mf (1 + a2)(t− x)2. A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 968 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) with δ > 0. From the above, we have |f(t)− f(x)| ≤ 6Mf (1 + a2)(t− x)2 + ( 1 + |t− x| δ ) ωa+1(f, δ), for x ∈ [0, a] and t ≥ 0. Thus |M (β,γ) n,α (f ;x)− f(x)| ≤ 6Mf (1 + a2)(M (β,γ) n,α (t− x)2;x) +ωa+1(f, δ) ( 1 + 1 δ (M (β,γ) n,α (t− x)2;x) 1 2 ) . Applying Cauchy-Schwarz’s inequality, we get |M (β,γ) n,α (f ;x)− f(x)| ≤ 6Mf (1 + a2)ξ(β,γ)n,α (a) + 2ωa+1 ( f, √ ξ (β,γ) n,α (a) ) , on choosing δ = √ ξ (β,γ) n,α (a). This completes the proof of theorem. 3.5. Weighted approximation In this section we give some weighted approximation properties of the operators M (β,γ) n,α . We do this for the following class of continuous functions defined on [0,∞). Let Bν [0,∞) denote the weighted space of real-valued functions f defined on [0,∞) with the property |f(x)| ≤Mfν(x) for all x ∈ [0,∞), where ν(x) = 1 + x2 is a weight function and Mf is a constant depending on the function f . We also consider the weighted subspace Cν [0,∞) of Bν [0,∞) given by Cν [0,∞) = {f ∈ Bν [0,∞) : f is continuous on [0,∞)} and C∗ν [0,∞) denotes the subspace of all functions f ∈ Cν [0,∞) for which lim |x|→∞ f(x) ν(x) exists finitely. It is obvious that C∗ν [0,∞) ⊂ Cν [0,∞) ⊂ Bν [0,∞). The space Bν [0,∞) is a normed linear space with the following norm: ‖ f ‖ν= sup x∈[0,∞) |f(x)| ν(x) . Theorem 7. For each f ∈ C∗ν [0,∞), we have lim n→∞ ‖M (β,γ) n,α (f)− f ‖ν= 0. A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 969 Proof. From [6], we know that it is sufficient to verify the following three conditions lim n→∞ ‖M (β,γ) n,α (ei)− ei ‖ν= 0, i = 0, 1, 2. (17) Since M (β,γ) n,α (1;x) = 1, the condition in (17) holds true for i = 0. By Lemma 2, we have ‖M (β,γ) n,α (t)− x ‖ν = sup x∈[0,∞) |M (β,γ) n,α (t;x)− x| 1 + x2 ≤ γ n+ γ sup x∈[0,∞) ( x 1 + x2 ) + β n+ γ sup x∈[0,∞) ( 1 1 + x2 ) ≤ β + γ n+ γ which implies that lim n→∞ ‖M (β,γ) n,α (t)− x ‖ν= 0. Again by Lemma 2, we have ‖M (β,γ) n,α (t2)− x2 ‖ν = sup x∈[0,∞) |M (β,γ) n,α (t2;x)− x2| 1 + x2 ≤ ∣∣∣∣ n3(α+ 1) α(n− 1)(n+ γ)2 − 1 ∣∣∣∣+ ∣∣∣∣2n2 + 2nβ(n− 1) (n− 1)(n+ γ)2 ∣∣∣∣+ β2 (n+ γ)2 , which implies that lim n→∞ ‖M (β,γ) n,α (t2)− x2 ‖ν= 0. This completes the proof of theorem. 3.6. Weighted Lp-approximation Let w be positive continuous function on real axis [0,∞) satisfying the condition∫ ∞ 0 x2pw(x)dx <∞. We denote by Lp,w[0,∞)(1 ≤ p <∞) the linear space of p-absolutely integrable on [0,∞) with respect to the weight function w Lp,w[0,∞) = { f : [0,∞)→ R, ‖f‖p,w = (∫ ∞ 0 |f(x)|pw(x)dx ) 1 p <∞ } . Theorem 8. [8] Let (Ln)n≥1 be a uniformly bounded sequence of positive linear operators from Lp,w[0,∞) into Lp,w[0,∞), satisfying the conditions lim n→∞ ‖ Ln(tk)− xk ‖p,w= 0, k = 0, 1, 2. (18) Then for every f ∈ Lp,w[0,∞) lim n→∞ ‖ Ln(f)− f ‖p,w= 0. A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 970 Now we choose w(x) = 1 (1+x2r)p , 1 ≤ p < ∞ and consider analogue weighted Lp-space [5]: Lp,2r[0,∞) = { f : [0,∞)→ R, ‖f‖p,2r = (∫ ∞ 0 ∣∣∣∣ f(x) 1 + x2r ∣∣∣∣p dx) 1 p <∞ } . Theorem 9. For every f ∈ Lp,2r[0,∞), r > 1, we have lim n→∞ ‖M (β,γ) n,α (f)− f ‖p,2r= 0. Proof. Using the Theorem 8, we see that it is sufficient to verify the three conditions (18). Since M (β,γ) n,α (1;x) = 1, the first condition is obvious for k = 0. By Lemma 2, for k = 1, we have(∫ ∞ 0 ∣∣∣∣∣M (β,γ) n,α (t;x)− x 1 + x2r ∣∣∣∣∣ p dx ) 1 p ≤ γ n+ γ (∫ ∞ 0 ∣∣∣∣ x 1 + x2r ∣∣∣∣p dx) 1 p + β n+ γ (∫ ∞ 0 ∣∣∣∣ 1 1 + x2r ∣∣∣∣p dx) 1 p which implies that lim n→∞ ‖M (β,γ) n,α (t)− x ‖p,2r= 0. For k = 2, we can write(∫ ∞ 0 ∣∣∣∣∣M (β,γ) n,α (t2;x)− x2 1 + x2r ∣∣∣∣∣ p dx ) 1 p ≤ ( n3(α+ 1) α(n− 1)(n+ γ)2 − 1 )(∫ ∞ 0 ∣∣∣∣ x2 1 + x2r ∣∣∣∣p dx) 1 p + 2n2 + 2nβ(n− 1) (n− 1)(n+ γ)2 (∫ ∞ 0 ∣∣∣∣ x 1 + x2r ∣∣∣∣p dx) 1 p + β2 (n+ γ)2 (∫ ∞ 0 ∣∣∣∣ 1 1 + x2r ∣∣∣∣p dx) 1 p which implies that lim n→∞ ‖M (β,γ) n,α (t2)− x2 ‖p,2r= 0. This completes the proof of theorem. 4. King type modification In this section, we discuss better convergence rates by King type operators. To make the convergence faster, King [26] proposed an approach to modify the classical Bernstein polynomial, so that the sequence preserve test functions e0 and e2, where ei(t) = ti, i = 0, 1, 2. After this approach many researcher contributed in this direction. As the operator M (β,γ) n,α (f ;x) defined in (3) preserve only the constant functions so further modification of these operators is proposed to be made so that the modified operators A. Kumar, D. Tapiawala, L. N. Mishra / Eur. J. Pure Appl. Math, 11 (4) (2018), 958-975 971 preserve the constant as well as linear functions. For this purpose the modification of (3) is defined as M̂ (β,γ) n,α (f ;x) = ∞∑ k=1 m (α) n,k(rn(x)) ∫ ∞ 0 bn,k(t)f ( nt+ β n+ γ ) dt + ( α α+ nrn(x) )α f ( β n+ γ ) (19) where rn(x) = (n+γ)x−β n and x ∈ In = [ β n+γ ,∞). Lemma 4. For every x ∈ In, we have (i) M̂ (β,γ) n,α (1;x) = 1, (ii) M̂ (β,γ) n,α (t;x) = x, (iii) M̂ (β,γ) n,α (t2;x) = n(α+ 1)x2 α(n− 1) + (2nα− 2αβ − 2nβ)x α(n− 1)(n+ γ) + nβ2 + αβ2 − 2nαβ α(n− 1)(n+ γ)2 . Consequently, for each x ∈ In, we have the following equalities M̂ (β,γ) n,α ((t− x);x) = 0 M̂ (β,γ) n,α ((t− x)2;x) = (n+ α)x2 α(n− 1) + (2nα− 2αβ − 2nβ)x α(n− 1)(n+ γ) + nβ2 + αβ2 − 2nαβ α(n− 1)(n+ γ)2 = λ(β,γ)n,α (x). (20) Theorem 10. For f ∈ CB(In), we have |M̂ (β,γ) n,α (f ;x)− f(x)| ≤M ′ω2 ( f, √ λ (β,γ) n,α (x) ) , where λ (β,γ) n,α (x) is given by (20) and M ′ is a positive constant. Proof. Let g ∈W 2 and x, t ∈ In. Using the Taylor’s expansion we have g(t) = g(x) + (t− x)g′(x) + ∫ t x (t− v)g′′(v)dv. Applying M̂ (β,γ) n,α on both sides and using Lemma ??, we get M̂ (β,γ) n,α (g;x)− g(x) = M̂ (β,γ) n,α (∫ t x (t− v)g′′(v)dv, x ) . REFERENCES 972 Obviously, we have ∣∣∣∣∫ t x (t− v)g′′(v)dv ∣∣∣∣ ≤ (t− x)2‖g′′‖. Therefore | M̂ (β,γ) n,α (g;x)− g(x) |≤ M̂ (β,γ) n,α ((t− x)2;x) ‖ g′′ ‖= λ(β,γ)n,α (x) ‖ g′′ ‖ . Since | M̂ (β,γ) n,α (f ;x) |≤ ‖f‖, we get | M̂ (β,γ) n,α (f ;x)− f(x) | ≤ | M̂ (β,γ) n,α (f − g;x) | + | (f − g)(x) | + | M̂ (β,γ) n,α (g;x)− g(x) | ≤ 2‖f − g‖+ λ(β,γ)n,α (x)‖g′′‖. Finally, taking the infimum over all g ∈W 2 and using (10) we obtain | M̂ (β,γ) n,α (f ;x)− f(x) |≤M ′ω2 ( f, √ λ (β,γ) n,α (x) ) , which proves the theorem. Theorem 11. Let f ∈ CB(In). 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