R E S E A R C H
Open Access
A short note on approximation properties of
Stancu generalization of
q-Durrmeyer
operators
Vishnu Narayan Mishra
*and Prashantkumar Patel
*Correspondence:
[email protected] Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology,
Ichchhanath Mahadev Road, Surat, Gujarat 395 007, India
Abstract
In the present paper, we introduce a simple Stancu generalization ofq-analogue of well-known Durrmeyer operators. We first estimate moments ofq-Durrmeyer-Stancu operators. We also establish the rate of convergence as well as Voronovskaja type asymptotic formula forq-Durrmeyer-Stancu operators.
1 Introduction
In the last decade, the applications ofq-calculus in the approximation theory is one of the main areas of research. To approximate Lebesgue integrable functions on the interval [, ], Durrmeyer introduced the integral modification of the well-known Bernstein poly-nomials. In , Derriennic [] first studied these operators in detail. After theq-analogue of Bernstein polynomials by Phillips [], Gupta and Heping [] introducedq-Durrmeyer operators. Several other researchers have studied in this direction and obtained different approximation properties of many operators [, ]. In the present article, we propose the q-analogue of the Stancu generalization of Durrmeyer operators and study the conver-gence behavior. We have used notations ofq-calculus as given in [–].
We setpnk(q;x) = n
k
qx
k( –x)n–k
q ,p∞k(q;x) = x
k
(–q)k[k]q!( –x)∞q .
Phillips [] proposed the followingq-Bernstein polynomials, which for each positive integernandf∈C[, ] are defined as
Bn,q(f;x) = n
k= f
[k]q
[n]q
pnk(q;x).
In , Derriennic [] introduced aq-analogue of the Durrmeyer operators and has es-tablished some approximation properties of theq-Durrmeyer operators. After couple of years, Gupta [] studiedq-analogue of Durrmeyer operators and discussed approxima-tion properties of the followingq-Durrmeyer operators: Forf ∈C[, ],
Dn,q(f;x) = [n+ ]q n
k=
q–kpnk(q;x)
f(t)pnk(q;qt)dqt
=
n
k=
Ank(f)pnk(q;x); ≤x≤. ()
Recently, Ibrahimet al.[, ] introduced Stancu generalization of certain operators and discussed its approximation properties. Motivated by such type operators, we introduce the Stancu type generalization of theq-Durrmeyer operators () for ≤α≤β, which is defined as follows:
Dαn,,βq(f;x) = [n+ ]q n
k=
q–kpnk(q;x)
f
[n]qt+α
[n]q+β
pnk(q;qt)dqt
=
n
k=
Aαnk,β(f)pnk(q;x); ≤x≤. ()
It can be easily verified that in caseq= ,α= andβ= , the operators defined in () reduce to the well-known Durrmeyer operators as defined in []. Throughout the present manuscript, the expressiongn(x)⇑g(x) means uniform convergence of a sequence{gn(x)}
tog(x).
The present note deals with the study ofq-Durrmeyer-Stancu operators{Dαn,,qβ(f)}for <
q< . First, we estimate the moments forq-Durrmeyer-Stancu operators. We also study the rate of convergence as well as asymptotic formula for these operators{Dαn,,qβ(f)}. We
establish a direct results in terms ofω(f,·).
2 Estimation of moments
In this section, we shall obtainDαn,,βq(ti;x),i= , , . . . .
Note that fors= , , . . . and by the definition ofq-Beta function [], we have
tspnk(q;qt)dqt=
qk[n]q![k+s]q!
[n+s+ ]q![k]q!
and
tsp∞k(q;qt)dqt= ( –q)s+
qk[k+s] q!
[k]q!
.
()
Lemma We have
Dαn,,βq(;x) = ,
Dαn,,βq(t;x) = [n]q+α[n+ ]q+qx[n]
q
[n+ ]q([n]q+β)
,
Dαn,,βqt;x=q [n]
q([n]q– )x+ ((q( +q)+ αq)[n]q+ αq[]q[n]q)x
([n]q+β)[n+ ]q[n+ ]q
+ α
([n]q+β)
+( +q+ αq )[n]
q+ α[]q[n]q
([n]q+β)[n+ ]q[n+ ]q
.
Lemma We have
δn(x) =Dnα,,βq(t–x,x) =
q[n]
q
[n+ ]q([n]q+β)
–
x+ [n]q+α[n+ ]q [n+ ]q([n]q+β)
,
γn(x) =Dαn,,qβ
(t–x),x
=q [n]
q–q[n]q– q[n]q[n+ ]q([n]q+β) + [n+ ]q[n+ ]q([n]q+β)
([n]q+β)[n+ ]q[n+ ]q
+q( +q) [n]
q+ qα[n]q[n+ ]q– ([n]q+ α[n+ ]q)[n+ ]q([n]q+β)
([n]q+β)[n+ ]q[n+ ]q
x
+( +q)[n]
q+ α[n]q[n+ ]q
([n]q+β)[n+ ]q[n+ ]q
.
Remark [] By simple computation, it can easily be verified that
Dn,q
tm;x= [n+ ]q! [n+m+ ]q!
n
k=
[k+ ]q[k+ ]q· · ·[k+m]qpnk(q;x), r≥.
Using [k+s]q= [s]q+qs[k]q, we get [k+ ]q[k+ ]q· · ·[k+m]q= m
s=([s]q+qs[k]q) = m
s=cs(m)[k]sq, wherecs(m) > ,s= , , , . . . ,mare constants independent ofk. Hence,
Dn,q(tm;x) =[n[+nm+]+]q!q! ms=cs(m) nk=[k]sqpnk(q;x) =[n[+nm+]+]q!q! ms=cs(m)[n]sqBn,q(ts;x).
Remark For allm∈N∪ {}, ≤α≤β, we have the following recursive relation for the images of the monomialstmunderDα,β
n,q(tm;x) in terms ofDn,q(tj;x);j= , , , . . . ,m, as
Dαn,,βqtm;x=
m
j=
m j
[n]jqαm–j
([n]q+β)m
Dn,q
tj,x
=
m
j=
m j
[n]jqαm–j
([n]q+β)m
[n+ ]q!
[n+j+ ]q! j
s=
cs(m)[n]sqBn,q
ts;x.
Sincecs(m) > fors= , , . . . ,mandBn,q(ts;x) is a polynomial of degree≤min(s,n) (see
[]), we getDαn,,qβ(tm;x) is a polynomial of degree≤min(m,n).
3 Convergence ofq-Durrmeyer-Stancu operators
Theorem Let qn∈(, ].Then the sequence{Dnα,,qβn(f)}convergence to f uniformly on[, ]
for each f ∈C[, ]if and only iflimn→∞qn= .
The proof of the above theorem follows along the lines of [, Theorem ], thus we omit the details.
Letq∈(, ) be fixed. We defineDα∞,,βq(f, ) =f() and forx∈(, )
Dα∞,,βq(f;x) = –q
∞
k=
p∞k(q;x)q–k
f
[n]qt+α
[n]q+β
p∞k(q;qt)dqt
= ∞
k=
Aα∞,βk(f)p∞k(q;x). ()
Using the fact that [], we have
∞
k=
p∞k(q;x) = ,
∞
k=
–qkp∞k(q;x) =x, and
∞
k=
Using () and (), it is easy to prove that
Dα∞,,βq(;x) = , Dα∞,,βq(t;x) = [n]q( +q(x– )) +α [n]q+β
,
Dα,β
∞,q
t;x=[n]
qqx+ ([n]q(q( +q)( –q)) + qα[n]q)x
([n]q+β)
+[n]
q( –q)( +q) + α( –q)[n]q+α
([n]q+β)
.
Forf ∈C[, ],t> , we define the modulus of continuityω(f,t) as follows:ω(f,t) =
sup{|f(x) –f(y)|:|x–y| ≤t,x,y∈[, ]}. We shall show the following theorem.
Theorem Let <q< then for each f ∈C[, ]the sequence{Dnα,,βq(f;x)}converges to
Dα∞,,βq(f;x)uniformly on[, ].Furthermore, Dαn,,qβ(f) –Dα∞,,βq(f) ≤Cqα,βω(f,qn).
The proof of the above theorem follows along the lines of [, Theorem ], thus we omit the details.
Remark We may observe that, for f(x) = x, we have Dα,β
n,q(f) –D∞,α,βq(f) qn
ω(f, √
qn), whereA(n)B(n) means thatA(n)B(n) andA(n)B(n), andA(n)B(n)
means that there exists a positive constantCindependent ofnsuch thatA(n)≤CB(n). Hence, the estimate of Theorem is sharp in the following sense: the sequenceqn in
Theorem cannot be replaced by any other sequence decreasing to zero more rapidly asn→ ∞.
Lemma [] Let L be a positive linear operator on C[, ],which reproduces linear func-tions.If L(t,x) >x∀x∈(, ),then L(f) =f if and only if f is linear.
Remark Since Dα∞,,βq(t,x) =
[n]q[(–q)(+q)+q(+q)(–q)x+q(–q)x+qx]+α[n]q(+q(x–))+α
[n]q+β > x
for <q< consequence of Lemma we have the following:
Theorem Let <q< be fixed and let f ∈C[, ].Then D∞,α,βq(f;x) =f(x)for all x∈[, ]
if and only if f is linear.
Remark Let <q< be fixed and letf∈C[, ]. Then the sequence{Dnαβ,q(f;x)}does not
approximatef(x) unlessfis linear. This is completely in contrast to the classical Bernstein polynomials, by which{Dn(f;x)}approximatesf(x) for anyf∈C[, ].
Theorem For any f ∈C[, ],{Dα∞,,βq(f)}converges to f uniformly on[, ]as q→–.
Next, we establish a Voronovskaja type asymptotic formula for the operatorsDαn,,qβn:
Theorem Let f be bounded and integrable on the interval[, ],second derivative of f
exists at a fixed point x∈[, ]and q=qn∈(, )such that qn→as n→ ∞,then
lim
n→∞[n]qn
Dα,β
n,qn(f;x) –f(x)
The proof of the above lemma follows along the lines of [, Theorem ]; thus, we omit the details.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
PP computed the moments of the modified operators, established the rate of convergence and Voronovskaja type asymptotic formula. VM conceived of the study and participated in its design and coordination. VM and PP contributed equally to this work. All the authors read and approved the final manuscript.
Acknowledgements
Dedicated to Prof. Hari M. Srivastava on the occasion of his 72th Birth Anniversary.
Received: 14 January 2013 Accepted: 19 March 2013 Published: 4 April 2013 References
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doi:10.1186/1687-1812-2013-84
Cite this article as:Mishra and Patel:A short note on approximation properties of Stancu generalization of
