JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2019/9607517 9607517 Research Article On ( p , q ) -Analogue of Gamma Operators http://orcid.org/0000-0001-9199-0502 Cheng Wen-Tao 1 Zhang Wen-Hui 2 Hudzik Henryk 1 School of Mathematical Sciences Anqing Normal University Anhui 246133 China aqtc.edu.cn 2 Qingdao Hengxing University of Science and Technology Shandong 266041 China 2019 1222019 2019 12 11 2018 15 01 2019 21 01 2019 1222019 2019 Copyright © 2019 Wen-Tao Cheng and Wen-Hui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, a kind of new analogue of Gamma type operators based on ( p , q ) -integers is introduced. The Voronovskaja type asymptotic formula of these operators is investigated. And some other results of these operators are studied by means of modulus of continuity and Peetre K - functional. Finally, some direct theorems concerned with the rate of convergence and the weighted approximation for these operators are also obtained.

National Natural Science Foundation of China 11626031
1. Introduction

In recent years, with the rapid development of q -calculus, the study of approximation theory with q -integer has been discussed widely. Afterwards, with the generalization from q -calculus to ( p , q ) -calculus, it has been used efficiently in many areas of sciences such as algebras [1, 2] and CAGD . And, recently, approximation by sequences of linear positive operators has been transferred to operators with ( p , q ) -integer. Some useful notations and definitions about q -calculus and ( p , q ) -calculus in this paper are reviewed in .

Let 0 < q < p 1 . For each nonnegative integer n , the ( p , q ) -integer [ n ] p , q and ( p , q ) -factorial [ n ] p , q ! are defined as (1) n p , q = p n - q n p - q n = 1,2 , and (2) n p , q ! = 1 p , q 2 p , q n p , q , n 1 1 , n = 0

Further, the ( p , q ) - power basis is defined as (3) x y p , q n = x + y p x + q y p 2 x + q 2 y p n - 1 x + q n - 1 y . And (4) x y p , q n = x - y p x - q y p 2 x - q 2 y p n - 1 x - q n - 1 y .

Let n be a nonnegative integer; the ( p , q ) -Gamma function is defined as (5) Γ p , q n + 1 = p q p , q n p - q n = n p , q ! , 0 < q < p 1 .

Aral and Gupta  proposed ( p , q ) -Beta function of second kind for m , n N as (6) B p , q m , n = 0 x m - 1 1 p x p , q m + n d p , q x . And the relationship between ( p , q ) -analogues of Beta and Gamma functions is as follows: (7) B p , q m , n = q Γ p , q m Γ p , q n p m + 1 q m - 1 m / 2 Γ p , q m + n . Particularly, when p = q = 1 , B ( m , n ) = Γ ( m ) Γ ( n ) / Γ ( m + n ) . It may be observed that, in ( p , q ) -setting, order is important, which is the reason why ( p , q ) -variant of Beta function does not satisfy commutativity property; that is, B p , q ( m , n ) B p , q ( n , m ) .

In , Mazhar studied some approximation properties of the Gamma operators as follows: (8) F - n f ; x = 2 n ! x n + 1 n ! n - 1 ! 0 t n - 1 x + t 2 n + 1 f t d t , n > 1 , x > 0 .

Recently, Mursaleen first applied ( p , q ) -calculus in approximation theory and introduced the ( p , q ) -analogue of Bernstein operators , ( p , q ) -Bernstein-Stancu operators , and ( p , q ) -Bernstein-Schurer operators  and investigated their approximation properties. And many well-known approximation operators with ( p , q ) -integer have been introduced, such as ( p , q ) -Bernstein-Stancu-Schurer-Kantorovich operators , ( p , q ) -Szász-Baskakov operators , and ( p , q ) -Baskakov-Beta operators . As we know, many researchers have studied approximation properties of the Gamma operators and their modifications (see , etc.). All this achievement motivates us to construct the ( p , q ) -analogue of the Gamma operators (8). First, we introduce ( p , q ) -analogue of Gamma operators as follows.

Definition 1.

For n N , n > 1 , x ( 0 , ) , and 0 < q < p 1 , the ( p , q ) -Gamma operators can be defined as (9) F n p , q f ; x = x n + 1 p n 2 q n 2 + n B p , q n , n + 1 0 t n - 1 p n q n x t p , q 2 n + 1 f t d p , q t

The paper is organized as follows. In the first section, we give the basic notations and the definition of ( p , q ) -Gamma operators. In the second section, we present the moments of the operators. In the third section, we obtain Voronovskaja type asymptotic formula. In the fourth section, we present a direct result of ( p , q ) -Gamma operators in terms of first- and second-order modulus of continuity. In the last section, we study the rate of convergence and the weighted approximation of the ( p , q ) -Gamma operators.

2. Auxiliary Results

In order to obtain the approximation properties of the operators F n p , q ( f ; x ) , we need the following lemma and remarks.

Lemma 2.

The following equalities hold:

F n p , q ( 1 ; x ) = 1 .

F n p , q ( t ; x ) = x , for n > 1 .

F n p , q ( t 2 ; x ) = [ n + 1 ] p , q x 2 / p q [ n - 1 ] p , q , for n > 2 .

F n p , q ( t 3 ; x ) = [ n + 1 ] p , q [ n + 2 ] p , q x 3 / ( p q ) 3 [ n - 1 ] p , q [ n - 2 ] p , q , for n > 3 .

F n p , q ( t 4 ; x ) = [ n + 1 ] p , q [ n + 2 ] p , q [ n + 3 ] p , q x 4 / ( p q ) 6 [ n - 1 ] p , q [ n - 2 ] p , q [ n - 3 ] p , q , for n > 4 .

Proof.

According to the properties of ( p , q ) -Beta function and ( p , q ) -Gamma function, we have (10) F n t k ; x = x n + 1 p n 2 q n 2 + n B p , q n , n + 1 0 t n + k - 1 p n q n x t p , q 2 n + 1 d p , q t = x n + 1 p n 2 q n 2 + n B p , q n , n + 1 0 1 x 2 n + 1 p n 2 n + 1 q n 2 n + 1 t n + k - 1 1 p t / x p n + 1 q n p , q 2 n + 1 d p , q t = x n + 1 p n 2 q n 2 + n B p , q n , n + 1 0 x n + k p n + 1 n + k q n n + k x 2 n + 1 p n 2 n + 1 q n 2 n + 1 t / x p n + 1 q n n + k - 1 1 p t / x p n + 1 q n p , q 2 n + 1 d p , q t x p n + 1 q n = x k p n k + k q n k B p , q n + k , n - k + 1 B p , q n , n + 1 = x k p n k + k q n k q n - 1 p n + 1 n / 2 q n + k - 1 p n + k + 1 n + k / 2 Γ p , q n + k Γ p , q n - k + 1 Γ p , q n Γ p , q n + 1 = x k p q - k k - 1 / 2 n + k - 1 p , q ! n - k p , q ! n - 1 p , q ! n p , q ! This proves Lemma 2.

Remark 3.

Let n > 2 , and x ( 0 , ) ; then, for 0 < q < p 1 , we have the central moments as follows:

F n p , q ( t - x ; x ) = 0 .

A ( x ) : = F n p , q ( ( t - x ) 2 ; x ) = q / p - 1 + p n - 2 [ 2 ] p , q / q [ n - 1 ] p , q x 2 .

Remark 4.

The sequences ( p n ) and ( q n ) satisfy 0 < q n < p n < 1 such that p n 1 , q n 1 , and p n n a , q n n b , and [ n ] p n , q n as n , where 0 a , b < 1 , and x ( 0 , ) ; then

lim n [ n - 1 ] p n , q n F n p n , q n ( ( t - x ) 2 ; x ) = ( a + b ) x 2 .

lim n [ n - 3 ] p n , q n F n p n , q n ( ( t - x ) 4 ; x ) = 0 .

Proof.

(1) Using Remark 3, (11) l i m n n - 1 p n , q n F n p n , q n t - x 2 ; x = l i m n p n n - 2 2 p n , q n q n - p n - q n n - 1 p n , q n p n x 2 = l i m n p n n - 2 2 p n , q n q n - p n n - 1 - q n n - 1 p n x 2 = 2 a - a - b x 2 = a + b x 2

(2) Let k = n - 3 ; we have (12) n + 1 p n , q n n + 2 p n , q n n + 3 p n , q n = q n 4 k p n , q n + p n k 4 p n , q n q n 5 k p n , q n + p n k 5 p n , q n q n 6 k p n , q n + p n k 6 p n , q n ~ q n 15 k p n , q n 3 + p n k q n 9 6 p n , q n + q n 10 5 p n , q n + q n 11 4 p n , q n k p n , q n 2 . Similarly, we have (13) n + 1 p n , q n n + 2 p n , q n n - 3 p n , q n ~ q n 9 k p n , q n 3 + p n k q n 5 4 p n , q n + q n 4 5 p n , q n k p n , q n 2 . (14) n + 1 p n , q n n - 2 p n , q n n - 3 p n , q n ~ q n 5 k p n , q n 3 + p n k q n 4 p n , q n + q n 4 k p n , q n 2 (15) n - 1 p n , q n n - 2 p n , q n n - 3 p n , q n ~ q n 3 k p n , q n 3 + p n k q n 2 p n , q n + q n 2 k p n , q n 2 Using Lemma 2, we obtain (16) F n p n , q n t - x 4 ; x ~ A n + 1 k p n , q n B n x 4 where A n = q n 15 - 4 p n 3 q n 12 + 6 p n 5 q n 10 - 3 p n 6 q n q and (17) B n = p n k q n 9 6 p n , q n + q n 10 5 p n , q n + q n 11 4 p n , q n - 4 p n 3 q n 3 q n 5 4 p n , q n + q n 4 5 p n , q n + 6 p n 5 q n 5 q n 4 p n , q n + q n 4 - 3 p n 6 q n 6 q n 2 p n , q n + q n 2 . Combining with (18) k p n , q n A n ~ k p n , q n q n 6 - 4 p n 3 q n 3 + 6 p n 5 q n - 3 p n 6 = k p n , q n 4 p n 3 p n 3 - q n 3 - 6 p n 5 p n - q n - p n 6 - q n 6 = k p n , q n 4 p n 3 3 p n , q n p n - q n - 6 p n 5 p n - q n - 6 p n , q n p n - q n = k p n , q n p n n - q n n n p n , q n 4 p n 3 3 p n , q n - 6 p n 5 - 6 p n , q n ~ p n n - q n n 4 p n 3 3 p n , q n - 6 p n 5 - 6 p n , q n ~ a - b 12 - 6 - 6 = 0 and (19) B n ~ 4 + 5 + 6 - 4 × 4 + 5 + 6 × 4 + 1 - 3 × 2 + 1 = 0 , we can obtain lim n [ n - 3 ] p n , q n F n p n , q n ( ( t - x ) 4 ; x ) = 0 .

3. Voronovskaja Type Theorem

We give a Voronovskaja type asymptotic formula for F n p , q ( f ; x ) by means of the second and fourth central moments.

Theorem 5.

Let f be bounded and integrable on the interval x ( 0 , ) ; second derivative of f exists at a fixed point x ( 0 , ) ; the sequences ( p n ) and ( q n ) satisfy 0 < q n < p n < 1 such that p n 1 , q n 1 , and p n n a , q n n b , and [ n ] p n , q n as n , where 0 a , b < 1 ; then (20) l i m n n - 3 p n , q n F n p n , q n f ; x - f x = a + b 2 x 2 f x .

Proof.

Let x ( 0 , ) be fixed. In order to prove this identity, we use Taylor’s expansion: (21) f t - f x = t - x f x + t - x 2 f x 2 + Φ p n , q n t , x , where Φ p n , q n ( x , t ) is bounded and lim t x Φ p n , q n ( t , x ) = 0 . By applying the operator F n p n , q n ( f ; x ) to the equality above, we obtain (22) F n p n , q n f ; x - f x = f x F n p n , q n t - x ; x + 1 2 f x F n p n , q n t - x 2 ; x + F n p n , q n Φ p n , q n t , x t - x 2 ; x = 1 2 f x F n p n , q n t - x 2 ; x + F n p n , q n Φ p n , q n t , x t - x 2 ; x . Since lim t x Φ p n , q n ( t , x ) = 0 , for all ϵ > 0 , there exists δ > 0 such that | t - x | < δ and it will imply | Φ p n , q n ( t , x ) | < ϵ for all fixed x ( 0 , ) as n is sufficiently large. Meanwhile, if | t - x | δ , then | Φ p n , q n ( t , x ) | C / δ 2 ( t - x ) 2 , where C > 0 is a constant. Using Remark 4, we have (23) l i m n n - 3 F n p n , q n t - x 2 ; x = l i m n n - 1 F n p n , q n t - x 2 ; x = a + b x 2 and (24) n - 3 p n , q n F n p n , q n Φ p n , q n t , x t - x 2 ; x ϵ n - 3 p n , q n F n p n , q n t - x 2 ; x + C δ 2 n - 3 p n , q n F n p n , q n t - x 4 ; x 0 n . The proof is completed.

4. Local Approximation

We denote the space of all real valued continuous bounded functions f defined on the interval [ 0 , + ) by C B [ 0 , + ) . The norm · on the space C B [ 0 , + ) is given by (25) f = s u p f x : x 0 , + .

Let us consider the following K -functional: (26) K f , δ = i n f g C B 2 0 , f - g + δ g , where δ > 0 and C B 2 0 , = g C B 0 , : g , g C B 0 , . By  (p. 177, Theorem 2.4), there exists an absolute constant C > 0 such that (27) K f , δ C ω 2 f , δ where (28) ω 2 f , δ = s u p 0 < h δ s u p x 0 , f x + 2 h - 2 f x + h + f x is the second-order modulus of smoothness of f . By (29) ω f , δ = s u p 0 < h δ s u p x 0 , f x + h - f x , we denote the usual modulus of continuity of f C B [ 0 , ) .

Our first result is a direct local approximation theorem for the operators F n p , q ( f ; x ) .

Theorem 6.

Let f C B [ 0 , + ) ; 0 < q < p 1 ; then, for every x ( 0 , ) and n > 2 , we have (30) F n p , q f ; x - f x C ω 2 f , A x , where C is some positive constant.

Proof.

For all g C B 2 [ 0 , ) , using Taylor’s expansion for x ( 0 , ) , we have (31) g t = g x + g x t - x + x t t - u g u d u . Applying the operators F n p , q to both sides of the equality above and using Remark 3, we get (32) F n p , q g ; x - g x = F n p , q x t t - u g u d u ; x F n p , q x t t - u g u d u ; x F n p , q g t - x 2 ; x A x g . Using F n p , q f ; x f , we have (33) F n p , q f ; x - f x F n p , q f - g ; x - f - g x + F n p , q g ; x - g x 2 f - g + A x g Lastly, taking infimum on both sides of the inequality above over all g C B 2 [ 0 , ) , (34) F n p , q f ; x - f x 2 K f ; A x for which we have the desired result by (27).

Theorem 7.

Let 0 < γ 1 and let E be any bounded subset of the interval [ 0 , ) . If f C B [ 0 , ) is locally in L i p ( γ ) , i . e . , the condition (35) f x - f t L x - t γ , t E a n d x 0 , holds, then, for each x ( 0 , ) , we have (36) F n p , q f ; x - f x L A x γ / 2 + 2 d x ; E γ , where L is a constant depending on γ and f ; and d ( x ; E ) is the distance between x and E defined by (37) d x ; E = i n f t - x : t E .

Proof.

From the properties of infimum, there is at least one point t 0 in the closure of E ; that is, t 0 E ¯ , such that (38) d x ; E = t 0 - x . Using the triangle inequality, we have (39) F n p , q f ; x - f x F n p , q f t - f x ; x F n p , q f t - f t 0 ; x + F n p , q f t 0 - f x ; x L F n p , q t - t 0 γ ; x + F n p , q t 0 - x γ ; x L F n p , q t - x γ ; x + 2 t 0 - x γ Choosing a 1 = 2 / γ and a 2 = 2 / 2 - γ and using the well-known Hölder inequality, (40) F n p , q f ; x - f x L F n p , q t - x γ a 1 ; x 1 / a 1 F n p , q 1 a 2 ; x 1 / a 2 + 2 t 0 - x γ L F n p , q t - x 2 ; x γ / 2 + 2 t 0 - x γ L A γ / 2 x + 2 d x ; E γ This completes the proof.

5. Rate of Convergence and Weighted Approximation

Let C ρ [ 0 , ) be the set of all functions defined on [ 0 , ) satisfying the condition | f ( x ) | C f ρ ( x ) , where C f > 0 is a constant depending only on f and ρ ( x ) is a weight function. Let C ρ [ 0 , ) be the space of all continuous functions in C ρ [ 0 , ) with the norm f ρ = sup x [ 0 , ) | f ( x ) | / ρ ( x ) and C ρ 0 [ 0 , ) = f C ρ [ 0 , ) : lim x | f ( x ) | / ρ ( x ) < . We consider ρ ( x ) = 1 + x 2 in the following two theorems. Meanwhile, we denote the modulus of continuity of f on the closed interval [ 0 , a ] , a > 0 , by (41) ω a f , δ = s u p t - x δ s u p x , t 0 , a f t - f x . Obviously, for the function f C ρ [ 0 , ) , the modulus of continuity ω a ( f , δ ) tends to zero. Then we establish the following theorem on the rate of convergence for the operators F n p , q ( f ; x ) .

Theorem 8.

Let f C ρ [ 0 , ) , 0 < q < p 1 , and ω a + 1 ( f , δ ) be its modulus of continuity on the finite interval [ 0 , a + 1 ] [ 0 , ) , where a > 0 . Then, for every n > 2 , x > 0 , (42) F n p , q f ; x - f x C 0 , a 4 C f 1 + a 2 A x + 2 ω a + 1 f , A x

Proof.

For all x [ 0 , a ] and t > a + 1 , we easily have ( t - x ) 2 ( t - a ) 2 1 ; therefore, (43) f t - f x f t + f x C f 2 + x 2 + t 2 = C f 2 + x 2 + x - t - x 2 C f 2 + 3 x 2 + 2 x - t 2 C f 4 + 3 x 2 t - x 2 4 C f 1 + a 2 t - x 2 . And, for all x [ 0 , a ] , t [ 0 , a + 1 ] , and δ > 0 , we have (44) f t - f x ω a + 1 f , t - x 1 + t - x δ ω a + 1 f , δ From (43) and (44), we get (45) f t - f x 4 C f 1 + a 2 t - x 2 + 1 + t - x δ ω a + 1 f , δ . By Schwarz’s inequality and Remark 3, we have (46) F n p , q f ; x - f x F n p , q f t - f x ; x 4 C f 1 + a 2 F n p , q t - x 2 ; x + F n p , q 1 + t - x δ ; x ω a + 1 f , δ 4 C f 1 + a 2 F n p , q t - x 2 ; x + ω a + 1 f , δ 1 + 1 δ F n p , q t - x 2 ; x 4 C f 1 + a 2 A x + ω a + 1 f , δ 1 + 1 δ A x By taking δ = A ( x ) , we get the proof of Theorem 8.

The following is a direct estimate in weighted approximation.

Theorem 9.

Let ( p n ) and ( q n ) satisfy 0 < q n < p n 1 such that p n 1 , q n 1 , p n n a , q n n b , and [ n ] p n , q n . Then, for f C ρ 0 [ 0 , ) , we have (47) l i m n F n p n , q n f ; x - f x ρ = 0 .

Proof.

Using the Korovkin theorem in , we see that it is sufficient to verify the following three conditions: (48) l i m n F n p n , q n t k ; x - x k ρ = 0 , k = 0,1 , 2 . Since F n p n , q n ( 1 ; x ) = 1 and F n p n , q n ( t ; x ) = x , (48) holds true for k = 0,1 . By Remark 3, for n > 2 , we have (49) l i m n F n p n , q n f ; x - f x ρ = s u p x 0 , q n p n - 1 + p n n - 2 2 p n , q n q n n - 1 p n , q n x 2 1 + x 2 q n p n - 1 + p n n - 2 2 p n , q n q n n - 1 p n , q n 0 n . Thus the proof is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant no. 11626031).

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