Global approximation theorems for general gamma type operators

GLOBAL APPROXIMATION THEOREMS FOR GENERAL GAMMA TYPE OPERATORS

arXiv:1508.06883v1 [math.GM] 21 Aug 2015

Alok Kumar Department of Computer Science Dev Sanskriti Vishwavidyalaya Haridwar Haridwar-249411, India [email protected] Dedicated to Prof. P. N. Agrawal Abstract. In this paper, we obtained some global approximation results for general Gamma type operators. Keywords: Gamma type operators, Global approximation, Positive linear operators. Mathematics Subject Classification(2010): 41A25, 26A15, 40A35.

1. Introduction For a measurable complex valued and locally bounded function defined on [0, ∞), Lupas and M¨ uller [12] defined and studied some approximation properties of linear positive operators {Gn } defined by   Z ∞ n du, Gn (f ; x) = gn (x, u)f u 0 where

xn+1 −xu n e u , x > 0. n! In [13], Mazhar gives an important modifications of the Gamma operators using the same gn (x, u) Z ∞Z ∞ Fn (f ; x) = gn (x, u)gn−1(u, t)f (t)dudt 0 0 Z (2n)!xn+1 ∞ tn−1 = f (t)dt, n > 1, x > 0. n!(n − 1)! 0 (x + t)2n+1 gn (x, u) =

Recently, Karsli [7] considered the following Gamma type linear and positive operators Z ∞Z ∞ Ln (f ; x) = gn+2 (x, u)gn (u, t)f (t)dudt 0 0 Z tn (2n + 3)!xn+3 ∞ f (t)dt, x > 0, = n!(n + 2)! (x + t)2n+4 0 1

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and obtained some approximation results. ¨ In [11], Karsli and Ozarslan obtained some local and global approximation results for the operators Ln (f ; x). In 2007, Mao [14] define the following generalised Gamma type linear and positive operators Z ∞Z ∞ Mn,k (f ; x) = gn (x, u)gn−k (u, t)f (t)dudt 0 0 Z tn−k (2n − k + 1)!xn+1 ∞ f (t)dt, x > 0, = n!(n − k)! (x + t)2n−k+2 0 which includes the operators Fn (f ; x) for k = 1 and Ln−2 (f ; x) for k = 2. Some approximation properties of Mn,k were studied in [8] and [9]. Several authors obtain the global approximation results for different operators (see [1], [3] and [4]). We can rewrite the operators Mn,k (f ; x) as Z ∞ Mn,k (f ; x) = Kn,k (x, t)f (t)dt, (1.1) 0

where Kn,k (x, t) =

(2n − k + 1)!xn+1 tn−k , x, t ∈ (0, ∞). n!(n − k)! (x + t)2n−k+2

In this paper, we study some global approximation results of the operators Mn,k . Let p ∈ N0 (set of non-negative integers), f ∈ Cp , where Cp is a polynomial weighted space with the weight function wp , w0 (x) = 1, wp (x) =

1 , p ≥ 1, 1 + xp

(1.2)

and Cp is the set of all real valued functions f for which wp f is bounded and uniformly continuous on [0, ∞). The norm on Cp is defined by ||f ||p = sup wp (x)|f (x)|, f ∈ Cp [0, ∞). x∈[0,∞)

We also consider the following Lipschitz classes: ωp2(f ; δ) = sup ||∆2h f ||p , h∈(0,δ]

∆2h f (x) = f (x + 2h) − 2f (x + h) + f (x), ωp1 (f ; δ) = sup{wp (x)|f (t) − f (x)| : |t − x| ≤ δ and t, x ≥ 0}, Lip2p α = {f ∈ Cp [0, ∞) : ωp2 (f ; δ) = O(δ α) as δ → 0+ }, where h > 0 and α ∈ (0, 2].

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2. Auxiliary Results In this section we give some preliminary results which will be used in the proofs of our main theorems. Let us consider em (t) = tm , ϕx,m (t) = (t − x)m , m ∈ N0 . Lemma 1. [8] For any m ∈ N0 (set of non-negative integers), m ≤ n − k Mn,k (tm ; x) =

[n − k + m]m m x [n]m

(2.1)

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 (2.1) we get (i) Mn,k (1; x) = 1, n−k+1 (ii) Mn,k (t; x) = x, n (n − k + 2)(n − k + 1) 2 (iii) Mn,k (t2 ; x) = x. n(n − 1)

Lemma 2. [8] Let m ∈ N0 and fixed x ∈ (0, ∞), then !   m X (n − m + j)!(n − k + m − j)! m xm . Mn,k (ϕx,m ; x) = (−1)j n!(n − k)! j j=0

Lemma 3. For m = 0, 1, 2, 3, 4, one has (i) Mn,k (ϕx,0; x) = 1, 1−k x, (ii) Mn,k (ϕx,1; x) = n k 2 − 5k + 2n + 4 2 x, (iii) Mn,k (ϕx,2; x) = n(n − 1) −k 3 + 12k 2 − 17k + n(18 − 12k) + 24 3 (iv) Mn,k (ϕx,3; x) = x, n(n − 1)(n − 2) k 4 − 22k 3 + k 2 (143 + 12n) − k(314 + 108n) + 12n2 + 268n + 192 4 x, (v) Mn,k (ϕx,4; x) = n(n − 1)(n − 2)(n − 3)
(vi) Mn,k (ϕx,m; x) = O n−[(m+1)/2] . Proof. Using Lemma 2, we get Lemma 3.



Theorem 1. For the operators Mn,k and for fixed p ∈ N0 , there exists a positive constant Np,k such that   1 wp (x)Mn,k ; x ≤ Np,k . (2.2) wp

Furthermore, for all f ∈ Cp [0, ∞), we have

kMn,k (f ; .)kp ≤ Np,k kf kp ,

which guarantees that Mn,k maps Cp [0, ∞) into Cp [0, ∞).

(2.3)

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Proof. For p = 0, (2.2) follows immediately. Using Lemma 1, we get   1 wp (x)Mn,k ;x = wp (x) (Mn,k (e0 ; x) + Mn,k (ep ; x)) wp   (n − p)!(n − k + p)! p x = wp (x) 1 + n!(n − k)! ≤ Np,k wp (x)(1 + xp ) = Np,k , where Np,k



 (n − p)!(n − k + p)! = max sup ,1 . n!(n − k)! n

Observe that for all f ∈ Cp and every x ∈ (0, ∞), we get Z wp(t) tn−k (2n − k + 1)!xn+1 ∞ |f (t)| dt wp (x) |Mn,k (f ; x)| ≤ wp (x) 2n−k+2 n!(n − k)! (x + t) wp(t) 0   1 ≤ kf kp wp (x)Mn,k ;x wp ≤ Np,k kf kp . Taking supremum over x ∈ (0, ∞), we get (2.3).



Lemma 4. For the operators Mn,k and fixed p ∈ N0 , there exists a positive constant Np,k such that   x2 ϕx,2 ; x ≤ Np,k . wp (x)Mn,k wp (t) n Proof. Using Lemma (3), we can write   ϕx,2 k 2 − 5k + 2n + 4 2 w0 (x)Mn,k ;x = x w0 (t) n(n − 1) x2 ≤ Np,k , n which gives the result for p = 0. Let p ≥ 1. Then using Lemma 1 and Lemma 3, we get   ϕx,2 Mn,k ;x = Mn,k (ep+2 ; x) − 2xMn,k (ep+1 ; x) + x2 Mn,k (ep ; x) + Mn,k (ϕx,2 ; x) wp (t) (n − p − 2)!(n − k + p + 2)! p+2 (n − p − 1)!(n − k + p + 1)! p+2 = x −2 x n!(n − k)! n!(n − k)! (n − p)!(n − k + p)! p+2 k 2 − 5k + 2n + 4 2 x + x + n!(n − k)! n(n − 1) x2 ≤ Np,k (1 + xp ), n where Np,k is a positive constant. Hence, the proof is completed. 

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3. Rate of Convergence Let p ∈ N0 . By Cp2 [0, ∞), we denote the space of all functions f ∈ Cp [0, ∞) such that f , f ′′ ∈ Cp [0, ∞). ′

Theorem 2. Let p ∈ N0 , n ∈ N and g ∈ Cp1 [0, ∞), there exists a positive constant Np,k such that x wp (x)|Mn,k (f ; x) − f (x)| ≤ Np,k kf ′kp √ n for all x ∈ (0, ∞). Proof. We have f (t) − f (x) = By using linearity of Mn,k we get

Z

t

Z

x

t

 f (v)dv; x . ′

(3.1)

  1 1 dv ′ . + ≤ kf kp |t − x| wp (v) wp (t) wp (x)

x

x

f ′ (v)dv.

x

Mn,k (f ; x) − f (x) = Mn,k Remark that Z Z t f ′ (v)dv ≤ kf ′ kp

t

From (3.1) we obtain

′



wp (x)|Mn,k (f ; x) − f (x)| ≤ kf kp Mn,k (|ϕx,1|; x) + wp (x)Mn,k Using Cauchy-Schwarz inequality, we can write



 |ϕx,1| ;x . wp (t)

Mn,k (|ϕx,1|; x) ≤ (Mn,k (|ϕx,2|; x))1/2 ,     1/2   1/2 |ϕx,1| 1 ϕx,2 Mn,k ; x ≤ Mn,k ;x Mn,k ;x . wp (t) wp (t) wp (t) Using Leema 3, Theorem 1 and Lemma 4, we obtain x wp (x)|Mn,k (f ; x) − f (x)| ≤ Np,k kf ′ kp √ . n  Lemma 5. Let p ∈ N0 , If

  1−k x + f (x), Hn,k (f ; x) = Mn,k (f ; x) − f x + n

(3.2)

then there exists a positive constant Np,k such that for all x ∈ (0, ∞) and n ∈ N, we have wp (x)|Hn,k (g; x) − g(x)| ≤ Np,k ||g ′′||p for any function g ∈ Cp2 .

x2 n

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Proof. From Lemma 1, we observe that the operators Hn,k are linear and reproduce the linear functions. Hence Hn,k (ϕx,1; x) = 0. Let g ∈ Cp2 . By the Taylor formula one can write Z t ′ g(t) − g(x) = (t − x)g (x) + (t − v)g ′′ (v)dv, t ∈ (0, ∞). x

Then, |Hn,k (g; x) − g(x)|

Since

and

we get

Z t  ′′ (t − v)g (v)dv; x = |Hn,k (g(t) − g(x)); x| = Hn,k x Z t  Z x+ 1−k x   n 1−k ′′ ′′ (t − v)g (v)dv; x − = Mn,k x+ x − v g (v)dv . n x x Z t   k g ′′ kp (t − x)2 1 1 ′′ (t − v)g (v)dv ≤ + 2 wp (x) wp (t) x

Z   2 x+ 1−k x n ||g ′′||p 1 − k 1−k ′′ x − v g (v)dv ≤ x , x+ 2wp (x) x n n

    2 kg ′′kp ϕx,2 kg ′′kp 1 − k wp (x)|Hn,k (g; x) − g(x)| ≤ Mn,k (ϕx,2 ; x) + wp (x)Mn,k ;x + x . 2 wp (t) 2 n

Hence by Lemma 4, we obtain

wp (x)|Hn,k (g; x) − g(x)| ≤ Np,k kg ′′ kp

x2 n

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



The next theorem is the main result of this section. Theorem 3. Let p ∈ N0 , n ∈ N and f ∈ Cp [0, ∞), then there exists a positive constant Np,k such that     x 1−k 2 1 x . wp (x) |Mn,k (f ; x) − f (x)| ≤ Np,k ωp f, √ + ωp f, n n

Furthermore, if f ∈ Lip2p α for some α ∈ (0, 2], then  2 α/2   x 1−k 1 wp (x) |Mn,k (f ; x) − f (x)| ≤ Np,k + ωp f, x , n n holds.

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Proof. Let p ∈ N0 , f ∈ Cp [0, ∞) and x ∈ (0, ∞) be fixed. We consider the Steklov means of f by fh and given by the formula Z Z 4 h/2 h/2 fh (x) = 2 {2f (x + s + t) − f (x + 2(s + t))}dsdt, h 0 0 for h > 0 and x ≥ 0. We have 4 f (x) − fh (x) = 2 h

Z

0

h/2

Z

0

h/2

∆2s+t f (x)dsdt,

which gives kf − fh kp ≤ ωp2 (f, h).

(3.3)

Furthermore, we have ′′

fh (x) = and


1 8∆2h/2 f (x) − ∆2h f (x) , 2 h ′′

kfh kp ≤

9 2 ω (f, h). h2 p

(3.4)

From (3.3) and (3.4) we conclude that fh ∈ Cp2 [0, ∞) if f ∈ Cp [0, ∞). Moreover |Mn,k (f ; x) − f (x)| ≤ Hn,k (|f (t) − fh (t); x|) + |f (x) − fh (x)|   1−k + |Hn,k (fh ; x) − fh (x)| + f x + x − f (x) , n

where Hn,k is defined in (3.2). Since fh ∈ Cp2 [0, ∞) by the above, it follows from Theorem 1 and Lemma 5 that x2 ′′ wp (x) |Mn,k (f ; x) − f (x)| ≤ (N + 1)kf − fh kp + Np,k kfh kp  n  1−k + wp (x) f x + x − f (x) . n

By (3.3) and (3.4), the last inequality yields that     1−k 1 x2 1 2 + ωp f, wp (x) |Mn,k (f ; x) − f (x)| ≤ Np,k ωp (f ; h) 1 + 2 x . h n n

Thus, choosing h = √xn , the first part of the proof is completed. The remainder of the proof can be easily obtained from the definition of the space Lip2p α.  Acknowledgements The author is extremely grateful to the referee for making valuable suggestions leading to the overall improvements in the paper.

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