Skip to main content

q-analogue of a new sequence of linear positive operators

Abstract

This paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [0,). An estimate for the rate of convergence and weighted approximation properties are also obtained.

MSC:41A25, 41A36.

1 Introduction

In the year 2003 Agrawal and Mohammad [1] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as

D n (f,x)=n k = 1 p n , k (x) 0 s n , k 1 (t)f(t)dt+ p n , 0 (x)f(0),x[0,),
(1.1)

where

p n , k (x)= ( n + k 1 k ) x k ( 1 + x ) n + k , s n , k (t)= e n t ( n t ) k k ! .

It is observed in [1] that these operators reproduce constant as well as linear functions. Later, some direct approximation results for the iterative combinations of these operators were studied in [14].

A lot of works on q-calculus are available in literature of different branches of mathematics and physics. For systematic study, we refer to the work of Ernst [5], Kim [10, 11], and Kim and Rim [9]. The application of q-calculus in approximation theory was initiated by Phillips [13], who was the first to introduce q-Bernstein polynomials and study their approximation properties. Very recently the q-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [2, 3] and [7] respectively. We recall some notations and concepts of q-calculus. All of the results can be found in [5] and [8]. In what follows, q is a real number satisfying 0<q<1.

For nN,

The q-binomial coefficients are given by

[ n k ] q = [ n ] q ! [ k ] q ! [ n k ] q ! ,0kn.

The q-Beta integral is defined by [12]

Γ q (t)= 0 1 1 q x t 1 E q (qx) d q x,t>0,
(1.2)

which satisfies the following functional equation:

Γ q (t+1)= [ t ] q Γ q (t), Γ q (1)=1.

For fC[0,)q>0 and each positive integer n, the q-Baskakov operators [2] are defined as

B n , q ( f , x ) = k = 0 [ n + k 1 k ] q q k ( k 1 ) 2 x k ( 1 + x ) q n + k f ( [ k ] q q k 1 [ n ] q ) = k = 0 p n , k q ( x ) f ( [ k ] q q k 1 [ n ] q ) ,
(1.3)

where

( 1 + x ) q n :={ ( 1 + x ) ( 1 + q x ) ( 1 + q n 1 x ) , n = 1 , 2 , , 1 , n = 0 .

Remark 1 The first three moments of the q-Baskakov operators are given by

As the operators D n (f,x) have mixed basis functions in summation and integration and have an interesting property of reproducing linear functions, we were motivated to study these operators further. Here we define the q-analogue of the operators as

D n q (f,x)= [ n ] q k = 1 p n , k q (x) 0 q / ( 1 q n ) q k s n , k 1 q (t)f ( t q k ) d q t+ p n , 0 q (x)f(0),
(1.4)

where x[0,) and

p n , k q (x)=[ n + k 1 k ] q q k ( k 1 ) 2 x k ( 1 + x ) q n + k , s n , k q (t)= E q ( [ n ] q t ) ( [ n ] q t ) k [ k ] q ! .

In case q=1, the above operators reduce to the operators (1.1). In the present paper, we estimate a local approximation theorem and the rate of convergence of these new operators as well as their weighted approximation properties.

2 Moment estimation

Lemma 1 The following equalities hold:

  1. (i)

    D n q (1,x)=1,

  2. (ii)

    D n q (t,x)=x,

  3. (iii)

    D n q ( t 2 ,x)= x 2 + x [ n ] q (1+q+ x q ).

Proof The operators D n q are well defined on the function 1,t, t 2 . Then for every x[0,), we obtain

D n q (1,x)= [ n ] q k = 1 p n , k q (x) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) d q t+ p n , 0 q (x).

Substituting [ n ] q t=qy and using (1.2), we have

D n q ( 1 , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k 1 [ k 1 ] q ! E q ( q y ) q d q y [ n ] q + p n , 0 q ( x ) = k = 1 p n , k q ( x ) + p n , 0 q ( x ) = B n , q ( 1 , x ) = 1 ,

where B n , q (f,x) is the q-Baskakov operator defined by (1.3).

Next, we have

D n q (t,x)= [ n ] q k = 1 p n , k q (x) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) t q k d q t.

Again substituting [ n ] q t=qy and using (1.2), we have

D n q ( t , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k [ k 1 ] q ! [ n ] q E q ( q y ) q d q y [ n ] q q k = k = 0 p n , k q ( x ) q [ k ] q [ n ] q q k = B n , q ( t , x ) = x .

Finally,

D n q ( t 2 , x ) = [ n ] q k = 0 p n , k q (x) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) t 2 q 2 k d q t.

Again substituting [ n ] q t=qy, using (1.2) and [ k + 1 ] q = [ k ] q + q k , we have

D n q ( t 2 , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k + 1 [ k 1 ] q ! [ n ] q 2 E q ( q y ) q 2 k q d q y [ n ] q = k = 1 p n , k q ( x ) [ k + 1 ] q [ k ] q [ n ] q 2 q 2 k 2 = k = 1 p n , k q ( x ) ( [ k ] q + q k ) [ k ] q [ n ] q 2 q 2 k 2 = B n , q ( t 2 , x ) + q [ n ] q B n , q ( t , x ) = x 2 + x [ n ] q ( 1 + q + x q ) .

 □

Remark 2 If we put q=1, we get the moments of a new sequence D n (f,x) considered in [1] as operators as

Lemma 2 Letq(0,1), then forx[0,)we have

D n q ( ( t x ) 2 , x ) = x ( x + q [ 2 ] q ) q [ n ] q .

3 Direct theorems

By C B [0,) we denote the space of real valued continuous bounded functions f on the interval [0,); the norm- on the space C B [0,) is given by

f= sup 0 x < | f ( x ) | .

The Peetre’s K-functional is defined by

K 2 (f,δ)=inf { f g + δ g : g W 2 } ,

where W 2 ={g C B [0,): g , g C B [0,)}. By [4], pp.177], there exists a positive constant C>0 such that K 2 (f,δ)C ω 2 (f, δ 1 / 2 )δ>0 and the second order modulus of smoothness is given by

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

Also, for f C B [0,) a usual modulus of continuity is given by

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

Theorem 1 Letf C B [0,)and0<q<1. Then for allx[0,)andnN, there exists an absolute constantC>0such that

| D n q ( f , x ) f ( x ) | C ω 2 ( f , x ( x + q [ 2 ] q ) q [ n ] q ) .

Proof Let g W 2 and x,t[0,). By Taylor’s expansion, we have

g(t)=g(x)+ g (x)(tx)+ x t (tu) g (u)du.

Applying Lemma 2, we obtain

D n q (g,x)g(x)= D n q ( x t ( t u ) g ( u ) d u , x ) .

Obviously, we have | x t (tu) g (u)du| ( t x ) 2 g . Therefore,

| D n q ( g , x ) g ( x ) | D n q ( ( t x ) 2 , x ) g = x ( x + q [ 2 ] q ) q [ n ] q g .

Using Lemma 1, we have

| D n q ( f , x ) | [ n ] q k = 1 p n , k q (x) 0 q / ( 1 q n ) q k s n , k 1 q (t) | f ( t q k ) | d q t+ p n , 0 q (x) | f ( 0 ) | f.

Thus

| D n q ( f , x ) f ( x ) | | D n q ( f g , x ) ( f g ) ( x ) | + | D n q ( g , x ) g ( x ) | 2 f g + x ( x + q [ 2 ] q ) q [ n ] q g .

Finally, taking the infimum over all g W 2 and using the inequality K 2 (f,δ)C ω 2 (f, δ 1 / 2 ), δ>0, we get the required result. This completes the proof of Theorem 1. □

We consider the following class of functions:

Let H x 2 [0,) be the set of all functions f defined on [0,) satisfying the condition |f(x)| M f (1+ x 2 ), where M f is a constant depending only on f. By C x 2 [0,), we denote the subspace of all continuous functions belonging to H x 2 [0,). Also, let C x 2 [0,) be the subspace of all functions f C x 2 [0,), for which lim | x | f ( x ) 1 + x 2 is finite. The norm on C x 2 [0,) is f x 2 = sup x [ 0 , ) | f ( x ) | 1 + x 2 . We denote the modulus of continuity of f on closed interval [0,a], a>0 as by

ω a (f,δ)= sup | t x | δ sup x , t [ 0 , a ] | f ( t ) f ( x ) | .

We observe that for function f C x 2 [0,), the modulus of continuity ω a (f,δ) tends to zero.

Theorem 2 Letf C x 2 [0,), q(0,1)and ω a + 1 (f,δ)be its modulus of continuity on the finite interval[0,a+1][0,), wherea>0. Then for everyn>2,

D n q ( f ) f C [ 0 , a ] 6 M f a ( 1 + a 2 ) ( 2 + a ) q [ n ] q +2ω ( f , a ( a + q [ 2 ] q ) q [ n ] q ) .

Proof For x[0,a] and t>a+1, since tx>1, we have

| f ( t ) f ( x ) | M f ( 2 + x 2 + t 2 ) M f ( 2 + 3 x 2 + 2 ( t x ) 2 ) 6 M f ( 1 + a 2 ) ( t x ) 2 .
(3.1)

For x[0,a] and ta+1, we have

|f(t)f(x)| ω a + 1 ( f , | t x | ) ( 1 + | t x | δ ) ω a + 1 (f,δ)
(3.2)

with δ>0.

From (3.1) and (3.2) we can write

|f(t)f(x)|6 M f ( 1 + a 2 ) ( t x ) 2 + ( 1 + | t x | δ ) ω a + 1 (f,δ)
(3.3)

for x[0,a] and t0. Thus

| D n q ( f , x ) f ( x ) | D n q ( | f ( t ) f ( x ) | , x ) 6 M f ( 1 + a 2 ) D n q ( ( t x ) 2 , x ) + ω a + 1 ( f , δ ) ( 1 + 1 δ D n q ( ( t x ) 2 , x ) ) 1 2 .

Hence, by using Schwarz inequality and Lemma 2, for every q(0,1) and x[0,a]

| D n q ( f , x ) f ( x ) | 6 M f ( 1 + a 2 ) x ( q [ 2 ] q + x ) q [ n ] q + ω a + 1 ( f , δ ) ( 1 + 1 δ x ( q [ 2 ] q + x ) q [ n ] q ) 6 M f a ( 1 + a 2 ) ( 2 + a ) q [ n ] q + ω a + 1 ( f , δ ) ( 1 + 1 δ a ( a + q [ 2 ] q ) q [ n ] q ) .

By taking δ= a ( q [ 2 ] q + a ) q [ n ] q we get the assertion of our theorem. □

4 Higher order moments and an asymptotic formula

Lemma 3 ([6])

Let0<q<1, we have

B n , q ( t 3 , x ) = 1 [ n ] q x + 1 + 2 q q 2 [ n + 1 ] q [ n ] q 2 x 2 + 1 q 3 [ n + 1 ] q [ n + 2 ] q [ n ] q 2 x 3 , B n , q ( t 4 , x ) = 1 [ n ] q 3 x + 1 q 3 ( 1 + 3 q + 3 q 2 ) [ n + 1 ] q [ n ] q 3 x 2 + 1 q 5 [ 2 ] q ( 1 + 3 q + 5 q 2 + 3 q 3 ) [ n + 1 ] q [ n + 2 ] q [ n ] q 3 x 3 + 1 q 6 [ 2 ] q [ 3 ] q [ 4 ] q ( 1 + 3 q + 5 q 2 + 6 q 3 + 5 q 4 + 3 q 5 + q 6 ) × [ n + 1 ] q [ n + 2 ] q [ n + 3 ] q [ n ] q 3 x 4 .

Now, we present higher order moments for the operators (1.4).

Lemma 4 Let0<q<1, we have

The proof of Lemma 4 can be obtained by using Lemma 3.

We consider the following classes of functions:

Theorem 3 Let q n (0,1), then the sequence{ D n q n (f)}converges to f uniformly on[0,A]for eachf C 2 [0,)if and only if lim n q n =1.

Theorem 4 Assume that q n (0,1), q n 1and q n n aasn. For anyf C 2 [0,)such that f , f C 2 [0,)the following equality holds

lim n [ n ] q n ( D n q n ( f ; x ) f ( x ) ) = ( x 2 + 2 x ) f (x)

uniformly on any[0,A], A>0.

Proof Let f, f , f C 2 [0,) and x[0,) be fixed. By using Taylor’s formula, we may write

f(t)=f(x)+ f (x)(tx)+ 1 2 f (x) ( t x ) 2 +r(t;x) ( t x ) 2 ,
(4.1)

where r(t;x) is the Peano form of the remainder, r(;x) C 2 [0,) and lim t x r(t;x)=0. Applying D n q n to (4.1), we obtain

[ n ] q n ( D n q n ( f ; x ) f ( x ) ) = 1 2 f (x) [ n ] q n D n q n ( ( t x ) 2 ; x ) + [ n ] q n D n q n ( r ( t ; x ) ( t x ) 2 ; x ) .

By the Cauchy-Schwarz inequality, we have

D n q n ( r ( t ; x ) ( t x ) 2 ; x ) D n q n ( r 2 ( t ; x ) ; x ) D n q n ( ( t x ) 4 ; x ) .
(4.2)

Observe that r 2 (x;x)=0 and r 2 (;x) C 2 [0,). Then it follows from Theorem 3 and Lemma 4, that

lim n D n q n ( r 2 ( t ; x ) ; x ) = r 2 (x;x)=0
(4.3)

uniformly with respect to x[0,A]. Now from (4.2), (4.3) and Remark 2, we get immediately

lim n [ n ] q n D n q n ( r ( t ; x ) ( t x ) 2 ; x ) =0.

Then, we get the following

 □

References

  1. Agrawal PN, Mohammad AJ: Linear combination of a new sequence of linear positive operators. Rev. Unión Mat. Argent. 2003, 44(1):33–41.

    MathSciNet  MATH  Google Scholar 

  2. Aral A, Gupta V: Generalized q -Baskakov operators. Math. Slovaca 2011, 61(4):619–634. 10.2478/s12175-011-0032-3

    MathSciNet  Article  MATH  Google Scholar 

  3. Aral A, Gupta V: On the Durrmeyer type modification of the q -Baskakov type operators. Nonlinear Anal. 2010, 72(3–4):1171–1180. 10.1016/j.na.2009.07.052

    MathSciNet  Article  MATH  Google Scholar 

  4. DeVore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.

    Book  MATH  Google Scholar 

  5. Ernst, T: The history of q-calculus and a new method. U.U.D.M Report 2000, 16, ISSN 1101–3591, Department of Mathematics, Upsala University (2000)

    Google Scholar 

  6. Finta Z, Gupta V: Approximation properties of q -Baskakov operators. Cent. Eur. J. Math. 2010, 8(1):199–211. 10.2478/s11533-009-0061-0

    MathSciNet  Article  MATH  Google Scholar 

  7. Gupta V, Radu C: Statistical approximation properties of q -Baskakov-Kantorovich operators. Cent. Eur. J. Math. 2009, 7(4):809–818. 10.2478/s11533-009-0055-y

    MathSciNet  Article  MATH  Google Scholar 

  8. Kac VG, Cheung P: Quantum Calculus. Springer, New York; 2002.

    Book  MATH  Google Scholar 

  9. Kim T, Rim S-H: A note on the q -integral and q -series. Adv. Stud. Contemp. Math. (Pusan) 2000, 2: 37–45.

    MathSciNet  MATH  Google Scholar 

  10. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15: 51–57.

    MathSciNet  Article  MATH  Google Scholar 

  11. Kim T: q -Bernoulli numbers associated with q -Stirling numbers. Adv. Differ. Equ. 2008., 2008:

    Google Scholar 

  12. Koornwinder TH: q -special functions, a tutorial. In Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Edited by: Gerstenhaber M, Stasheff J. Am. Math. Soc., Providence; 1992.

    Google Scholar 

  13. Phillips GM: Bernstein polynomials based on the q -integers. Ann. Numer. Math. 1997, 4: 511–518.

    MathSciNet  MATH  Google Scholar 

  14. Wang X: The iterative approximation of a new sequence of linear positive operators. J. Jishou Univ. Nat. Sci. Ed. 2005, 26(2):72–78.

    Google Scholar 

Download references

Acknowledgements

The work was done while the first author visited Division of General Education-mathematics, Kwangwoon University, Seoul, South Korea for collaborative research during June 15-25, 2010.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Taekyun Kim.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed in preparing the manuscript equally.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Gupta, V., Kim, T. & Lee, SH. q-analogue of a new sequence of linear positive operators. J Inequal Appl 2012, 144 (2012). https://doi.org/10.1186/1029-242X-2012-144

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-144

Keywords

  • Durrmeyer type operators
  • weighted approximation
  • rate of convergence
  • q-integral