3. Approximation Properties

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International Journal of Advances in Applied Mathematics and Mechanics

Approximation properties of general gamma type operators in polynomial weighted space

Research Article

Alok Kumar∗, Artee, D. K. Vishwakarma

Department of Computer Science, Dev Sanskriti Vishwavidyalaya, Haridwar-249411, Uttarakhand, India

Received 12 November 2016; accepted (in revised version) 19 December 2016

Abstract: In this paper we give direct approximation theorems for general Gamma type operators in polynomial weighted spaces of functions of one variable. The results are given in terms of some Ditzian-Totik moduli of smoothness.

MSC: 41A25 • 26A15 • 40A35

Keywords: Gamma type operators • Rate of convergence • Steklov means • Polynomial weighted space

1. Introduction

In [1], Lupas and M ¨uller defined and studied some approximation properties of a sequence of positive and linear operators {G_n} defined as

G_n( f ; x) = Z _∞

0

g_n(x, u) f³n u

´ d u,

where

gn(x, u) =xⁿ⁺¹

n! e^−xuuⁿ, x ∈ (0,∞).

Approximation properties of {Gn} in some function spaces were studied in many papers (see [1–4]). The above operators were modified by several researchers (see [5–7]), which showed that new operators have similar approximation properties to {Gn} (see [8–14]).

In 2007, Mao [15] defined the following operators

M_n,k( f ; x) =(2n − k + 1)!

n!(n − k)! xⁿ⁺¹ Z _∞

0

t^n−k

(x + t)^2n−k+2f (t )d t , x, t ∈ (0,∞) (1)

for any f for which the above integral is convergent.

Some approximation properties of {M_n,k} were studied in these papers (see [16–19]).

Recently, Alok Kumar [20] obtained the following result.

∗ Corresponding author.

E-mail addresses: [email protected](Alok Kumar),[email protected](Artee),[email protected] (D. K. Vish- wakarma).

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Lemma 1.1 ([20]).

If r^{t h}derivative f^{(r )}(r = 0,1,2...) exists continuously, then we get M^{(r )}

n,k( f ; x) = βnx^n+1−r Z _∞

0

t^n−k+r

(x + t)^2n−k+2f^{(r )}(t )d t , x ∈ (0,∞), where

βn=(2n − k + 1)!

n!(n − k)! .

The Voronovskaja type theorem and the local rate of convergence for operators M_n,k^{(r )} were given in [20].

The aim of this paper is to study approximation properties of M_n,k^{(r )} in C_p, p ∈ N0(set of non-negative integers), where C_pis a polynomial weighted space with the weight functionµp

µ0(x) = 1, µp(x) = 1

1 + x^p, p ≥ 1, (2)

and C_p is the set of all real valued continuous functions f for which fµp is bounded and uniformly continuous on [0, ∞).

The norm on C_pis defined by

°°f°

°p= sup

x∈[0,∞)µp(x)| f (x)|, f ∈ Cp.

We also consider the modulus of smoothness of f ∈ Cp, ω²p( f ;δ) := sup

h∈(0,δ]

°

°∆²_hf°

°p,

and the modulus of continuity of f ∈ Cp, ωp( f ;δ) := sup

h∈(0,δ]

°

°∆hf°

°p, where

∆²_hf (x) := f (x + 2h) − 2f (x + h) + f (x), ∆hf (x) := f (x + h) − f (x) for x, h ∈ [0,∞).

2. Auxiliary results

In this section we give some preliminary results which will be used in the main part of this paper.

Let

b(n, k, r ) = βnx^n+1−r Z _∞

0

t^n−k+r

(x + t)^2n−k+2d t =(n − r )!(n − k + r )!

n!(n − k)! . We define the sequence of linear and positive operators {M_n,k,r^∗ } as

M_n,k,r^∗ (g ; x) = βn

b(n, k, r )x^n+1−r Z_∞

0

t^n−k+r

(x + t)^2n−k+2g (t )d t , (3)

for each x, t ∈ (0,∞),r ∈ N0(see [20]).

Let us consider

e_m(t ) = t^m, ϕx,m(t ) = (t − x)^m, m ∈ N0, x, t ∈ (0,∞).

Lemma 2.1 ([20]).

For any m ∈ N0, m + r ≤ n and r ≤ n we have

M_n,k,r^∗ (em; x) =(n − r − m)!(n − k + r + m)!

(n − r )!(n − k + r )! x^m and

M_n,k,r^∗ (ϕx,m; x) = Ãm

X

j =0

(−1)^j Ãm

j

!(n − r − m + j )!(n − k + r + m − j )!

(n − r )!(n − k + r )!

! x^m

for each x ∈ (0,∞).

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Lemma 2.2 ([20]).

For m = 0,1,2,3,4, one has (i) M_n,k,r^∗ (ϕx,0; x) = 1,

(ii) M_n,k,r^∗ (ϕx,1; x) =2r − k + 1 n − r x,

(iii) M_n,k,r^∗ (ϕx,2; x) =4r²+ 4r (2 − k) + 2n + k²− 5k + 4 (n − r )(n − r − 1) x², (iv) M_n,k,r^∗ (ϕx,3; x) = c_n,k,r

(n − r )(n − r − 1)(n − r − 2)x³, (v) M_n,k,r^∗ (ϕx,4; x) = d_n,k,r

(n − r )(n − r − 1)(n − r − 2)(n − r − 3)x⁴,

where c_n,k,r= 8r³+ r²(36 − 2k) + r (51 + 14n − 42k + 6k²) − k³+ 12k²− 34k − n²+ n(17 − 6k − 6k²+ 2kr ) + 21 and

d_n,k,r= 16r⁴+r³(128−32k)+r²(348+48n −216k +24k²)+r (366+177n +k(6n²−54n −440)+120k²−8k³)+k⁴+k³(4n − 22) + 139k²− k(245 + 116n) + 24n²+ 131n + 100.

Let p, r ∈ N0. By C_p^r, we denote the space of all functions f ∈ Cpsuch that f⁰, f⁰⁰... f^{(r )}∈ Cp.

Theorem 2.1.

For the operator M^{(r )}

n,kand for fixed p ∈ N0, n ∈ N and f ∈ C^rp, there exists a positive constantK_p,k,r¹ depending only on the parameters p, k and r such that

°

° 1

b(n, k, r )M_n,k^{(r )}( f ; .)

°

°p≤ Kp,k,r¹

°°f^{(r )}°

°p (4)

which guarantees that M^{(r )}_n,kmaps C_p^r into C_p^r.

Proof. For p = 0, the proof follows immediately.

Let p ∈ N . ByLemma 2.2, we can find a positive constantK_p,k,r¹ such that µp(x)M_n,k,r^∗

µ 1 µp(t ); x

¶

= µp(x){M_n,k,r^∗ (e₀; x) + M_n,k,r^∗ (e_p; x)}

= µp(x)

½

1 +(n − r − p)!(n − k + r + p)!

(n − r )!(n − k + r )! x^p

¾

≤ K_p,k,r¹ µp(x){1 + x^p} = K_p,k,r¹ , where

K_p,k,r¹ = max

½ sup

n

(n − r − p)!(n − k + r + p)!

(n − r )!(n − k + r )! , 1

¾ .

Observe that for every f ∈ Cp^{(r )}and x ∈ (0,∞), we get

µp(x)

¯

¯ 1

b(n, k, r )M_n,k^{(r )}( f ; x)

¯

¯≤ µp(x) βn

b(n, k, r )x^n+1−r Z_∞

0

t^n−k+r

(x + t)^2n−k+2| f^{(r )}(t )|µp(t ) µp(t )d t

≤ k f^{(r )}kpµp(x)M_n,k,r^∗ µ 1

µp(t ); x

¶

≤ K_p,k,r¹ k f^{(r )}kp. Taking supremum over x ∈ (0,∞), we get desired result.

Lemma 2.3.

For the operators M_n,k,r^∗ and for fixed p, r ∈ N0, there exists a positive constantK_p,k,r² such that

µp(x)M_n,k,r^∗ µ ϕx,2

µp(t ); x

¶

≤ K_p,k,r² x² n for all x ∈ (0,∞) and n ∈ N .

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Proof. Let p = 0. UsingLemma 2.2, we can write µ0(x)M_n,k,r^∗

µϕx,2

µ0(t ); x

¶

=4r²+ 4r (2 − k) + 2n + k²− 5k + 4 (n − r )(n − r − 1) x²

=n(4r²+ 4r (2 − k) + 2n + k²− 5k + 4) (n − r )(n − r − 1)

x² n

≤ Kk,r²

x² n . Now, let p > 0. Then, we get fromLemma 2.2

M_n,k,r^∗ µ ϕx,2

µp(t ); x

¶

= Mn,k,r^∗ (e_p+2; x) − 2xM_n,k,r^∗ (e_p+1; x) + x²M_n,k,r^∗ (e_p; x) + M_n,k,r^∗ (ϕx,2; x)

=(n − r − p − 2)!(n − k + r + p)!

(n − r )!(n − k + r )! x^p+2[(n − r − p)(n − r − p − 1)

− 2(n − r − p − 1)(n − k + r + p + 1) + (n − k + r + p + 2) (n − k + r + p + 1)] +4r²+ 4r (2 − k) + 2n + k²− 5k + 4

(n − r )(n − r − 1) x²

= µ

1 +

½

1 + 8r p + 4p²+ 8p − 4kp − 1 4r²+ r (8 − 4k) + 2n + k²− 5k + 4

¾ (n − r − p − 2)!(n − k + r + p)!

(n − r − 2)!(n − k + r )! x^p

¶

4r²+ r (8 − 4k) + 2n + k²− 5k + 4 (n − r )(n − r − 1) x²

≤ Kp,k,r²

x²

n (1 + x^p).

Hence, the proof is completed.

3. Approximation Properties

Theorem 3.1.

For every p, r ∈ N0and g ∈ Cp¹, there exists a positive constantK_p,k,r³ depending only on the parameters p, k and r such that

µp(x)|M_n,k,r^∗ (g ; x) − g (x)| ≤ K_p,k,r³ kg⁰kp

px n for all x ∈ (0,∞) and n ∈ N .

Proof. Let x ∈ (0,∞) be fixed. We have g (t ) − g (x) =

Z t x

g⁰(v)d v, t ∈ (0,∞).

By using the linearity of M_n,k,r^∗ and M_n,k,r^∗ (1; x) = 1, we get M_n,k,r^∗ (g ; x) − g (x) = M_n,k,r^∗

µZ t x

g⁰(v)d v; x

¶

. (5)

Observe that

¯

¯ Z t

x

g⁰(v)d v

¯

¯≤ kg⁰kp

¯

¯ Z t

x

d v µp(v)

¯

¯≤ kg⁰kp|t − x|

µ 1

µp(t )+ 1 µp(x)

¶ . Using (5) we obtain

µp(x)|M_n,k,r^∗ (g ; x) − g (x)| ≤ kg⁰kp

µ

M_n,k,r^∗ (|ϕx,1|; x) + µp(x)M_n,k,r^∗ µ |ϕx,1| µp(t ); x

¶¶

. Applying the Cauchy-Schwarz inequality we can write

M_n,k,r^∗ (|ϕx,1|; x) ≤q

M_n,k,r^∗ (ϕx,2; x) ×q M_n,k,r^∗ ¡

ϕx,0; x¢,

M_n,k,r^∗ µ |ϕx,1| µp(t ); x

¶

≤ s

M_n,k,r^∗ µ 1

µp(t ); x

¶

× s

M_n,k,r^∗ µ ϕx,2

µp(t ); x

¶ . Finally, usingLemma 2.2,Lemma 2.3andTheorem 2.1, we obtain

µp(x)|M_n,k,r^∗ (g ; x) − g (x)| ≤ K_p,k,r³ kg⁰kp

px n.

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Theorem 3.2.

Let p, r ∈ N0and n ∈ N . If

W_n,k,r( f ; x) = M_n,k,r^∗ ( f ; x) − f µ

x +2r − k + 1 n − r x

¶

+ f (x), (6)

then there exists a positive constantK_p,k,r⁴ depending only on the parameters p, k and r such that for all x ∈ (0,∞), we have

µp(x)¯

¯W_n,k,r( f ; x) − f (x)¯

¯≤ K_p,k,r⁴ k f⁰⁰kp

x² n for any function f ∈ Cp².

Proof. FromLemma 2.2, we observe that the operators Wn,k,r are linear and reproduce the linear functions.

Hence

W_n,k,r(ϕx,1; x) = 0.

For f ∈ Cp²and t ∈ (0,∞), we have

f (t ) = f (x) + (t − x)f⁰(x) + Z t

x (t − v)f⁰⁰(v)d v.

Then,

|Wn,k,r( f ; x) − f (x)| =|Wn,k,r( f (t ) − f (x); x)| =

¯

¯ W_n,k,r

µZ t

x (t − v)f⁰⁰(v)d v; x

¶¯

¯

=

¯

¯ M_n,k,r^∗

µZ t

x (t − v)f⁰⁰(v)d v; x

¶

−

Z _x+^{2r −k+1}

n−r x x

µ

x +2r − k + 1 n − r x − v

¶

f⁰⁰(v)d v

¯

¯ .

Notice that

¯

¯ Z t

x (t − v) f⁰⁰(v)d v

¯

¯≤k f⁰⁰kp(t − x)² 2

µ 1

µp(x)+ 1 µp(t )

¶

and

¯

Z _x+^{2r −k+1}

n−r x x

µ

x +2r − k + 1 n − r x − v

¶

f⁰⁰(v)d v

¯

≤ k f⁰⁰kp

2µp(x)

µ 2r − k + 1 n − r x

¶2

.

Using the above inequality, we get

µp(x)|Wn,k,r( f ; x) − f (x)| ≤k f⁰⁰kp

2 µ

M_n,k,r^∗ (ϕx,2; x) + µp(x)M_n,k,r^∗ µ ϕx,2

µp(t ); x

¶¶

+k f⁰⁰kp

2

µ 2r − k + 1 n − r x

¶2

.

Hence, byLemma 2.3we obtain

µp(x)|Wn,k,r( f ; x) − f (x)| ≤ K_p,k,r⁴ k f⁰⁰kp

x² n for every f ∈ Cp².

Theorem 3.3.

Let p, r ∈ N0and n ∈ N . If f ∈ C^{(r )}p , then there exists a positive constantK_p,k,r⁵ such that

µp(x)

¯

¯ 1

b(n, k, r )M_n,k^{(r )}( f ; x) − f^{(r )}(x)

¯

¯≤ K_p,k,r⁵ ω²p

µ f^{(r )}, x

pn

¶ + ωp

µ

f^{(r )},2r − k + 1 n − r x

¶ ,

for all x ∈ (0,∞).

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Proof. Let f ∈ Cp^r. We consider the Steklov means ˜f_h^{(r )}, h > 0 of f^{(r )}as (see p. 317, [21])

f˜_h^{(r )}(x) = 4 h²

Z h/2 0

¡2f^{(r )}(x + s + t) − f^{(r )}(x + 2(s + t))¢ d sd t,

for x, h ∈ (0,∞). We have

f^{(r )}(x) − ˜f_h^{(r )}(x) = 4 h²

Z_h/2

0

Z_h/2

0 ∆²_s+tf^{(r )}(x)d sd t , which gives

°

° f

(r )− ˜f_h^{(r )}°

°

°p≤ ω²p¡ f^{(r )}, h¢ . (7)

Remark that f˜_h^{(r +2)}(x) = 1

h²¡8∆²_h/2f^{(r )}(x) − ∆²_hf^{(r )}(x)¢ and

°

° f˜_h^{(r +2)}

°

°p≤ 9

h²ω²p¡ f^{(r )}, h¢ . (8)

From (7) and (8), we conclude that ˜f_h^{(r )}∈ Cp²if f^{(r )}∈ Cp. Observe that

¯

¯M

∗

n,k,r( f^{(r )}; x) − f^{(r )}(x)¯

¯

¯ ≤ Wn,k,r

³¯

¯

¯ f

(r )− ˜f_h^{(r )}¯

¯

¯ ; x

´ +

¯

¯ f

(r )(x) − ˜f_h^{(r )}(x)¯

¯

¯ +

¯

¯Wn,k,r³ ˜f_h^{(r )}; x´

− ˜f_h^{(r )}(x)¯

¯

¯ +

¯

¯ f^{(r )}

µ

x +2r − k + 1 n − r x

¶

− f^{(r )}(x)

¯

¯ ,

where W_n,k,ris defined in (6). Since ˜f_h^{(r )}∈ Cp², it follows fromTheorem 2.1andTheorem 3.2that

µp(x)

¯

¯ 1

b(n, k, r )M_n,k^{(r )}( f ; x) − f^{(r )}(x)

¯

¯= µp(x)

¯

¯M

∗

n,k,r( f^{(r )}; x) − f^{(r )}(x)

¯

≤ (K + 5)

°

° f

(r )− ˜f_h^{(r )}

°

°_p+ K_p,k,r⁴

°

° f˜_h^{(r +2)}

°

°_p x²

n + µp(x)

¯

¯ f^{(r )}

µ

x +2r − k + 1 n − r x

¶

− f^{(r )}(x)

¯

¯ Using (7) and (8), we get

µp(x)

¯

¯ 1

b(n, k, r )M_n,k^{(r )}( f ; x) − f^{(r )}(x)

¯

¯≤ Kp,k,r⁵ ω²p( f , h)

½ 1 + 1

h² x²

n

¾ + ωp

µ

f^{(r )},2r − k + 1 n − r x

¶ .

Thus, choosing h =^p^x_n we get the desired result.

Acknowledgement

The author(s) are very grateful to the referee for making valuable comments leading to the overall improvement of the paper.

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