Global approximation theorems for general Gamma type Operators

(1)

Journal homepage:www.ijaamm.com

International Journal of Advances in Applied Mathematics and Mechanics

Global approximation theorems for general Gamma type Operators

Research Article

Alok Kumar∗, D. K. Vishwakarma

Department of Computer Science, Dev Sanskriti Vishwavidyalaya, Haridwar-249411, Uttarakhand, India Dedicated to Prof. P. N. Agrawal

Received 17 September 2015; accepted (in revised version) 16 November 2015

Abstract: In the present paper we obtained some direct approximation theorems for general Gamma type operators in polynomial weighted spaces.

MSC: 41A25 • 26A15 • 40A35

Keywords:Gamma type operators • Global approximation • Polynomial weighted space • Steklov means

1. Introduction

For a measurable complex valued and locally bounded function defined on [0, ∞), Lupas and M ¨uller [1] defined and studied some approximation properties of Gamma operators {G_n} defined by

G_n( f ; x) = Z _∞

0

g_n(x, u) f³n u

´ d u,

where

g_n(x, u) =xⁿ⁺¹

n! e^−xuuⁿ, x > 0.

In [2], Mazhar gives an important modifications of the Gamma operators using the same g_n(x, u)

F_n( f ; x) = Z _∞

0

Z_∞

0

g_n(x, u)g_n−1(u, t ) f (t )d ud t

=(2n)!xⁿ⁺¹ n!(n − 1)!

Z _∞

0

tⁿ⁻¹

(x + t)²ⁿ⁺¹f (t )d t , n > 1, x > 0.

Recently, by using the techniques due to Mazhar, ˙I zgi and B ¨uy ¨ukyazici [3], Karsli [4] independently considered the following Gamma type linear and positive operators

L_n( f ; x) = Z _∞

0

Z _∞

0

g_n+2(x, u)g_n(u, t ) f (t )d ud t

=(2n + 3)!xⁿ⁺³ n!(n + 2)!

Z _∞

0

tⁿ

(x + t)²ⁿ⁺⁴f (t )d t , x > 0, and obtained some approximation results.

In [5], Karsli and ¨Ozarslan obtained some local and global approximation results for the operators L_n( f ; x). Global

∗ Corresponding author.

E-mail address:[email protected](Alok Kumar), [email protected](D. K. Vishwakarma)

(2)

approximation results for different operators were examined in many papers, for example in [6], [7] and [8].

In 2007, Mao [9] define the following generalised Gamma type linear and positive operators M_n,k( f ; x) =

Z _∞

0

Z _∞

0

gn(x, u)g_n−k(u, t ) f (t )d ud t

=(2n − k + 1)!xⁿ⁺¹ n!(n − k)!

Z _∞

0

t^n−k

(x + t)^2n−k+2f (t )d t , x > 0, which includes the operators Fn( f ; x) for k = 1 and Ln−2( f ; x) for k = 2.

Some approximation properties of M_n,kwere studied in [10], [11] and [12].

We can rewrite the operators Mn,k( f ; x) as M_n,k( f ; x) =

Z _∞

0

K_n,k(x, t ) f (t )d t , (1)

where

K_n,k(x, t ) =(2n − k + 1)!xⁿ⁺¹ n!(n − k)!

t^n−k

(x + t)^2n−k+2, x, t ∈ (0,∞).

In this paper, we study some global approximation results of the operators M_n,k. Let p ∈ N0(set of non-negative integers), f ∈ Cp, where C_pis a polynomial weighted space with the weight function w_p,

w₀(x) = 1, wp(x) = 1

1 + x^p, p > 0, (2)

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

The norm on C_pis defined by the formula k f kp= sup

x∈[0,∞)

wp(x)| f (x)|.

We also consider the following Lipschitz classes:

ω²p( f ;δ) = sup

h∈(0,δ]k∆²hf kp,

∆²_hf (x) = f (x + 2h) − 2f (x + h) + f (x),

ω¹p( f ;δ) = sup

h∈(0,δ]

k∆hf kp,

∆hf (x) = f (x + h) − f (x),

Li p²_pα = {f ∈ Cp[0, ∞) : ω²p( f ;δ) = O(δ^α) asδ → 0⁺}, where h > 0 and α ∈ (0,2].

From the above it follows that

δ→0lim⁺ω¹p( f ;δ) = 0, lim

δ→0⁺ω²p( f ;δ) = 0, for every f ∈ Cp[0, ∞).

2. Auxiliary results

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

Let us consider

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

Lemma 2.1 ([11]).

For any m ∈ N0(set of non-negative integers), m ≤ n − k M_n,k(t^m; x) =[n − k + m]m

[n]m

x^m (3)

where n, k ∈ N and [x]m= x(x − 1)...(x − m + 1), [x]0= 1, x ∈ R.

In particular for m = 0, 1, 2... in (3) we get

(3)

(i) M_n,k(1; x) = 1, (ii) M_n,k(t ; x) =n − k + 1

n x,

(iii) M_n,k(t²; x) =(n − k + 2)(n − k + 1) n(n − 1) x².

Lemma 2.2 ([11]).

Let m ∈ N0and fixed x ∈ (0,∞), then

M_n,k(ϕx,m; x) = Ãm

X

j =0

(−1)^j Ãm

j

!(n − m + j )!(n − k + m − j )!

n!(n − k)!

! x^m.

Lemma 2.3.

For m = 0,1,2,3,4, one has (i) M_n,k(ϕx,0; x) = 1, (ii) M_n,k(ϕx,1; x) =1 − k

n x,

(iii) M_n,k(ϕx,2; x) =k²− 5k + 2n + 4 n(n − 1) x²,

(iv) M_n,k(ϕx,3; x) =−k³+ 12k²− 17k + n(18 − 12k) + 24 n(n − 1)(n − 2) x³,

(v) M_n,k(ϕx,4; x) =k⁴− 22k³+ k²(143 + 12n) − k(314 + 108n) + 12n²+ 268n + 192 n(n − 1)(n − 2)(n − 3) x⁴, (vi) M_n,k(ϕx,m; x) = O¡n^−[(m+1)/2]¢.

Proof. UsingLemma 2.2, we getLemma 2.3.

Theorem 2.1.

For the operators M_n,k and for fixed p ∈ N0, there exists a positive constant N_p,k depending only on the parameters p and k such that

w_p(x)M_n,k

µ 1

w_p(t ); x

¶

≤ Np,k. (4)

Moreover for every f ∈ Cp[0, ∞), we have

kMn,k( f ; .)kp≤ Np,kk f kp, (5)

which shows that M_n,kis a linear positive operator from the space C_p[0, ∞) into Cp[0, ∞).

Proof. For p = 0, (4) follows immediately. UsingLemma 2.1, we get w_p(x)M_n,k

µ 1

w_p(t ); x

¶

= wp(x)¡M_n,k(e₀; x) + Mn,k(e_p; x)¢

= wp(x) µ

1 +(n − p)!(n − k + p)!

n!(n − k)! x^p

¶

≤ Np,kw_p(x)(1 + x^p) = Np,k, where

N_p,k= max

½ sup

n

(n − p)!(n − k + p)!

n!(n − k)! , 1

¾ . Observe that for all f ∈ Cpand every x ∈ (0,∞), we get

w_p(x)¯

¯M_n,k( f ; x)¯

¯≤ wp(x)(2n − k + 1)!xⁿ⁺¹ n!(n − k)!

Z _∞

0

t^n−k

(x + t)^2n−k+2| f (t )|wp(t ) w_p(t )d t

≤ k f kpw_p(x)Mn,k

µ 1

wp(t ); x

¶

≤ Np,kk f kp. Taking supremum over x ∈ (0,∞), we get (5).

(4)

Lemma 2.4.

For the operators M_n,kand fixed p ∈ N0, there exists a positive constant N_p,kdepending only on the parameters p and k such that

w_p(x)M_n,k µ ϕx,2

w_p(t ); x

¶

≤ Np,k

x² n .

Proof. UsingLemma 2.3, we can write

w₀(x)M_n,k µ ϕx,2

w₀(t ); x

¶

=k²− 5k + 2n + 4 n(n − 1) x²

≤ Nk

x² n , which gives the result for p = 0.

Let p ≥ 1. Then usingLemma 2.1andLemma 2.3, we get

M_n,k µ ϕx,2

w_p(t ); x

¶

= Mn,k(e_p+2; x) − 2xMn,k(e_p+1; x) + x²M_n,k(ep; x) + Mn,k(ϕx,2; x)

=(n − p − 2)!(n − k + p + 2)!

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

n!(n − k)! x^p+2 +(n − p)!(n − k + p)!

n!(n − k)! x^p+2+k²− 5k + 2n + 4 n(n − 1) x²

≤ Np,k

x²

n (1 + x^p),

where N_p,k= µ

1 +¡k²− 5k + 6p + 4p²− 4kp¢ (n − p − 2)!(n − k + p)!

(n − 2)!(k²− 5k + 2n + 4)x^p

¶k²− 5k + 2n + 4

n − 1 is a positive constant. This completes the proof.

3. Direct results

The proof of direct theorems will follow from Jackson type inequality, the Steklov means and appropriate esti- mates of the moments of the operators.

Let p ∈ N0. By C_p²[0, ∞), we denote the space of all functions f ∈ Cp[0, ∞) such that f⁰, f⁰⁰∈ Cp[0, ∞).

Theorem 3.1.

Let p ∈ N0and f ∈ Cp¹[0, ∞), there exists a positive constant Np,kdepending only on the parameters p and k such that

wp(x)|Mn,k( f ; x) − f (x)| ≤ Np,kk f⁰kp

px n for all x ∈ (0,∞) and n ∈ N .

Proof. Let x ∈ (0,∞) be fixed. Then for f ∈ C¹p[0, ∞) and t ∈ (0,∞), we have

f (t ) − f (x) = Z t

x

f⁰(v)d v.

By using linearity of M_n,kwe get

M_n,k( f ; x) − f (x) = Mn,k

µZ t x

f⁰(v)d v; x

¶

. (6)

Remark that

¯

¯ Z t

x

f⁰(v)d v

¯

¯≤ k f⁰kp

¯

¯ Z t

x

d v w_p(v)

¯

¯≤ k f⁰kp|t − x|

µ 1

w_p(t )+ 1 w_p(x)

¶ .

From (6) we obtain

w_p(x)|Mn,k( f ; x) − f (x)| ≤ kf⁰kp

½

M_n,k(|ϕx,1|; x) + wp(x)M_n,kµ |ϕx,1| w_p(t ); x

¶¾ .

(5)

Using Cauchy-Schwarz inequality, we can write M_n,k(|ϕx,1|; x) ≤¡M_n,k(ϕx,2; x)¢1/2

×¡M_n,k(ϕx,0; x)¢1/2

,

M_n,kµ |ϕx,1| w_p(t ); x

¶

≤ µ

M_n,k µ 1

w_p(t ); x

¶¶1/2

× µ

M_n,k µ ϕx,2

w_p(t ); x

¶¶1/2

.

UsingLemma 2.3,Theorem 2.1andLemma 2.4, we obtain wp(x)|Mn,k( f ; x) − f (x)| ≤ Np,kk f⁰kp

px n.

Lemma 3.1.

Let p ∈ N0. If

T_n,k( f ; x) = Mn,k( f ; x) − f µ

x +1 − k n x

¶

+ f (x), (7)

then there exists a positive constant N_p,ksuch that for all x ∈ (0,∞) and n ∈ N , we have

wp(x)|Tn,k(g ; x) − g (x)| ≤ Np,kkg⁰⁰kp

x² n for any function g ∈ C²p.

Proof. FromLemma 2.1, we observe that the operators T_n,kare linear and reproduce the linear functions.

Hence

T_n,k(ϕx,1; x) = 0.

Let g ∈ Cp². By the Taylor formula one can write

g (t ) − g (x) = (t − x)g⁰(x) + Zt

x (t − v)g⁰⁰(v)d v, t ∈ (0,∞).

Then,

|Tn,k(g ; x) − g (x)| = |Tn,k(g − g (x)); x| =

¯

¯ T_n,k

µZ t

x (t − v)g⁰⁰(v)d v; x

¶¯

¯

=

¯

¯ M_n,k

µZ t

x (t − v)g⁰⁰(v)d v; x

¶

− Z _x+^1−k

n x x

µ

x +1 − k n x − v

¶

g⁰⁰(v)d v

¯

¯ .

Since

¯

¯ Z t

x (t − v)g⁰⁰(v)d v

¯

¯≤∥ g⁰⁰∥p(t − x)² 2

µ 1

wp(x)+ 1 wp(t )

¶

and

¯

¯ Z _x+^1−k_n x

x

µ

x +1 − k n x − v

¶

g⁰⁰(v)d v

¯

≤ kg⁰⁰kp

2wp(x) µ 1 − k

n x

¶2

,

we get

w_p(x)|Tn,k(g ; x) − g (x)| ≤kg⁰⁰kp

2 µ

M_n,k(ϕx,2; x) + wp(x)M_n,k µ ϕx,2

w_p(t ); x

¶¶

+kg⁰⁰kp

2

µ 1 − k n x

¶2

.

Hence byLemma 2.4, we obtain

w_p(x)|Tn,k(g ; x) − g (x)| ≤ Np,kkg⁰⁰kp

x² n for any function g ∈ Cp². The Lemma is proved.

(6)

Theorem 3.2.

Let p ∈ N0, n ∈ N and f ∈ Cp[0, ∞), there exists a positive constant Np,kdepending only on the parameters p and k such that

w_p(x)¯

¯M_n,k( f ; x) − f (x)¯

¯≤ Np,kω²p

µ f , x

pn

¶ + ω¹p

µ f ,1 − k

n x

¶ .

Furthermore, if f ∈ Li p²pα for some α ∈ (0,2], then

w_p(x)¯

¯M_n,k( f ; x) − f (x)¯

¯≤ Np,k

µx² n

¶α/2

+ ω¹p

µ f ,1 − k

n x

¶ ,

holds.

Proof. Let p ∈ N0, f ∈ Cp[0, ∞) and x ∈ (0,∞) be fixed. We consider the Steklov means of f by fhand given by the formula

f_h(x) = 4 h²

Z h/2 0

Zh/2

0 {2 f (x + s + t) − f (x + 2(s + t))}d sd t, for h, x ∈ (0,∞). We have

f (x) − fh(x) = 4 h²

Z h/2 0

Z h/2

0 ∆²_s+tf (x)d sd t , which gives

k f − fhkp≤ ω²p( f , h). (8)

Furthermore, we have

f_h⁰⁰(x) = 1

h²¡8∆²_h/2f (x) − ∆²_hf (x)¢ , and

k f_h⁰⁰kp≤ 9

h²ω²p( f , h). (9)

From (8) and (9) we conclude that f_h∈ C²p[0, ∞) if f ∈ Cp[0, ∞).

Moreover

|Mn,k( f ; x) − f (x)| ≤ Tn,k(| f (t) − fh(t ); x|) + | f (x) − fh(x)| + |Tn,k( f_h; x) − fh(x)| +

¯

¯ f

µ

x +1 − k n x

¶

− f (x)

¯

¯ ,

where Tn,kis defined in (7).

Since f_h∈ Cp²[0, ∞) by the above, it follows fromTheorem 2.1andLemma 3.1that

w_p(x)¯

¯M_n,k( f ; x) − f (x)¯

¯≤ (N + 1)k f − fhkp+ Np,kk f_h⁰⁰kp

x² n w_p(x)

¯

¯ f

µ

x +1 − k n x

¶

− f (x)

¯

¯ .

By (8) and (9), the last inequality yields that

w_p(x)¯

¯M_n,k( f ; x) − f (x)¯

¯≤ Np,kω²p( f ; h) µ

1 + 1 h²

x² n

¶ + ω¹p

µ f ,1 − k

n x

¶ .

Thus, choosing h = x

pn, the first part of the proof is completed.

The proof of second part can be easily obtained from the definition of the space Li p²_pα.

4. Acknowledgment

The author(s) are very thankful to Head of Department Computer Science, Dev Sanskriti Vishwavidyalaya, Haridwar, Uttarakhand, India for providing necessary facilities and information. Author(s) would also wish to express his gratitude to his parents for their moral support.

(7)

References

[1] A. Lupas, M. M ¨uller, Approximationseigenschaften der GammaoperatÃ ˝uren, Mathematische Zeitschrift 98 (1967) 208–226.

[2] S. M. Mazhar, Approximation by positive operators on infinite intervals, Math. Balkanica 5 (2) (1991) 99–104.

[3] A. ˙I zgi, I. B ¨uy ¨ukyazici, Approximation and rate of approximation on unbounded intervals, Kastamonu Edu. J.

Okt. 11 (2003) 451–460(in Turkish).

[4] H. Karsli, Rate of convergence of a new Gamma type operators for the functions with derivatives of bounded variation, Math. Comput. Modell. 45 (5-6) (2007) 617–624.

[5] H. Karsli, M. A. ¨Ozarslan, Direct local and global approximation results for operators of gamma type, Hacet. J.

Math. Stat. 39 (2010) 241–253.

[6] M. Becker, Global approximation theorems for Sz ´asz-Mirakyan and Baskakov operators in polynomial weight spaces, Indiana University Mathematics Journal 27 (1) (1978) 127–142.

[7] M. Felten, Local and global approximation theorems for positive linear operators, J. Approx. Theory 94 (1998) 396–419.

[8] Z. Finta, Direct local and global approximation theorems for some linear positive operators, Analysis in Theory and Applications 20 (4) (2004) 307–322.

[9] L. C. Mao, Rate of convergence of Gamma type operator, J. Shangqiu Teachers Coll. 12 (2007) 49–52.

[10] A. Kumar, Voronovskaja type asymptotic approximation by general Gamma type operators, Int. J. of Mathematics and its Applications 3 (4-B) (2015) 71–78.

[11] H. Karsli, On convergence of general Gamma type operators, Anal. Theory Appl. 27 (3) (2011) 288–300.

[12] H. Karsli, P. N. Agrawal, M. Goyal, General Gamma type operators based on q-integers, Appl. Math. Comput. 251 (2015) 564–575.

[13] A. ˙I zgi, Voronovskaya type asymptotic approximation by modified gamma operators, Appl. Math. Comput. 217 (2011) 8061–8067.

[14] H. Karsli, V. Gupta, A. Izgi, Rate of pointwise convergence of a new kind of gamma operators for functions of bounded variation, Appl. Math. Letters 22 (2009) 505–510.

[15] R. A. DeVore, G. G. Lorentz, Constructive Approximation. Springer, Berlin 1993.

Submit your manuscript to IJAAMM and benefit from:

Ï Regorous peer review

Ï Immediate publication on acceptance Ï Open access: Articles freely available online Ï High visibility within the field

Ï Retaining the copyright to your article

—————————————————————————————

Submit your next manuscript at Ï [email protected]