Direct estimates for certain integral type Operators - European Journal

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 approximation 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 wellknown Sz´ asz operators depending on α ∈ R as ∞ X k (α) (α) Gn (f ; x) = mn,k (x)f , x ∈ [0, ∞) (1) n k=0

where (α)

mn,k (x) =

k ( nx (α)k α ) , . α+k k! (1 + nx α )

∗

Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3305

Email addresses: [email protected] (A. Kumar), [email protected] (D. Tapiawala), [email protected], l n [email protected] (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 (α) operator Gn 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: α Z ∞ ∞ X α (α) Mn(α) (f ; x) = bn,k (t)f (t)dt + f (0), (2) mn,k (x) α + nx 0 k=1

where 1 tk−1 . , B(n + 1, k) (1 + t)k+n+1

bn,k (t) =

and B(m, n) being the Beta function defined as B(m, n) =

Γ(m)Γ(n) , m, n > 0. Γ(m + n)

They obtain different approximation properties for these operators. For the different values 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 operators, 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): (β,γ) Mn,α (f ; x)

=

∞ X k=1

(α) mn,k (x)

Z

∞

bn,k (t)f

0

nt + β n+γ

dt +

α α + nx

α β f . (3) n+γ

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 investigate weighted approximation theorems. Lastly we study King type modification of the operators (3).


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2. Moment estimates In the sequel, we shall need the following auxiliary results which will be necessary to prove our main results. (α)

Lemma 1. [11] For the operators Mn (f ; x), we have (i) Mn(α) (1; x) = 1, (ii) Mn(α) (t; x) = x, (iii) Mn(α) (t2 ; x) =

x[nx(α + 1) + 2α] . α(n − 1) (β,γ)

Lemma 2. For the operators Mn,α (f ; x), we have (β,γ) (i) Mn,α (1; x) = 1,

nx + β , n+γ 2 2n + 2nβ(n − 1) β2 n3 (α + 1) 2 (β,γ) 2 x + x + . (iii) Mn,α (t ; x) = α(n − 1)(n + γ)2 (n − 1)(n + γ)2 (n + γ)2 (β,γ) (ii) Mn,α (t; x) =

Proof. For x ∈ [0, ∞), in view of Lemma 1, we have (β,γ) Mn,α (1; x) = 1.

The first order moment is given by (β,γ) Mn,α (t; x) =

n β nx + β Mn(α) (t; x) + = . n+γ n+γ n+γ

The second order moment is given by (β,γ) 2 Mn,α (t ; x)

2 2 n 2nβ β (α) 2 (α) = Mn (t ; x) + M (t; x) + n+γ (n + γ)2 n n+γ 2 3 n (α + 1) 2n + 2nβ(n − 1) β2 2 = x + x + . α(n − 1)(n + γ)2 (n − 1)(n + γ)2 (n + γ)2

Lemma 3. For f ∈ CB [0, ∞) (space of all real valued bounded functions on [0, ∞) endowed with norm k f kCB [0,∞) = sup |f (x)|), x∈[0,∞)

(β,γ) (f ) k≤k f k . k Mn,α


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Proof. In view of (3) and Lemma 2, we get (β,γ) (β,γ) kMn,α (f )k ≤ kf kMn,α (1; x) = kf k.

Remark 1. For every x ∈ [0, ∞), we have (β,γ) Mn,α ((t − x); x) =

β − γx n+γ

and (β,γ) Mn,α

2

(t − x) ; x

=

2 2n + 2βγ(1 − n) β2 n2 (n + α) + αγ 2 (n − 1) 2 x + x + α(n − 1)(n + γ)2 (n − 1)(n + γ)2 (n + γ)2

(β,γ) = ξn,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. (β,γ) Theorem 1. Let f ∈ C[0, ∞). Then lim Mn,α (f ; x) = f (x), uniformly in each compact n→∞

subset of [0, ∞). Proof. In view of Lemma 2, we get (β,γ) lim Mn,α (ei ; x) = xi , i = 0, 1, 2,

n→∞

uniformly in each compact subset of [0, ∞). Applying Bohman-Korovkin theorem, it (β,γ) follows that lim Mn,α (f ; x) = f (x), uniformly in each compact subset of [0, ∞). n→∞

3.1. Voronovskaja type theorem In this section we prove Voronvoskaja type asymptotic theorem for the operators (β,γ) Mn,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 1 (β,γ) lim n Mn,α (f ; x) − f (x) = (β − γx)f 0 (x) + 2x + (l + 1)x2 f 00 (x). n→∞ 2 Proof. Let x ∈ [0, ∞) be fixed. Using Taylor’s expansion formula of function f , it follows 1 f (t) = f (x) + (t − x)f 0 (x) + f 00 (x)(t − x)2 + r(t, x)(t − x)2 , 2

(4)


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where r(t, x) is a bounded function and lim r(t, x) = 0. t→x

Applying

(β,γ) Mn,α (f ; x)

on both sides of (4), we get

1 (β,γ) (β,γ) (β,γ) n Mn,α (f ; x) − f (x) = nf 0 (x)Mn,α ((t − x); x) + nf 00 (x)Mn,α (t − x)2 ; x 2 (β,γ) 2 +nMn,α (t − x) r(t, x); x . In view of Remark 1, we have (β,γ) lim nMn,α ((t − x); x) = β − γx

(5)

(β,γ) lim nMn,α (t − x)2 ; x = 2x + (l + 1)x2 .

(6)

n→∞

and n→∞

Now, we shall show that (β,γ) lim nMn,α n→∞

2 r(t, x)(t − x) ; x = 0.

By using Cauchy-Schwarz inequality, we have 1/2 1/2 (β,γ) 2 (β,γ) 2 (β,γ) Mn,α r(t, x)(t − x) ; x ≤ Mn,α (r (t, x); x) Mn,α ((t − x)4 ; x) . (7) We observe that r2 (x, x) = 0 and r2 (., x) ∈ CB [0, ∞). Then, it follows that (β,γ) 2 lim Mn,α (r (t, x); x) = r2 (x, x) = 0.

n→∞

(8)

Now, from (7) and (8) we obtain (β,γ) lim nMn,α r(t, x)(t − x)2 ; x = 0.

n→∞

(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 {k f − g k +δ k g 00 k},

g∈W 2

where δ > 0 and W 2 = {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}. By, p. 177, Theorem 2.4 in [2], there exists an absolute constant M > 0 such that √ K2 (f, δ) ≤ M ω2 (f, δ), (10)


963

where √ ω2 (f, δ) =

sup | f (x + 2h) − 2f (x + h) + f (x) |

sup

√ 0 0, we have (β,γ)

(β,γ) |Mn,α (f ; x)

− f (x)| ≤ M

ξn,α (x) a1 x2 + a2 x

!r/2 . (a ,a2 )

Proof. First we prove the theorem for r = 1. Then, for f ∈ LipM1 we have

(1), and x > 0,

(β,γ) (β,γ) |Mn,α (f ; x) − f (x)| ≤ Mn,α (|f (t) − f (x)|; x) |t − x| (β,γ) ≤ M Mn,α ;x (t + a1 x2 + a2 x)1/2


≤

(a1

x2

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M M (β,γ) (|t − x|; x). + a2 x)1/2 n,α

Applying Cauchy-Schwarz inequality, we get 1/2 M (β,γ) 2 M ((t − x) ; x) n,α (a1 x2 + a2 x)1/2 !1/2 (β,γ) ξn,α (x) ≤ M . a1 x2 + a2 x

(β,γ) |Mn,α (f ; x) − f (x)| ≤

Thus the result holds for r = 1. (a ,a ) Now, we prove that the result is true for 0 < r < 1. Then, for f ∈ LipM1 2 (r), and x > 0, we get (β,γ) |Mn,α (f ; x) − f (x)| ≤ 1 r

Taking p =

and q =

p p−1 ,

x2

(a1

M M (β,γ) (|t − x|r ; x). + a2 x)r/2 n,α

applying the H¨olders inequality, we have r M (β,γ) M (|t − x|; x) . n,α (a1 x2 + a2 x)r/2

(β,γ) |Mn,α (f ; x) − f (x)| ≤

Finally by Cauchy-Schwarz inequality, we get (β,γ)

(β,γ) |Mn,α (f ; x)

ξn,α (x) a1 x2 + a2 x

− f (x)| ≤ M

!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 ω ¯ φ (f, δ) =

sup

sup

| f (x + hφ(x)) − f (x) |

0 0 such that √ √ (15) M −1 ω2,φ (f, δ) ≤ K2,φ (f, δ) ≤ M ω2,φ (f, δ). 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 M (β,γ) | Mn,α (f ; x) − f (x) |≤ 4K2,φ f, +ω ¯ φ f, , 2n n for n sufficiently large. Proof. Let g ∈ W 2 (φ). Applying Taylor’s expansion, we may write Z

0

g(t) = g(x) + (t − x)g (x) +

t

(t − v)g 00 (v)dv.

x (β,γ)

Applying M n,α on both sides of the above equation, we get (β,γ)

|M n,α (g; x) − g(x)| ≤ ≤

Z nx+β Z t n+γ 00 |t − v||g (v)|dv ; x + x x 2 2 00 k φ g k (β,γ) β − γx ξn,α (x) + . 2 φ (x) n+γ

(β,γ) Mn,α

nx + β 00 n + γ − v |g (v)|dv (16)

In view of Remark 1, it follows that there exist a positive constant M > 0 such that (β,γ) ξn,α (x) M 1 β − γx 2 M ≤ , ≤ 2. φ2 (x) n φ2 (x) n + γ n Thus, (β,γ) |M n,α (g; x)

2 00

− g(x)| ≤ M k φ g k

1 1 + n n2

≤

2M k φ2 g 00 k . n

Now, (β,γ) |Mn,α (f ; x) − f (x)| ≤

(β,γ) |M n,α (f

− g; x)| + |(f − g)(x)| +

(β,γ) |M n,α (g; x)

nx + β − g(x)| + f − f (x) n+γ


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2M nx + β 2 00 ≤ 4kf −g k+ − f (x) . k φ g k + f n n+γ Also, we obtain nx + β f − f (x) = f n+γ

x + φ(x)

≤ sup f f,

M n

!

− f (x)

!

− f (x)

−x

φ(x)

x + φ(x) √

≤ ω ¯φ

nx+β n+γ

β−γx n+γ

φ(x)

! .

Using the above equations, we get

M (β,γ) 2 00 | Mn,α (f ; x) − f (x) | ≤ 4 k f − g k + kφ g k +ω ¯φ 2n

√

M f, 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

sup |f (t) − f (x)|.

|t−x|≤δ x,t∈[0,a]

We observe that for a function f ∈ CB [0, ∞), the modulus of continuity ωa (f, δ) tends to zero. (β,γ) Now, we give a rate of convergence theorem for the operators Mn,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 q (β,γ) (β,γ) 2 (β,γ) |Mn,α (f ; x) − f (x)| ≤ 6Mf (1 + a )ξn,α (a) + 2ωa+1 f, ξn,α (a) , (β,γ)

where ξn,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 .


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For x ∈ [0, a] and t ≤ a + 1, we have |t − x| ωa+1 (f, δ) |f (t) − f (x)| ≤ ωa+1 (f, |t − x|) ≤ 1 + δ with δ > 0. From the above, we have |t − x| |f (t) − f (x)| ≤ 6Mf (1 + a )(t − x) + 1 + ωa+1 (f, δ), δ 2

2

for x ∈ [0, a] and t ≥ 0. Thus (β,γ) (β,γ) |Mn,α (f ; x) − f (x)| ≤ 6Mf (1 + a2 )(Mn,α (t − x)2 ; x) 1 1 (β,γ) 2 2 +ωa+1 (f, δ) 1 + (Mn,α (t − x) ; x) . δ

Applying Cauchy-Schwarz’s inequality, we get (β,γ) |Mn,α (f ; x)

− f (x)| ≤ 6Mf (1 + a

2

(β,γ) )ξn,α (a)

q (β,γ) + 2ωa+1 f, ξn,α (a) ,

q (β,γ) on choosing δ = ξn,α (a). This completes the proof of theorem.

3.5. Weighted approximation (β,γ)

In this section we give some weighted approximation properties of the operators Mn,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)| ≤ Mf ν(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 f (x) Cν∗ [0, ∞) denotes the subspace of all functions f ∈ Cν [0, ∞) for which lim exists |x|→∞ ν(x) finitely. It is obvious that Cν∗ [0, ∞) ⊂ Cν [0, ∞) ⊂ Bν [0, ∞). The space Bν [0, ∞) is a normed linear space with the following norm: |f (x)| . x∈[0,∞) ν(x)

k f kν = sup

Theorem 7. For each f ∈ Cν∗ [0, ∞), we have (β,γ) lim k Mn,α (f ) − f kν = 0.

n→∞


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Proof. From [6], we know that it is sufficient to verify the following three conditions (β,γ) lim k Mn,α (ei ) − ei kν = 0, i = 0, 1, 2.

(17)

n→∞ (β,γ)

Since Mn,α (1; x) = 1, the condition in (17) holds true for i = 0. By Lemma 2, we have (β,γ)

k

(β,γ) Mn,α (t)

− x kν

|Mn,α (t; x) − x| = sup 1 + x2 x∈[0,∞) β γ x + ≤ sup 2 n + γ x∈[0,∞) 1 + x n+γ ≤

sup x∈[0,∞)

1 1 + x2

β+γ n+γ

(β,γ) which implies that lim k Mn,α (t) − x kν = 0. n→∞ Again by Lemma 2, we have (β,γ)

(β,γ) 2 k Mn,α (t ) − x2 kν

|Mn,α (t2 ; x) − x2 | 1 + x2 x∈[0,∞) 2 2n + 2nβ(n − 1) β2 n3 (α + 1) + − 1 ≤ + (n − 1)(n + γ)2 (n + γ)2 , α(n − 1)(n + γ)2 =

sup

(β,γ) 2 which implies that lim k Mn,α (t ) − x2 kν = 0. n→∞ This completes the proof of theorem.

3.6. Weighted Lp -approximation Let w be positive continuous function on real axis [0, ∞) satisfying the condition Z ∞ x2p w(x)dx < ∞. 0

We denote by Lp,w [0, ∞)(1 ≤ p < ∞) the linear space of p-absolutely integrable on [0, ∞) with respect to the weight function w ) ( Z ∞ 1 p |f (x)|p w(x)dx