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Gupta et al.Journal of Inequalities and Applications2012,2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
q-analogue of a new sequence of linear positive operators 1 2* 3 Vijay Gupta , Taekyun Kim and Sang-Hun Lee
* Correspondence: tkkim@kw.ac.kr 2 Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea Full list of author information is available at the end of the article
Open Access
Abstract This paper deals with Durrmeyer type generalization ofq-Baskakov type operators using the concept ofq-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:Primary 41A25; 41A36 Keywords:Durrmeyer type operators; weighted approximation; rate of convergence;q-integral
1 Introduction In the year  Agrawal and Mohammad [] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as
Dn(f,x) =n pn,k(x)sn,k–(t)f(t)dt+pn,(x)f(), k=
where   k n+k– x pn,k(x) = , n+k k( +x)
k (nt) nt sn,k(t) =e. k!
It is observed in [] 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 []. A lot of works onq-calculus are available in literature of different branches of mathe-matics and physics. For systematic study, we refer to the work of Ernst [], Kim [, ], and Kim and Rim []. The application ofq-calculus in approximation theory was initiated by Phillips [], who was the first to introduceq-Bernstein polynomials and study their approximation properties. Very recently theq-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [, ] and [] respectively. We recall some notations and concepts ofq-calculus. All of the results can be found in [] and []. In what follows,qis a real number satisfying  <q< .
©2012 Gupta et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Gupta et al.Journal of Inequalities and Applications2012, 2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
n  –q [n]q:= ,  –q [n]q[n– ]q∙ ∙ ∙[]q, [n]q! := ,
n= , , . . . , n= .
Theq-binomial coefficients are given by   n[n]q! = , kn. k[k]q![nk]q! q
Theq-Beta integral is defined by []
–q t– q(t) =x Eq(–qx)dqx,
t> ,
which satisfies the following functional equation:
q(t+ ) = [t]qq(t),
q() = .
ForfC[,),q>  and each positive integern, theq-Baskakov operators [] are defined as     k n+k– x[k]q k(k–) Bn,q(f,x) =q f n+k k– k( +x)q[n]q q k=q (.)   [k] q q =p(x)f, n,k k– q[n]q k=
where   n– ( +x)( +qx)∙ ∙ ∙ +q x, n ( +x) := q ,
n= , , . . . , n= .
Remark The first three moments of theq-Baskakov operators are given by
Bn,q(,x) = ,
Bn,q(t,x) =x,    x  Bn,qt,x=x+  +x. [n]qq
As the operatorsDn(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 theq-analogue of the operators as
n q/(–q)   q q q qkk (f,x) = [n]qp(x)q s(t)f+p(x Dn,k n,k–tq dqtn,)f(), n k=
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Gupta et al.Journal of Inequalities and Applications2012, 2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
wherex[,) and   k qn+k– x k(k–) p(x) =q, n,k n+k k( +x)q q
k q([n]qt) s(t) =Eq–[n]qt. n,k [k]q!
In caseq= , the above operators reduce to the operators (.). In the present paper, we estimate a local approximation theorem and the rate of convergence of these new opera-tors as well as their weighted approximation properties.
2 Moment estimation Lemma The following equalities hold: q (i)Dn(,x) = , q (ii)Dn(t,x) =x, q  x x (iii)Dn(t,x) =x+ ( +q+ ). [n]qq
q ProofThe operatorsDnare well defined on the function ,t,t. Then for everyx[,), we obtain
n q/(–q)k–   ([n]qt)q q qk D(,x)x). n= [n]qp(x)q Eq–[n]qt dqt+pn,( n,k [k– ]q! k=
Substituting [n]qt=qyand using (.), we have
/(–q)k– qk(qy) qq dqyq D(,x) = [n] +p n qp(x)q Eq(–qy)n,(x) n,k [k– ]q! [n]q k= q q =p(x) +p(x) =Bn,q(,x) = , n,k n, k=
whereBn,q(f,x) is theq-Baskakov operator defined by (.). Next, we have
n q/(–q)k– q q([n]qt)  kk D(t,x) = [n]qp(x)q Eq–[n]qt tq dqt. n n,k [k– ]q! k=
Again substituting [n]qt=qyand using (.), we have
/(–q)k q q(qy)q dqy k t,x) = [n] Dn(qp(x)q Eq(–qy) n,k k [k– ]q![n]q[n]qq k= [k] q q =p(x)q=Bn,q(t,x) =x. n,k k [n]qq k=
n q/(–q)k–   q([n]qt)  q –k –k Dt,x= [n]qp n n,(x)q Eq–[n]qt t q dqt. k [k– ]q! k=
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Gupta et al.Journal of Inequalities and Applications2012, 2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
k Again substituting [n]qt=qy, using (.) and [k+ ]q= [k]q+q, we have
/(–q)k+ q(qy)q dqy q –k–k Dt,x= [n]qp(x)q Eq(–qy)q n n,k [k– ]q![n] [n]q q k= q[k+ ]q[k]q =p(x) n,k  k– [n]q q k= k q([k]q+q)[k]q =p(x) n,k  k– [n]q q k=    q x x   =Bn,qt,x+Bn,q(t,x) =x+  +q+ . [n]q[n]qq
Remark If we putq= , we get the moments of a new sequenceDn(f,x) considered in [] as operators as
Dn(tx,x) = ,  x(x+ ) Dn(tx) ,x= . n
Lemma Let q(, ), then for x[,)we have
x(x+q[]q) qD(tx) ,x= . n q[n]q
3 Direct theorems ByCB[,) we denote the space of real valued continuous bounded functionsfon the interval [,); the norm- ∙ on the spaceCB[,) is given by
  f=supf(x) . x<
The Peetre’sK-functional is defined by
    K(f,δ) =inffg+δg:gW,
  whereW={gCB[,) :g,gCB[,)}. By [, pp.], there exists a positive / constantC>  such thatK(f,δ)Cω(f,δ),δ>  and the second order modulus of smoothness is given by
  ωf,δ=sup supf(x+ h) – f(x+h) +f(x) . x<<hδ
Also, forfCB[,) a usual modulus of continuity is given by
  ω(f,δ) =sup supf(x+h) –f(x) . <hδx<
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Gupta et al.Journal of Inequalities and Applications2012, 2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
Theorem Let fCB[,)and <q< . Then for all x[,)and nN , there exists an absolute constant C> such that   x(x+q[]q) q   D(f,x) –f(x)Cωf, . n q[n]q
ProofLetgWandx,t[,). By Taylor’s expansion, we have t   g(t) =g(x) +g(x)(tx() + tu)g(u)du. x
Applying Lemma , we obtain
  t q q D(g,x) –g(x) =D(tu)g(u)du,x. n n x t  Obviously, we have|(tu)g(u)du| ≤(tx)g. Therefore, x
  x(x+q[]q) q q        D(g,x) –g(x)D(tx) ,x g=g. n n q[n]q
Using Lemma , we have
n q/(–q)   q q q qkk       )[n] Dn(f,xqp(x)q s(t)df tq qt+p(x)f()≤ f. n,k n,k–n, k=
q q q       D(f,x) –f(x)D(fg,x) – (fg)(x) +D(g,x) –g(x) n n n x(x+q[]q)    fg+g. q[n]q
 / Finally, taking the infimum over allgWand using the inequalityK(f,δ)Cω(f,δ), δ> , we get the required result. This completes the proof of Theorem .
We consider the following class of functions: LetH[,) be the set of all functionsfdefined on [,) satisfying the condition x |f(x)| ≤Mf( +x), whereMfis a constant depending only onf. ByCx[,), we denote the subspace of all continuous functions belonging toH[,). Also, letC[,) be x x f(x) the subspace of all functionsfCx[,), for whichlim|x|→∞is finite. The norm on +x |f(x)| C[,)sup isfx=x[,. We denote the modulus of continuity offon closed ) x+x interval [,a],a>  as by
  ωa(f,δ) =sup supf(t) –f(x) . |tx|≤δx,t[,a]
We observe that for functionfC[,), the modulus of continuityωa(f,δ) tends to x zero.
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Gupta et al.Journal of Inequalities and Applications2012, 2012:144 http://www.journalofinequalitiesandapplications.com/content/2012/1/144
Theorem Let fCx[,), q(, )andωa+(f,δ)be its modulus of continuity on the finite interval[,a+ ][,), where a> . Then for every n> ,   Mfa( +a)( +a)a(a+q[]q) q   D(f) –f+ ωf, . n C[,a] q[n]qq[n]q
ProofForx[,a] andt>a+ , sincetx> , we have
      f(t) –f(x)Mf +x+t     Mf + x+ (tx)     Mf +a(tx) .
Forx[,a] andta+ , we have    |tx|   f(t) –f(x)ωa+f,|tx| ≤ +ωa+(f,δ) δ
withδ> . From (.) and (.) we can write    |tx|     f(t) –f(x)Mf +a(tx +) + ωa+(f,δ) δ
forx[,a] andt. Thus
  q q     D(f,x) –f(x)Df(t) –f(x) ,x n n     q(tx) ,x Mf +aDn     q(tx) +ωa+(f,δ+)  Dn,x. δ
Hence, by using Schwarz inequality and Lemma , for everyq(, ) andx[,a]
Mf( +a)x(q[]q+x) q   D(f,x) –f(x)n q[n]q   x(q[]q+x) +ωa+(f,δ)  + δq[n]q   Mfa( +a)( +a) a(a+q[]q) +ωa+(f,δ)  + . q[n]qδq[n]q
a(q[]q+a) By takingδget the assertion of our theorem.= we q[n]q
4 Higher order moments and an asymptotic formula Lemma ([])Let <q< , we have
    + q[n+ ]q [n+ ]q[n+ ]q   Bn,qt,x=x+x+x,     [n]qq[n]q[n] q q
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