Skip to main content

Convergence of the q-Stancu-Szász-Beta type operators

Abstract

In this paper, we study on q-Stancu-Szász-Beta type operators. We give these operators convergence properties and obtain a weighted approximation theorem in the interval [0,).

MSC:41A25, 41A36.

1 Introduction

In [1], Mahmudov constructed q-Szász operators and obtained rate of global convergence in the frame of weighted spaces and a Voronovskaja type theorem for these operators. In [2], Gupta and Mahmudov studied on the q-analog of the Szász-Beta type operators. In [3], Yüksel and Dinlemez gave a Voronovskaja type theorem for q-analog of a certain family Szász-Beta type operators. In [4], Govil and Gupta introduced the q-analog of certain Beta-Szász-Stancu operators. They estimated the moments and established direct results in terms of modulus of continuity and an asymptotic formula for the q-operators. In [514], interesting generalization about q-calculus were given. Our aims are to give approximation properties and a weighted approximation theorem for q-Stancu-Szász-Beta type operators. We use without further explanation the basic notations and formulas, from the theory of q-calculus as set out in [1519]. Let A>0 and f be a real valued continuous function defined on the interval [0,). For 0<q1, q-Stancu-Szász-Beta type operators are defined as

B n , q ( α , β ) (f,x)= k = 0 s n , k q (x) 0 / A b n , k q (t)f ( [ n ] q t + α [ n ] q + β ) d q t,
(1.1)

where

s n , k q (x)= ( [ n ] q x ) k e [ n ] q x [ k ] q !

and

b n , k q (x)= q k 2 x k B q ( k + 1 , n ) ( 1 + x ) q n + k + 1 .

If we write q=1 and α=β=0 in (1.1), then the operators B n , q ( α , β ) (f,x) are reduced to Szász-Beta type operators studied in [2023].

2 Auxiliary results

For the sake of brevity, the notation F s q (n)= i = 1 s [ n i ] q and G β q (n)=( [ n ] q +β) will be used throughout the article. Now we are ready to give the following lemma for the Korovkin test functions.

Lemma 1 Let e m (t)= t m , m=0,1,2, we get

(i) B n , q ( α , β ) ( e 0 , x ) = 1 , (ii) B n , q ( α , β ) ( e 1 , x ) = [ n ] q 2 x q 2 G β q ( n ) F 1 q ( n ) + [ n ] q q G β q ( n ) F 1 q ( n ) + α G β q ( n ) , (iii) B n , q ( α , β ) ( e 2 , x ) = [ n ] q 4 x 2 q 6 G β q ( n ) 2 F 2 q ( n ) + { [ n ] q 3 q 5 G β q ( n ) 2 F 2 q ( n ) B n , q ( α , β ) ( e 2 , x ) = + ( 1 + [ 2 ] q ) [ n ] q 3 q 4 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q 2 q 2 G β q ( n ) 2 F 1 q ( n ) } x B n , q ( α , β ) ( e 2 , x ) = + [ 2 ] q [ n ] q 2 q 3 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q q G β q ( n ) 2 F 1 q ( n ) + α 2 G β q ( n ) 2 .

Proof Using the q-Gamma and q-Beta functions in [15, 24], we obtain the following equality:

q k 2 0 / A 1 B ( k + 1 , n ) t k + m ( 1 + t ) q n + k + 1 d q t = [ m + k ] q ! [ n m 1 ] q ! q { 2 k 2 ( k + m ) ( k + m + 1 ) } / 2 [ k ] q ! [ n 1 ] q ! .
(2.1)

Then, using (2.1), for m=0, we get

B n , q ( α , β ) ( e 0 , x ) = e [ n ] q x k = 0 ( [ n ] q x ) k [ k ] q ! q k ( k 1 ) / 2 = e [ n ] q x E q [ n ] q x = 1 ,

and the proof of (i) is finished. With a direct computation, we obtain (ii) as follows:

B n , q ( α , β ) ( e 1 , x ) = [ n ] q G β q ( n ) F 1 q ( n ) k = 1 ( [ n ] q x ) k [ k 1 ] q ! q k ( k 3 ) 2 / 2 e [ n ] q x + [ n ] q G β q ( n ) F 1 q ( n ) k = 0 ( [ n ] q x ) k [ k ] q ! q k ( k 1 ) 2 / 2 e [ n ] q x + α G β q ( n ) k = 0 ( [ n ] q x ) k [ k ] q ! q k ( k 1 ) / 2 e [ n ] q x = [ n ] q 2 x q 2 G β q ( n ) F 1 q ( n ) E q [ n ] q x e [ n ] q x + [ n ] q q G β q ( n ) F 1 q ( n ) E q [ n ] q x e [ n ] q x + α G β q ( n ) E q [ n ] q x e [ n ] q x = [ n ] q 2 x q 2 G β q ( n ) F 1 q ( n ) + [ n ] q q G β q ( n ) F 1 q ( n ) + α G β q ( n ) .

Using the equality

[ n ] q = [ s ] q + q s [ n s ] q ,0sn,
(2.2)

we get

B n , q ( α , β ) ( e 2 , x ) = [ n ] q 4 x 2 q 6 G β q ( n ) 2 F 2 q ( n ) + { [ n ] q 3 q 5 G β q ( n ) 2 F 2 q ( n ) + ( 1 + [ 2 ] q ) [ n ] q 3 q 4 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q 2 q 2 G β q ( n ) 2 F 1 q ( n ) } x + [ 2 ] q [ n ] q 2 q 3 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q q G β q ( n ) 2 F 1 q ( n ) + α 2 G β q ( n ) 2 ,

and so we have the proof of (iii). □

To obtain our main results we need to compute the second moment.

Lemma 2 Let q(0,1) and n>2. Then we have the following inequality:

B n , q ( α , β ) ( ( t x ) 2 , x ) ( 2 ( 1 q 4 ) q 6 + 164 ( α + β + 1 ) 2 [ n ] q q 6 F 2 q ( n ) ) x(x+1)+ 6 ( α + 1 ) 2 q 3 G β q ( n ) .

Proof From the linearity of the B n , q ( α , β ) operators and Lemma 1, we write the second moment as

B n , q ( α , β ) ( ( t x ) 2 , x ) = { [ n ] q 4 q 6 G β q ( n ) 2 F 2 q ( n ) 2 [ n ] q 2 q 2 G β q ( n ) F 1 q ( n ) + 1 } x 2 + { { 1 + ( 1 + [ 2 ] q ) q } [ n ] q 3 q 5 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q 2 q 2 G β q ( n ) 2 F 1 q ( n ) 2 [ n ] q q G β q ( n ) F 1 q ( n ) 2 α G β q ( n ) } x + [ 2 ] q [ n ] q 2 q 3 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q q G β q ( n ) 2 F 1 q ( n ) + α 2 G β q ( n ) 2 { [ n ] q 4 q 6 G β q ( n ) 2 F 2 q ( n ) 2 [ n ] q 2 q 2 G β q ( n ) F 1 q ( n ) + 1 + { 1 + ( 1 + [ 2 ] q ) q } [ n ] q 3 q 5 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q 2 q 2 G β q ( n ) 2 F 1 q ( n ) } x ( x + 1 ) + [ 2 ] q [ n ] q 2 q 3 G β q ( n ) 2 F 2 q ( n ) + 2 α [ n ] q q G β q ( n ) 2 F 1 q ( n ) + α 2 G β q ( n ) 2 { [ n ] q 4 ( 1 + q 6 ) 2 q 4 [ n 2 ] q 4 + 2 β q 6 [ n ] q [ n 1 ] q [ n 2 ] q q 6 G β q ( n ) 2 F 2 q ( n ) + ( q + q 2 + [ 2 ] q q 2 ) [ n ] q 3 q 6 G β q ( n ) 2 F 2 q ( n ) + q 6 β 2 [ n 1 ] q [ n 2 ] q q 6 G β q ( n ) 2 F 2 q ( n ) + 2 α q 4 [ n ] q 2 [ n 2 ] q q 6 G β q ( n ) 2 F 2 q ( n ) } x ( x + 1 ) + { [ 2 ] q + 2 α q 2 + α 2 q 3 } [ n ] q q 3 G β q ( n ) F 2 q ( n ) .

From (2.2), we have

B n , q ( α , β ) ( ( t x ) 2 , x ) { [ n 2 ] q 4 ( q 14 + q 8 2 q 4 ) q 6 G β q ( n ) 2 F 2 q ( n ) + ( 1 + q 6 ) { 4 [ 2 ] q q 6 [ n 2 ] q 3 + 6 [ 2 ] q 2 q 4 [ n 2 ] q 2 + 4 [ 2 ] q 3 q 2 [ n 2 ] q + [ 2 ] q 4 } q 6 G β q ( n ) 2 F 2 q ( n ) + ( q + q 2 + [ 2 ] q q 2 + 2 β q 6 + 2 α q 4 ) [ n ] q 3 + β 2 q 6 [ n ] q 2 q 6 G β q ( n ) 2 F 2 q ( n ) } x ( x + 1 ) + ( [ 2 ] q + q 2 ) ( [ 2 ] q + 2 α q 2 + α 2 q 3 ) q 3 G β q ( n ) F 1 q ( n ) ( 2 ( 1 q 4 ) q 6 + 164 ( α + β + 1 ) 2 [ n ] q q 6 F 2 q ( n ) ) x ( x + 1 ) + 6 ( α + 1 ) 2 q 3 G β q ( n ) .

And the proof of Lemma 2 is now finished. □

3 Direct estimates

Now in our considerations, C B [0,) denotes the set of all bounded-continuous functions from [0,) to . C B [0,) is a normed space with the norm f B =sup{|f(x)|:x[0,)}. We denote the first modulus of continuity on the finite interval [0,b], b>0,

ω [ 0 , b ] (f,δ)= sup 0 < h δ , x [ 0 , b ] | f ( x + h ) f ( x ) | .
(3.1)

The Peetre K-functional is defined by

K 2 (f,δ)=inf { f g B + δ g B : g W 2 } ,δ>0,

where W 2 ={g C B [0,): g , g C B [0,)}. By Theorem 2.4 in [25], p.177, there exists a positive constant C such that

K 2 (f,δ)C ω 2 (f, δ ),
(3.2)

where

ω 2 (f, δ )= sup 0 < h δ sup x [ 0 , ) | f ( x + 2 h ) 2 f ( x + h ) f ( x ) | .

Gadzhiev proved the weighted Korovkin-type theorems in [26]. We give the Gadzhiev results in weighted spaces. Let ρ(x)=1+ x 2 and the weighted spaces C ρ [0,) denote the space of all continuous functions f, satisfying |f(x)| M f ρ(x), where M f is a constant depending only on f. C ρ [0,) is a normed space with the norm f ρ =sup{ | f ( x ) | ρ ( x ) :x R + {0}} and C ρ [0,) denotes the subspace of all functions f C ρ [0,) for which lim | x | | f ( x ) | ρ ( x ) exists finitely.

Thus we are ready to give direct results. The following lemma is routine and its proof is omitted.

Lemma 3 Let

B ¯ n , q ( α , β ) (f,x)= B n , q ( α , β ) (f,x)f ( D n , q ( α , β ) ( x ) ) +f(x).
(3.3)

Then the following assertions hold for the operators (3.3):

(i) B ¯ n , q ( α , β ) ( 1 , x ) = 1 , (ii) B ¯ n , q ( α , β ) ( t , x ) = x , (iii) B ¯ n , q ( α , β ) ( t x , x ) = 0 ,

where D n , q ( α , β ) (x)= [ n ] q 2 x q 2 G β q ( n ) F 1 q ( n ) + [ n ] q q G β q ( n ) F 1 q ( n ) + α G β q ( n ) .

Lemma 4 Let q(0,1) and n>2. Then for every x[0,) and f C B [0,), we have the inequality

| B ¯ n , q ( α , β ) ( f , x ) f ( x ) | δ n , q ( α , β ) (x) f B ,

where δ n , q ( α , β ) (x)=( 2 ( 1 q 4 ) q 6 + 263 ( α + β + 1 ) 2 q 6 F 1 q ( n ) )x(x+1)+ 5 ( α + 1 ) 2 q 3 G β q ( n ) .

Proof Using Taylor’s expansion

f(t)=f(x)+(tx) f (x)+ x t (tu) f (u)du

and Lemma 3, we obtain

B ¯ n , q ( α , β ) (f,x)f(x)= B ¯ n , q ( α , β ) ( x t ( t u ) f ( u ) d u , x ) .

Then, using Lemma 1 and the inequality

| x t ( t u ) f ( u ) d u | f B ( t x ) 2 2 ,

we get

| B ¯ n , q ( α , β ) ( f , x ) f ( x ) | | B n , q ( α , β ) ( x t ( t u ) f ( u ) d u , x ) x D n , q ( α , β ) ( x ) { D n , q ( α , β ) ( x ) u } f ( u ) d u | f B 2 { ( 2 ( 1 q 4 ) q 6 + 164 ( α + β + 1 ) 2 [ n ] q q 6 F 2 q ( n ) + ( [ n ] q 2 q 2 G β q ( n ) F 1 q ( n ) 1 ) 2 + 2 [ n ] q 3 q 3 G β q ( n ) 2 F 1 q ( n ) 2 + 2 [ n ] q 2 α q 2 G β q ( n ) 2 F 1 q ( n ) ) x ( x + 1 ) + ( [ n ] q + α q [ n 1 ] q q G β q ( n ) F 1 q ( n ) ) 2 + 6 ( α + 1 ) 2 q 3 G β q ( n ) F 1 q ( n ) } f B 2 { ( 4 ( 1 q 4 ) q 6 + 526 ( α + β + 1 ) 2 q 6 F 1 q ( n ) ) x ( x + 1 ) + 10 ( α + 1 ) 2 q 3 G β q ( n ) } .

And the proof of the Lemma 4 is now completed. □

Theorem 1 Let ( q n )(0,1) a sequence such that q n 1 as n. Then for every n>2, x[0,) and f C B [0,), we have the inequality

| B n , q n ( α , β ) ( f , x ) f ( x ) | 2M ω 2 ( f , δ n , q n ( α , β ) ( x ) ) +w ( f , η n , q n ( α , β ) ( x ) ) ,

where η n , q n ( α , β ) (x)=( [ n ] q n 2 q n 2 G β q n ( n ) F 1 q n ( n ) 1)x+ [ n ] q n q n G β q n ( n ) F 1 q n ( n ) + α G β q n ( n ) .

Proof Using (3.3) for any g W 2 , we obtain the following inequality:

| B n , q n ( α , β ) ( f , x ) f ( x ) | | B ¯ n , q n ( α , β ) ( f g , x ) ( f g ) ( x ) + B ¯ n , q n ( α , β ) ( g , x ) g ( x ) | + | f ( [ n ] q n 2 q n 2 G β q n ( n ) F 1 q n ( n ) x + [ n ] q n q n G β q n ( n ) F 1 q n ( n ) + α G β q n ( n ) ) f ( x ) | .

From Lemma 4, we get

| B n , q n ( α , β ) ( f , x ) f ( x ) | 2 f g B + δ n , q n ( α , β ) ( x ) g + | f ( [ n ] q n 2 q n 2 G β q n ( n ) F 1 q n ( n ) x + [ n ] q n q n G β q n ( n ) F 1 q n ( n ) + α G β q n ( n ) ) f ( x ) | .

By using equality (3.1) we have

| B n , q n ( α , β ) ( f , x ) f ( x ) | 2 f g B + δ n , q n ( α , β ) (x) g B +w ( f , η n , q n ( α , β ) ( x ) ) .

Taking the infimum over g W 2 on the right-hand side of the above inequality and using the inequality (3.2), we get the desired result. □

Theorem 2 Let ( q n )(0,1) a sequence such that q n 1 as n. Then f C ρ [0,), and we have

lim n B n , q n ( α , β ) ( f ) f ρ =0.

Proof From Lemma 1, it is obvious that B n , q n ( α , β ) ( e 0 ) e 0 ρ =0. Since | [ n ] q n 2 q n 2 G β q n ( n ) F 1 q n ( n ) x+ [ n ] q n q n G β q n ( n ) F 1 q n ( n ) + α G β q n ( n ) x|(x+1)o(1) and x + 1 1 + x 2 is positive and bounded from above for each x0, we obtain

B n , q n ( α , β ) ( e 1 ) e 1 ρ x + 1 1 + x 2 o(1).

And then lim n B n , q n ( α , β ) ( e 1 ) e 1 ρ =0.

Similarly for every n>2, we write

B n , q n ( α , β ) ( e 2 ) e 2 ρ = sup x [ 0 , ) { | ( [ n ] q n 4 q n 6 G β q n ( n ) 2 F 2 q n ( n ) 1 ) x 2 1 + x 2 + { ( 1 + ( 1 + [ 2 ] q n ) q n ) [ n ] q n 3 + 2 α q 2 [ n ] q n 2 [ n 1 ] q n q n 5 G β q n ( n ) 2 F 2 q n ( n ) } x + [ 2 ] q n [ n ] q n 2 q n 3 G β q n ( n ) 2 F 2 q n ( n ) 1 + x 2 + + 2 α q n 2 [ n ] q n [ n 2 ] q n q n 3 G β q n ( n ) 2 F 2 q n ( n ) + α 2 G β q n ( n ) 2 | 1 + x 2 } sup x [ 0 , ) 1 + x + x 2 1 + x 2 o ( 1 ) ,

we get lim n B n , q n ( α , β ) ( e 2 ) e 2 ρ =0. Thus, from AD Gadzhiev’s theorem in [26], we obtain the desired result of Theorem 2. □

References

  1. Mahmudov NI: q -Szász operators which preserve x 2 . Math. Slovaca 2013, 63: 1059.

    Article  MathSciNet  MATH  Google Scholar 

  2. Gupta V, Mahmudov NI: Approximation properties of the q -Szasz-Mirakjan-Beta operators. Indian J. Ind. Appl. Math. 2012, 3: 41.

    Google Scholar 

  3. Govil NK, Gupta V: q -Beta-Szász-Stancu operators. Adv. Stud. Contemp. Math. 2012, 22: 117.

    MathSciNet  MATH  Google Scholar 

  4. Yüksel İ, Dinlemez Ü: On the approximation by the q -Szász-Beta type operators. Appl. Math. Comput. 2014, 235: 555.

    Article  MathSciNet  Google Scholar 

  5. Dinlemez Ü, Yüksel İ, Altın B: A note on the approximation by the q -hybrid summation integral type operators. Taiwan. J. Math. 2014, 18: 781.

    Article  MathSciNet  Google Scholar 

  6. Doğru O, Gupta V: Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q -integers. Georgian Math. J. 2005, 12: 415.

    MathSciNet  MATH  Google Scholar 

  7. Doğru O, Gupta V: Korovkin-type approximation properties of bivariate q -Meyer-König and Zeller operators. Calcolo 2006, 43: 51. 10.1007/s10092-006-0114-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Gupta V, Heping W: The rate of convergence of q -Durrmeyer operators for0<q<1. Math. Methods Appl. Sci. 2008, 31: 1946. 10.1002/mma.1012

    Article  MathSciNet  MATH  Google Scholar 

  9. Gupta V, Aral A: Convergence of the q -analogue of Szász-Beta operators. Appl. Math. Comput. 2010, 216: 374. 10.1016/j.amc.2010.01.018

    Article  MathSciNet  MATH  Google Scholar 

  10. Gupta V, Karsli H: Some approximation properties by q -Szász-Mirakyan-Baskakov-Stancu operators. Lobachevskii J. Math. 2012, 33: 175. 10.1134/S1995080212020138

    Article  MathSciNet  MATH  Google Scholar 

  11. Lupaş A: A q -analogue of the Bernstein operator. Seminar on Numerical and Statistical Calculus 1987, 85-92.

    Google Scholar 

  12. Phillips GM: Bernstein polynomials based on the q -integers. Ann. Numer. Math. 1997, 4: 511.

    MathSciNet  MATH  Google Scholar 

  13. Yüksel İ: Approximation by q -Phillips operators. Hacet. J. Math. Stat. 2011, 40: 191.

    MathSciNet  MATH  Google Scholar 

  14. Yüksel I: Approximation by q -Baskakov-Schurer-Szász type operators. AIP Conf. Proc. 2013, 1558: 1136. 10.1063/1.4825708

    Article  Google Scholar 

  15. De Sole A, Kac VG: On integral representations of q -gamma and q -Beta functions. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 2005, 16: 11.

    MathSciNet  MATH  Google Scholar 

  16. Gupta V, Agarwal RP: Convergence Estimates in Approximation Theory. Springer, Cham; 2014.

    Book  MATH  Google Scholar 

  17. Jackson FH: On q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193.

    MATH  Google Scholar 

  18. Kac VG, Cheung P Universitext. In Quantum Calculus. Springer, New York; 2002.

    Chapter  Google Scholar 

  19. Koelink HT, Koornwinder TH: q -Special functions, a tutorial. Contemp. Math. 134. In Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990). Am. Math. Soc., Providence; 1992:141-142.

    Chapter  Google Scholar 

  20. Deo N: Direct result on the Durrmeyer variant of Beta operators. Southeast Asian Bull. Math. 2008, 32: 283.

    MathSciNet  MATH  Google Scholar 

  21. Deo N: Direct result on exponential-type operators. Appl. Math. Comput. 2008, 204: 109. 10.1016/j.amc.2008.06.005

    Article  MathSciNet  MATH  Google Scholar 

  22. Gupta V, Srivastava GS, Sahai A: On simultaneous approximation by Szász-Beta operators. Soochow J. Math. 1995, 21: 1.

    MathSciNet  MATH  Google Scholar 

  23. Jung HS, Deo N, Dhamija M: Pointwise approximation by Bernstein type operators in mobile interval. Appl. Math. Comput. 2014, 214: 683.

    Article  MathSciNet  Google Scholar 

  24. Aral A, Gupta V, Agarwal RP: Applications of q-Calculus in Operator Theory. Springer, New York; 2013.

    Book  MATH  Google Scholar 

  25. De Vore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.

    Book  Google Scholar 

  26. Gadzhiev AD: Theorems of the type of P. P. Korovkin type theorems. Mat. Zametki 1976 English Translation, Math. Notes 20, 996 (1976) , 20: 781. English Translation, Math. Notes 20, 996 (1976).

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like thank the referee for many helpful comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ülkü Dinlemez.

Additional information

Competing interests

The author declares to have no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dinlemez, Ü. Convergence of the q-Stancu-Szász-Beta type operators. J Inequal Appl 2014, 354 (2014). https://doi.org/10.1186/1029-242X-2014-354

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-354

Keywords