R E S E A R C H
Open Access
q
-analogue of a new sequence of linear
positive operators
Vijay Gupta
1, Taekyun Kim
2*and Sang-Hun Lee
3*Correspondence: [email protected] 2Department of Mathematics,
Kwangwoon University, Seoul 139-701, S. Korea
Full list of author information is available at the end of the article
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
∞
k=
pn,k(x)
∞
sn,k–(t)f(t)dt+pn,(x)f(), x∈[,∞), (.)
where
pn,k(x) =
n+k– k
xk
( +x)n+k, sn,k(t) =e
–nt(nt)k
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< .
Forn∈N,
[n]q:=
–qn
–q,
[n]q! :=
⎧ ⎨ ⎩
[n]q[n– ]q· · ·[]q, n= , , . . . ,
, n= .
Theq-binomial coefficients are given by
n k
q
= [n]q! [k]q![n–k]q!
, ≤k≤n.
Theq-Beta integral is defined by []
q(t) =
–q
xt–Eq(–qx)dqx, t> , (.)
which satisfies the following functional equation:
q(t+ ) = [t]qq(t), q() = .
Forf ∈C[,∞), q> and each positive integer n, theq-Baskakov operators [] are defined as
Bn,q(f,x) =
∞
k=
n+k– k
q
qk(k–) x
k
( +x)n+k q
f
[k]q
qk–[n]
q
= ∞
k=
pqn,k(x)f
[k]q
qk–[n]
q
,
(.)
where
( +x)nq:=
⎧ ⎨ ⎩
( +x)( +qx)· · · +qn–x, n= , , . . . ,
, n= .
Remark The first three moments of theq-Baskakov operators are given by
Bn,q(,x) = ,
Bn,q(t,x) =x,
Bn,q
t,x=x+ x [n]q
+ qx
.
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
Dq
n(f,x) = [n]q
∞
k=
pqn,k(x)
q/(–qn)
wherex∈[,∞) and
pqn,k(x) = n+k– k
q
qk(k–) x
k
( +x)n+k q
, sqn,k(t) =Eq
–[n]qt
([n]qt)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:
(i) Dqn(,x) = ,
(ii) Dqn(t,x) =x,
(iii) Dqn(t,x) =x+[nx]q( +q+xq).
Proof The operatorsDqnare well defined on the function ,t,t. Then for everyx∈[,∞),
we obtain
Dq
n(,x) = [n]q
∞
k=
pqn,k(x)
q/(–qn)
q–k([n]qt)
k–
[k– ]q!
Eq
–[n]qt
dqt+pqn,(x).
Substituting [n]qt=qyand using (.), we have
Dq
n(,x) = [n]q
∞
k=
pqn,k(x)
/(–q)
q–k (qy)
k–
[k– ]q!
Eq(–qy)
q dqy
[n]q
+pqn,(x)
= ∞
k=
pqn,k(x) +pqn,(x) =Bn,q(,x) = ,
whereBn,q(f,x) is theq-Baskakov operator defined by (.).
Next, we have
Dq
n(t,x) = [n]q
∞
k=
pqn,k(x)
q/(–qn)
q–k([n]qt)
k–
[k– ]q!
Eq
–[n]qt
tq–kdqt.
Again substituting [n]qt=qyand using (.), we have
Dq
n(t,x) = [n]q
∞
k=
pqn,k(x)
/(–q)
q–k (qy)
k
[k– ]q![n]q
Eq(–qy)
q dqy
[n]qqk
= ∞
k=
pqn,k(x)q [k]q [n]qqk
=Bn,q(t,x) =x.
Finally,
Dq n
t,x= [n]q
∞
k=
pqn,k(x)
q/(–qn)
q–k([n]qt)
k–
[k– ]q!
Eq
–[n]qt
Again substituting [n]qt=qy, using (.) and [k+ ]q= [k]q+qk, we have
Dq n
t,x= [n]q
∞
k=
pqn,k(x)
/(–q)
q–k (qy)
k+
[k– ]q![n]q
Eq(–qy)q–k
q dqy
[n]q
= ∞
k=
pqn,k(x)[k+ ]q[k]q [n]
qqk–
= ∞
k=
pqn,k(x)([k]q+q
k)[k] q
[n]
qqk–
=Bn,q
t,x+ q [n]qBn,q
(t,x) =x+ x [n]q
+q+x q
.
Remark If we putq= , we get the moments of a new sequenceDn(f,x) considered in
[] as operators as
Dn(t–x,x) = ,
Dn
(t–x),x=x(x+ ) n .
Lemma Let q∈(, ), then for x∈[,∞)we have
Dq n
(t–x),x=x(x+q[]q) q[n]q
.
3 Direct theorems
ByCB[,∞) we denote the space of real valued continuous bounded functionsf on the
interval [,∞); the norm- · on the spaceCB[,∞) is given by
f= sup
≤x<∞
f(x).
The Peetre’sK-functional is defined by
K(f,δ) =inf
f–g+δg:g∈W∞,
where W
∞ ={g∈ CB[,∞) :g,g∈CB[,∞)}. 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
ω
f,√δ= sup
<h≤√δ
sup
≤x<∞
f(x+ h) – f(x+h) +f(x).
Also, forf ∈CB[,∞) a usual modulus of continuity is given by
ω(f,δ) = sup
<h≤δ
sup
≤x<∞
Theorem Let f∈CB[,∞)and <q< . Then for all x∈[,∞)and n∈N , there exists
an absolute constant C> such that
Dq
n(f,x) –f(x)≤Cω
f,
x(x+q[]q)
q[n]q
.
Proof Letg∈W
∞andx,t∈[,∞). By Taylor’s expansion, we have
g(t) =g(x) +g(x)(t–x) +
t
x
(t–u)g(u)du.
Applying Lemma , we obtain
Dq
n(g,x) –g(x) =Dqn
t
x
(t–u)g(u)du,x
.
Obviously, we have|xt(t–u)g(u)du| ≤(t–x)g. Therefore,
Dq
n(g,x) –g(x)≤Dqn
(t–x),xg=x(x+q[]q) q[n]q
g.
Using Lemma , we have
Dq
n(f,x)≤[n]q
∞
k=
pqn,k(x)
q/(–qn)
q–ksqn,k–(t)ftq–kdqt+pqn,(x)f()≤ f.
Thus
Dq
n(f,x) –f(x)≤Dqn(f–g,x) – (f–g)(x)+Dnq(g,x) –g(x)
≤f –g+x(x+q[]q) q[n]q
g.
Finally, taking the infimum over allg∈W∞ and 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:
LetHx[,∞) be the set of all functions f defined on [,∞) satisfying the condition
|f(x)| ≤Mf( +x), whereMf is a constant depending only onf. ByCx[,∞), we denote
the subspace of all continuous functions belonging toHx[,∞). Also, letC∗
x[,∞) be
the subspace of all functionsf ∈Cx[,∞), for whichlim|x|→∞ f(x)
+x is finite. The norm on
Cx∗[,∞) isfx=supx∈[,∞)|f(x)|
+x. We denote the modulus of continuity off on closed
interval [,a],a> as by
ωa(f,δ) = sup
|t–x|≤δ
sup
x,t∈[,a]
f(t) –f(x).
We observe that for functionf ∈Cx[,∞), the modulus of continuityωa(f,δ) tends to
Theorem Let f∈Cx[,∞), q∈(, )andωa+(f,δ)be its modulus of continuity on the
finite interval[,a+ ]⊂[,∞), where a> . Then for every n> ,
Dq
n(f) –fC[,a]≤
Mfa( +a)( +a)
q[n]q
+ ω
f,
a(a+q[]q)
q[n]q
.
Proof Forx∈[,a] andt>a+ , sincet–x> , we have
f(t) –f(x)≤Mf
+x+t
≤Mf
+ x+ (t–x) (.)
≤Mf
+a(t–x).
Forx∈[,a] andt≤a+ , we have
f(t) –f(x)≤ωa+
f,|t–x|≤
+|t–x| δ
ωa+(f,δ) (.)
withδ> .
From (.) and (.) we can write
f(t) –f(x)≤Mf
+a(t–x)+
+|t–x| δ
ωa+(f,δ) (.)
forx∈[,a] andt≥. Thus
Dq
n(f,x) –f(x)≤Dqnf(t) –f(x),x
≤Mf
+aDqn(t–x),x
+ωa+(f,δ)
+ δD
q n
(t–x),x
.
Hence, by using Schwarz inequality and Lemma , for everyq∈(, ) andx∈[,a]
Dq
n(f,x) –f(x)≤
Mf( +a)x(q[]q+x)
q[n]q
+ωa+(f,δ)
+ δ
x(q[]q+x)
q[n]q
≤Mfa( +a)( +a)
q[n]q
+ωa+(f,δ)
+ δ
a(a+q[]q)
q[n]q
.
By takingδ=
a(q[]q+a)
q[n]q we get the assertion of our theorem.
4 Higher order moments and an asymptotic formula Lemma ([]) Let <q< , we have
Bn,q
t,x= [n]q
x+ + q q
[n+ ]q
[n]
q
x+ q
[n+ ]q[n+ ]q
[n]
q
Bn,q
t,x= [n]
q
x+ q
+ q+ q[n+ ]q [n]
q
x
+ q[]
q
+ q+ q+ q[n+ ]q[n+ ]q [n]
q
x
+ q[]
q[]q[]q
+ q+ q+ q+ q+ q+q
×[n+ ]q[n+ ]q[n+ ]q
[n]
q
x.
Now, we present higher order moments for the operators (.).
Lemma Let <q< , we have
Dq n
t,x=[n+ ]q[n+ ]q q[n]
q
x+
+ q q
[n+ ]q
[n]
q
+( +q)q [n]q
+( +q) [n]q
x
+
[n]
q
+( +q)q [n]
q
+( +q)q
[n]
q x, Dq n
t,x
=
( + q+ q+ q+ q+ q+q)
q[]
q[]q[]q
[n+ ]q[n+ ]q[n+ ]q
[n]
q
x
+
( + q+ q+ q)
q[]
q
[n+ ]q[n+ ]q
[n]
q
+q( + q+q
)
[n]
q
[n+ ]q[n+ ]q
x
+
( + q+ q)
q
[n+ ]q
[n]q
+( + q)( + q+q
)
q[n]
q
[n+ ]q
+q
( + q+ q+q)
[n]
q
+q( + q+ q
+q)
[n]
q x + [n]q
+q( + q+q
)
[n]
q
+q
( + q+ q+q)
[n]
q
+q
( + q+ q+q)
[n]
q
x.
The proof of Lemma can be obtained by using Lemma . We consider the following classes of functions:
Cm[,∞) :=
f ∈C[,∞) :∃Mf > f(x)<Mf
+xmandfm:= sup x∈[,∞)
|f(x)| +xm
,
Cm∗[,∞) :=
f ∈Cm[,∞) : lim x→∞
|f(x)| +xm <∞
, m∈N.
Theorem Let qn∈(, ), then the sequence{Dnqn(f)}converges to f uniformly on[,A]
Theorem Assume that qn∈(, ), qn→and qnn→a as n→ ∞. For any f ∈C∗[,∞)
such that f,f∈C∗[,∞)the following equality holds
lim
n→∞[n]qn
Dqn
n (f;x) –f(x)
=x+ xf(x)
uniformly on any[,A], A> .
Proof Letf,f,f∈C∗[,∞) andx∈[,∞) be fixed. By using Taylor’s formula, we may write
f(t) =f(x) +f(x)(t–x) + f
(x)(t–x)+r(t;x)(t–x), (.)
wherer(t;x) is the Peano form of the remainder,r(·;x)∈C∗[,∞) andlimt→xr(t;x) = .
ApplyingDqn
n to (.), we obtain
[n]qn
Dqn
n (f;x) –f(x)
= f
(x)[n]
qnDqnn
(t–x);x+ [n]qnDqnn
r(t;x)(t–x);x.
By the Cauchy-Schwarz inequality, we have
Dqn
n
r(t;x)(t–x);x≤
Dqn
n
r(t;x);xDqn
n
(t–x);x. (.)
Observe that r(x;x) = and r(·;x)∈C∗
[,∞). Then it follows from Theorem and
Lemma , that
lim
n→∞D qn
n
r(t;x);x=r(x;x) = (.)
uniformly with respect tox∈[,A]. Now from (.), (.) and Remark , we get immedi-ately
lim
n→∞[n]qnD qn
n
r(t;x)(t–x);x= .
Then, we get the following
lim
n→∞[n]qn
Dqn
n (f;x) –f(x)
= lim
n→∞
f
(x)[n]
qnDqnn
(t–x);x+ [n]qnDqnn
r(t;x)(t–x);x
=x+ xf(x).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed in preparing the manuscript equally.
Author details
1School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi 110078, India. 2Department of Mathematics, Kwangwoon University, Seoul 139-701, S. Korea.3Division of General
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.
Received: 7 May 2012 Accepted: 4 June 2012 Published: 22 June 2012 References
1. Agrawal, PN, Mohammad, AJ: Linear combination of a new sequence of linear positive operators. Rev. Unión Mat. Argent.44(1), 33-41 (2003)
2. Aral, A, Gupta, V: Generalizedq-Baskakov operators. Math. Slovaca61(4), 619-634 (2011)
3. Aral, A, Gupta, V: On the Durrmeyer type modification of theq-Baskakov type operators. Nonlinear Anal.72(3-4), 1171-1180 (2010)
4. DeVore, RA, Lorentz, GG: Constructive Approximation. Springer, Berlin (1993)
5. Ernst, T: The history ofq-calculus and a new method. U.U.D.M Report 2000, 16, ISSN 1101-3591, Department of Mathematics, Upsala University (2000)
6. Finta, Z, Gupta, V: Approximation properties ofq-Baskakov operators. Cent. Eur. J. Math.8(1), 199-211 (2010) 7. Gupta, V, Radu, C: Statistical approximation properties ofq-Baskakov-Kantorovich operators. Cent. Eur. J. Math.7(4),
809-818 (2009)
8. Kac, VG, Cheung, P: Quantum Calculus. Universitext. Springer, New York (2002)
9. Kim, T, Rim, S-H: A note on theq-integral andq-series. Adv. Stud. Contemp. Math. (Pusan)2, 37-45 (2000) 10. Kim, T:q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys.15,
51-57 (2008)
11. Kim, T:q-Bernoulli numbers associated withq-Stirling numbers. Adv. Differ. Equ.2008, Art. ID 743295 (2008) 12. Koornwinder, TH:q-special functions, a tutorial. In: Gerstenhaber, M, Stasheff, J (eds.) Deformation Theory and
Quantum Groups with Applications to Mathematical Physics. Contemp. Math., vol. 134. Am. Math. Soc., Providence (1992)
13. Phillips, GM: Bernstein polynomials based on theq-integers. Ann. Numer. Math.4, 511-518 (1997)
14. Wang, X: The iterative approximation of a new sequence of linear positive operators. J. Jishou Univ. Nat. Sci. Ed.26(2), 72-78 (2005)
doi:10.1186/1029-242X-2012-144