Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441http://www.journalofinequalitiesandapplications.com/content/2014/1/441
R E S E A R C H Open Access
Szász-Baskakov type operators based onq-integersPurshottam N Agrawal1, Harun Karsli2 and Meenu Goyal1*
*Correspondence:[email protected] of Mathematics,Indian Institute of TechnologyRoorkee, Roorkee, 247667, IndiaFull list of author information isavailable at the end of the article
AbstractIn the present paper, we introduce the q-analog of the Stancu variant ofSzász-Baskakov operators. We establish the moments of the operators by using theq-derivatives and then prove the basic convergence theorem. Next, the Voronovskajatype theorem and some direct results for the above operators are discussed. Also, therate of convergence and weighted approximation by these operators in terms ofmodulus of continuity are studied. Then we obtain a point-wise estimate using theLipschitz type maximal function. Lastly, we study the A-statistical convergence ofthese operators and also, in order to obtain a better approximation, we study a Kingtype modification of the above operators.MSC: 41A25; 26A15; 40A35
Keywords: Szász-Baskakov operators; rate of convergence; modulus of continuity;weighted approximation; point-wise estimates; Lipschitz type maximal function;statistical convergence
1 IntroductionIn recent years, there has been intensive research on the approximation of functions bypositive linear operators introduced by making use of q-calculus. Lupas [] and Phillips []pioneered the study in this direction by introducing q-analogs of the well-known Bern-stein polynomials. Derriennic [] discussed modified Bernstein polynomials with Jacobiweights. Gupta [] and Gupta and Heping [] proposed the q-analogs of usual and dis-cretely defined Durrmeyer operators and studied their approximation properties. In [],Stancu introduced the positive linear operators P(α,β)
n : C[, ] → C[, ] by modifying theBernstein polynomials as
P(α,β)n (f ; x) =
n∑k=
pn,k(x)f(
k + α
n + β
),
where pn,k(x) =(n
k)xk( – x)n–k , x ∈ [, ] is the Bernstein basis function and α, β are any
two real numbers satisfying ≤ α ≤ β . Recently, Buyukyazici [] introduced the Stancutype generalization of certain q-Baskakov operators and studied some local direct resultsfor these operators. Subsequently, several authors (cf. [–] etc.) have considered such amodification for some other sequences of positive linear operators.
©2014 Agrawal et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.
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To approximate Lebesgue integrable functions defined on [,∞), Gupta et al. [] in-troduced the following positive linear operators:
Mn(f ; x) =∞∑ν=
qn,ν(x)∫ ∞
bn,ν(t)f (t) dt, (.)
where bn,ν(x) = B(ν+,n)
xν
(+x)n+ν+ , qn,ν(x) = e–nx (nx)νν! , x ∈ [,∞), and studied the asymptotic
approximation and error estimation in simultaneous approximation. Subsequently, forf ∈ Cγ [,∞) := {f ∈ C[,∞) : |f (t)| ≤ Mf ( + tγ ) for some Mf > ,γ > }, Gupta [] in-troduced the following operators:
Sn(f ; x) =∞∑ν=
qn,ν(x)∫ ∞
bn,ν–(t)f (t) dt + e–nxf (), (.)
by considering the value of the function at zero explicitly, and studied an estimate of errorin terms of the higher order modulus of continuity in simultaneous approximation fora linear combination of the operators (.), introduced by May []. Later on, Gupta andNoor [] discussed some direct results in a simultaneous approximation for the operators(.).
In [], Aral introduced Szász-Mirakjan operators based on q-integers and establishedsome direct results and gave two representations of the rth q-derivative of these operators.Aral and Gupta [] studied the basic convergence theorem for the rth order q-derivatives,a Voronovskaja type theorem, and some more properties of these operators. Agratini andDogru [] constructed Szász-King type operators and investigated the statistical con-vergence and the rate of local and global convergence for functions with a polynomialgrowth. Örkcü and Dogru [] proposed two different modifications of q-Szász-MirakjanKantorovich operators and obtained the rate of convergence in terms of the modulus ofcontinuity. In [], they introduced Kantorovich type generalization of q-Szász-Mirakjanoperators and discussed their A-statistical approximation properties.
Let the space Cγ [,∞) be endowed with the norm ‖f ‖γ = supt∈[,∞)|f (x)|
(+tγ ) then for f ∈Cγ [,∞), < q < , ≤ α ≤ β and each positive integer n, we introduce the followingStancu type modification of the operators (.) based on q-integers:
B(α,β)n,q (f ; x) =
∞∑ν=
qn,ν(q, x)qν–∫ ∞/A
bn,ν–(q, t)f
(qν[n]qt + α
[n]q + β
)dqt
+ eq(–[n]qx
)f(
α
[n]q + β
), (.)
where qn,ν(q, x) = eq(–[n]qx)[n]νqxν
[ν]q ! and bn,ν(q, t) = tνqν(ν–)/
(+t)n+ν+Bq(ν+,n) .
For α = β = , we denote B(α,β)n,q (f ; x) by Bn,q(f ; x).
Clearly, if q → – and α = β = , the operators defined by (.) reduce to the operatorsgiven by (.).
The purpose of the present paper is to study the basic convergence theorem, Voronovs-kaja type asymptotic formula, local approximation, rate of convergence, weighted approx-imation, point-wise estimation and A-statistical convergence of the operators (.). Fur-
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 3 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
ther, to obtain a better approximation we also propose a modification of these operatorsby using a King type approach.
We recall that the q-analog of beta function of second kind [, ] is defined by
Bq(t, s) = K(A, t)∫ ∞/A
xt–
( + x)t+sq
dqx, (.)
where K (x, t) = x+ xt( +
x )tq( + x)–t
q , and (a + b)sq =
∏s–i=(a + qib), s ∈ Z
+.In particular, for any positive integers n, m, we have
K (x, n) = qn(n–)
, K (x, ) = (.)
and
Bq(m, n) =�q(m)�q(n)�q(m + n)
.
2 Moment estimatesLemma For Bn,q(tm; x), m = , , , one has
() Bn,q(; x) = ;() Bn,q(t; x) = [n]qx
[n–]q, for n > ;
() Bn,q(t; x) = q[n]q
[n–]q[n–]qx + []q[n]q
q[n–]q[n–]qx, for n > .
Proof We observe that Bn,q are well defined for the functions , t, t. Thus, for every x ∈[,∞), using (.) and (.), we obtain
Bn,q(; x) =∞∑ν=
qn,ν(q, x)qν–∫ ∞/A
bn,ν–(q, t) dqt + eq
(–[n]qx
)
=∞∑ν=
qn,ν(q, x) = .
Next, for f (t) = t, again applying (.) and (.), we get
Bn,q(t; x) =∞∑ν=
qn,ν(q, x)qν–∫ ∞/A
bn,ν–(q, t)qνt dqt
=[n]q
[n – ]qx.
Proceeding similarly, we have
Bn,q(t; x
)=
[]q[n]q
q[n – ]q[n – ]qx +
q[n]q
[n – ]q[n – ]qx,
by using [ν + ]q = []q + q[ν]q. �
Lemma For the operators B(α,β)n,q (f ; x) as defined in (.), the following equalities hold:
() B(α,β)n,q (; x) = ;
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 4 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
() B(α,β)n,q (t; x) = [n]
q([n]q+β)[n–]q
x + α[n]q+β
, for n > ;
() B(α,β)n,q (t; x) = ( [n]q
[n]q+β){ (q[n]
q+α[n–]q)[n–]q[n–]q
x + []q[n]qq[n–]q[n–]q
x} + ( α[n]q+β
), for n > .
Proof This lemma is an immediate consequence of Lemma . Hence the details of its proofare omitted. �
Lemma For f ∈ CB[,∞) (the space of all bounded and uniform continuous functionson [,∞) endowed with norm ‖f ‖ = sup{|f (x)| : x ∈ [,∞)}), one has
∥∥B(α,β)n,q (f ; x)
∥∥ ≤ ‖f ‖.
Proof In view of (.) and Lemma , the proof of this lemma easily follows. �
Remark For every q ∈ (, ), we have
B(α,β)n,q
((t – x); x
)=
([n]q – ([n]q + β)[n – ]q)x + α[n – ]q
([n]q + β)[n – ]q, n > ,
and
B(α,β)n,q
((t – x); x
)=
{ +
q[n]q
([n]q + β)[n – ]q[n – ]q+ (α – )
[n]q
([n]q + β)[n – ]q
}x
+{ []q[n]
q
q([n]q + β)[n – ]q[n – ]q–
α
[n]q + β
}x +
α
([n]q + β) , n >
=: γn,q(x),
say.
3 Main resultsTheorem Let < qn < and A > . Then for each f ∈ Cγ [,∞), the sequence {B(α,β)
n,qn (f ; x)}converges to f uniformly on [, A] if and only if limn→∞ qn = .
Proof First, we assume that limn→∞ qn = .We have to show that {B(α,β)
n,qn (f ; x)} converges to f uniformly on [, A].From Lemma , we see that
B(α,β)n,qn (; x) → , B(α,β)
n,qn (t; x) → x, B(α,β)n,qn
(t; x
) → x,
uniformly on [, A] as n → ∞.
Therefore, the well-known property of the Korovkin theorem implies that {B(α,β)n,qn (f ; x)}
converges to f uniformly on [, A] provided f ∈ Cγ [,∞).We show the converse part by contradiction. Assume that qn does not converge to .
Then it must contain a subsequence {qnk } such that qnk ∈ (, ), qnk → a ∈ [, ) as k → ∞.
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Thus, [nk +s]qnk
= –qnk–(qnk )nk +s → ( – a) as k → ∞. Choosing n = nk , q = qnk in B(α,β)
n,qn (t; x),from Lemma , we have
B(α,β)n,qn
(t; x
) → (a + α( – a))x
( + ( – a)β) +( – a)x
a( + ( – a)β) +( – a)α
( + ( – a)β) �= x as k → ∞,
which leads us to a contradiction. Hence, limn→∞ qn = . This completes the proof. �
Theorem (Voronovskaja type theorem) Let f ∈ Cγ [,∞) and qn ∈ (, ) be a sequencesuch that qn → and qn
n → as n → ∞. Suppose that f ′′(x) exists at a point x ∈ [,∞),then we have
limn→∞[n]qn
(B(α,β)
n,qn (f ; x) – f (x))
= (α – βx)f ′(x) + ( – α)x( – x)f ′′(x).
Proof By Taylor’s formula we have
f (t) = f (x) + (t – x)f ′(x) +
f ′′(x)(t – x) + r(t, x)(t – x), (.)
where r(t, x) is the Peano form of the remainder and limt→x r(t, x) = .Applying B(α,β)
n,q (f , x) to the both sides of (.), we have
[n]qn
(B(α,β)
n,qn (f ; x) – f (x))
= [n]qn f ′(x)B(α,β)n,qn
((t – x); x
)+
[n]qn f ′′(x)B(α,β)n,qn
((t – x); x
)+ [n]qn B(α,β)
n,q((t – x)r(t, x); x
).
In view of Remark , we have
limn→∞[n]qn B(α,β)
n,qn
((t – x); x
)= α – βx (.)
and
limn→∞[n]qn B(α,β)
n,qn
((t – x); x
)= x( – x)( – α). (.)
Now, we shall show that
[n]qn B(α,β)n,qn
(r(t, x)(t – x); x
) →
when n → ∞.By using the Cauchy-Schwarz inequality, we have
B(α,β)n,qn
(r(t, x)(t – x); x
) ≤√
B(α,β)n,qn
(r(t, x); x
)√B(α,β)
n,qn
((t – x); x
). (.)
We observe that r(x, x) = and r(·, x) ∈ Cγ [,∞). Then it follows from Theorem that
limn→∞ B(α,β)
n,qn
(r(t, x); x
)= r(x, x) = , (.)
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 6 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
in view of the fact that B(α,β)n,qn ((t – x); x) = O(
[n]qn
). Now, from (.) and (.), we get
limn→∞ B(α,β)
n,qn
(r(t, x)(t – x); x
)= , (.)
and from (.), (.), and (.), we get the required result. �
Theorem (Voronovskaja type theorem) Let f ∈ Cγ [,∞) and qn ∈ (, ) be a sequencesuch that qn → and qn
n → as n → ∞. If f ′′(x) exists on [,∞), then
limn→∞[n]qn
(B(α,β)
n,qn (f ; x) – f (x))
= (α – βx)f ′(x) + ( – α)x( – x)f ′′(x)
holds uniformly on [, A], where A > .
Proof Let x ∈ [, A]. The remainder part of the proof of this theorem is similar to that ofthe proof of the previous theorem. So we omit it. �
3.1 Local approximationFor CB[,∞), let us consider the following K-functional:
K(f , δ) = infg∈W
{‖f – g‖ + δ∥∥g ′′∥∥}
, (.)
where δ > and W = {g ∈ CB[,∞) : g ′′ ∈ CB[,∞)}. By [] there exists an absoluteconstant C > such that
K(f , δ) ≤ Cω(f ,√
δ), (.)
where
ω(f ,√
δ) = sup<h≤√
δ
supx∈[,∞)
∣∣f (x + h) – f (x + h) + f (x)∣∣ (.)
is the second order modulus of smoothness of f . By
ω(f , δ) = sup<h≤δ
supx∈[,∞)
∣∣f (x + h) – f (x)∣∣,
we denote the usual modulus of continuity of f ∈ CB[,∞).
Theorem Let f ∈ CB[,∞) and q ∈ (, ). Then, for every x ∈ [,∞) and n ≥ , we have
∣∣B(α,β)n,q
(f (t); x
)– f (x)
∣∣ ≤ Cω(f , δn(x)
)+ ω
(f ,
([n]qqn– – β[n – ]q)x + α[n – ]q
([n]q + β)[n – ]q
),
where C is an absolute constant and
δn(x) =(
B(α,β)n,q
((t – x); x
)+
[([n]qqn– – β[n – ]q)x + α[n – ]q
([n]q + β)[n – ]q
])/
.
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Proof For x ∈ [,∞), we consider the auxiliary operators B(α,β)n,q defined by
B(α,β)n,q (f ; x) = B(α,β)
n,q (f ; x) – f( [n]
qx([n]q + β)[n – ]q
+α
[n]q + β
)+ f (x). (.)
From Lemma , we observe that the operators B(α,β)n,q are linear and reproduce the linear
functions.Hence
B(α,β)n,q
((t – x); x
)= . (.)
Let g ∈ W . By Taylor’s theorem, we have
g(t) = g(x) + g ′(x)(t – x) +∫ t
x(t – u)g ′′(u) du, t ∈ [,∞).
Applying B(α,β)n,q to the both sides of the above equation and using (.), we have
B(α,β)n,q (g; x) = g(x) + B(α,β)
n,q
(∫ t
x(t – u)g ′′(u) du; x
).
Thus, by (.) we get
∣∣B(α,β)n,q (g; x) – g(x)
∣∣≤ B(α,β)
n,q
(∫ t
x|t – u|∣∣g ′′(u)
∣∣du; x)
+∫ [n]qx
([n]q+β)[n–]q + α[n]q+β
x
∣∣∣∣ [n]qx
([n]q + β)[n – ]q+
α
[n]q + β– u
∣∣∣∣∣∣g ′′(u)∣∣du
≤[
B(α,β)n,q
((t – x); x
)+
( [n]qx
([n]q + β)[n – ]q+
α
[n]q + β– x
)]∥∥g ′′∥∥≤ δ
n(x)∥∥g ′′∥∥. (.)
On other hand, by (.) and Lemma , we have
∣∣B(α,β)n,q (f ; x)
∣∣ ≤ ∣∣B(α,β)n,q (f ; x)
∣∣ + ‖f ‖ ≤ ‖f ‖. (.)
Using (.) and (.) in (.), we obtain
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ ∣∣B(α,β)n,q (f – g; x)
∣∣ +∣∣(f – g)(x)
∣∣ +∣∣B(α,β)
n,q (g; x) – (g)(x)∣∣
+∣∣∣∣f
( [n]qx
([n]q + β)[n – ]q+
α
[n]q + β
)– f (x)
∣∣∣∣≤ ‖f – g‖ + δ
n(x)∥∥g ′′∥∥ +
∣∣∣∣f( [n]
qx + α[n – ]q
([n]q + β)[n – ]q
)– f (x)
∣∣∣∣.Hence, taking the infimum on the right hand side over all g ∈ W and using (.), we getthe required result. �
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Theorem Let f ∈ C[,∞), qn ∈ (, ) and ωa+(f , δ) be its modulus of continuity on thefinite interval [, a + ] ⊂ [,∞), where a > . Then, for every n > ,
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤ Mf( + a)γn,qn (x) + ωa+
(f ,
√γn,qn (x)
),
where γn,qn (x) is defined in Remark .
Proof From [], for x ∈ [, a] and t ∈ [,∞), we get
∣∣f (t) – f (x)∣∣ ≤ Mf
( + a)(t – x) +
( +
|t – x|δ
)ωa+(f , δ), δ > .
Thus, by applying the Cauchy-Schwarz inequality, we have
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤ Mf( + a)(B(α,β)
n,qn (t – x); x)
+ ωa+(f , δ)(
+δ
(B(α,β)
n,qn (t – x); x)
)
= Mf( + a)γn,qn (x) + ωa+
(f ,
√γn,qn (x)
),
on choosing δ =√
γn,qn (x). This completes the proof of the theorem. �
3.2 Weighted approximationLet C∗
[,∞) := {f ∈ C[,∞) : limx→∞ |f (x)|+x < ∞}. Throughout the section, we assume
that {qn} is a sequence in (, ) such that qn → .
Theorem For each f ∈ C∗ [,∞), we have
limn→∞
∥∥B(α,β)n,qn (f ) – f
∥∥ = .
Proof Making use of the Korovkin type theorem on weighted approximation [], we seethat it is sufficient to verify the following three conditions:
limn→∞
∥∥B(α,β)n,qn
(tk ; x
)– xk∥∥
= , k = , , . (.)
Since B(α,β)n,qn (; x) = , the condition in (.) holds for k = .
By Lemma , we have for n >
∥∥B(α,β)n,qn (t; x) – x
∥∥
≤∣∣∣∣ [n]qn qn–
n – β[n – ]qn
([n]qn + β)[n – ]qn
∣∣∣∣ supx∈[,∞)
x + x +
α
[n]qn + βsup
x∈[,∞)
+ x
≤∣∣∣∣ [n]qn qn–
n – β[n – ]qn
([n]qn + β)[n – ]qn
∣∣∣∣ +α
[n]qn + β,
which implies that the condition in (.) holds for k = .
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Similarly, we can write for n >
∥∥B(α,β)n,qn
(t; x
)– x∥∥
≤∣∣∣∣ qn[n]
qn
[n – ]qn [n – ]qn+
α
[n – ]qn–
∣∣∣∣+
[]qn [n]qn
qn[n – ]qn [n – ]qn+
α
([n]qn + β) ,
which implies that limn→∞ ‖B(α,β)n,qn (t; x) – x‖ = , (.) holds for k = . �
Now, we present a weighted approximation theorem for functions in C∗ [,∞). Such
type of results are proved in [] for classical Szász operators.
Theorem For each f ∈ C∗ [,∞) and α > , we have
limn→∞ sup
x∈[,∞)
|B(α,β)n,qn (f ; x) – f (x)|
( + x)+α= .
Proof Let x ∈ [,∞) be arbitrary but fixed. Then
supx∈[,∞)
|B(α,β)n,qn (f ; x) – f (x)|
( + x)+α≤ sup
x≤x
|B(α,β)n,qn (f ; x) – f (x)|
( + x)+α+ sup
x>x
|B(α,β)n,qn (f ; x) – f (x)|
( + x)+α
≤ ∥∥B(α,β)n,qn (f ) – f
∥∥C[,x] + ‖f ‖ sup
x>x
|B(α,β)n,qn ( + t; x)|( + x)+α
+ supx>x
|f (x)|( + x)+α
. (.)
Since |f (x)| ≤ ‖f ‖( + x), we have supx>x|f (x)|
(+x)+α ≤ ‖f ‖(+x
)α .Let ε > be arbitrary. We can choose x to be so large that
‖f ‖
( + x)α
<ε
. (.)
In view of Theorem , we obtain
‖f ‖ limn→∞
|B(α,β)n,qn ( + t; x)|( + x)+α
= + x
( + x)+α‖f ‖ =
‖f ‖
( + x)α≤ ‖f ‖
( + x)α
<ε
. (.)
Using Theorem , we can see that the first term of the inequality (.), implies that
∥∥B(α,β)n,qn (f ) – f
∥∥C[,x] <
ε
, as n → ∞. (.)
Combining (.)-(.), we get the desired result. �
For f ∈ C∗ [,∞), the weighted modulus of continuity is defined as
(f , δ) = supx≥,<h≤δ
|f (x + h) – f (x)| + (x + h) .
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Lemma [] If f ∈ C∗ [,∞) then
(i) (f , δ) is monotone increasing function of δ,(ii) limδ→+ (f , δ) = ,
(iii) for any λ ∈ [,∞), (f ,λδ) ≤ ( + λ)(f , δ).
Theorem If f ∈ C∗ [,∞), then for sufficiently large n we have
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤ K( + x+λ
)(f , δn), x ∈ [,∞),
where λ ≥ , δn = max{αn,βn,γn}, αn, βn, γn being
( qn[n]qn + α[n – ]qn [n]
qn
[n – ]qn [n – ]qn ([n]qn + β) + )
,( [n]
qn
qn[n – ]qn [n – ]qn ([n]qn + β)
)and
α
([n]qn + β) ,
respectively, and K is a positive constant independent of f and n.
Proof From the definition of (f , δ) and Lemma , we have
∣∣f (t) – f (x)∣∣ ≤ (
+(x + |t – x|))( +
|t – x|δ
)(f , δ)
≤ ( + (x + t))( +
|t – x|δ
)(f , δ)
:= φx(t)(
+δψx(t)
)(f , δ),
where φx(t) = + (x + t) and ψx(t) = |t – x|. Then we obtain
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤(
B(α,β)n,qn (φx; x) +
δn
B(α,β)n,qn (φxψx; x)
)(f , δn).
Now, applying the Cauchy-Schwarz inequality to the second term on the right hand side,we get
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣≤
(B(α,β)
n,qn (φx; x) +δn
√B(α,β)
n,qn
(φ
x ; x)√
B(α,β)n,qn
(ψ
x ; x))
(f , δn). (.)
From Lemma ,
+ x B(α,β)
n,qn
( + t; x
)=
+ x +
( qn[n]qn + α[n – ]qn [n]
qn
[n – ]qn [n – ]qn ([n]qn + β)
)x
+ x
+[]qn [n]
qn
qn[n – ]qn [n – ]qn ([n]qn + β)x
+ x +α
([n]qn + β)
+ x
≤ + C, for sufficiently large n, (.)
where C is a positive constant.
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 11 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
From (.), there exists a positive constant K such that B(α,β)n,qn (φx; x) ≤ K( + x), for
sufficiently large n.Proceeding similarly,
+x B(α,β)n,qn ( + t; x) ≤ + C, for sufficiently large n, where C is a
positive constant.
So there exists a positive constant K, such that√
B(α,β)n,qn (φ
x ; x) ≤ K( + x), where x ∈[,∞) and n is large enough. Also, we get
B(α,β)n,qn
(ψ
x ; x)
={ qn[n]
qn + α[n – ]qn [n]qn
[n – ]qn [n – ]qn ([n]qn + β) + –[n]
qn
([n]qn + β)[n – ]qn
}x
+{ [n]
qn
qn[n – ]qn [n – ]qn ([n]qn + β) –α
[n]qn + β
}x +
α
([n]qn + β)
≤ αnx + βnx + γn.
Hence, from (.), we have
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤ ( + x)(K +
δn
K√
αnx + βnx + γn
)(f , δn).
If we take δn = max{αn,βn,γn}, then we get
∣∣B(α,β)n,qn (f ; x) – f (x)
∣∣ ≤ ( + x)(K + K
√x + x +
)(f , δn)
≤ K( + x+λ
)(f , δn), for sufficiently large n and x ∈ [,∞).
Hence, the proof is completed. �
3.3 Point-wise estimatesIn this section, we establish some point-wise estimates of the rate of convergence of theoperators B(α,β)
n,q . First, we give the relationship between the local smoothness of f and localapproximation.
We know that a function f ∈ C[,∞) is in LipM η on E, η ∈ (, ], E be any boundedsubset of the interval [,∞) if it satisfies the condition
∣∣f (t) – f (x)∣∣ ≤ M|t – x|η, t ∈ [,∞) and x ∈ E,
where M is a constant depending only on α and f .
Theorem Let f ∈ C[,∞) ∩ LipM η, E ⊂ [,∞) and α ∈ (, ]. Then we have
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ M(γ η/
n,q (x) + dη(x, E)), x ∈ [,∞),
where M is a constant depending on η and f , and d(x, E) is the distance between x and Edefined as
d(x, E) = inf{|t – x| : t ∈ E
}.
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 12 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
Proof Let E be the closure of E in [,∞). Then there exists at least one point x ∈ E suchthat
d(x, E) = |x – x|.
By our hypothesis and the monotonicity of B(α,β)n,q , we get
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ B(α,β)n,q
(∣∣f (t) – f (x)∣∣; x
)+ B(α,β)
n,q(∣∣f (x) – f (x)
∣∣; x)
≤ M{
B(α,β)n,q
(|t – x|η; x)
+ |x – x|η}
≤ M{
B(α,β)n,q
(|t – x|η; x)
+ |x – x|η}
.
Now, applying Hölder’s inequality with p = η
and q = –
p , we obtain
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ M{[
B(α,β)n,q
(|t – x|; x)]η/ + dη(x, E)
},
from which the desired result immediate. �
Next, we obtain the local direct estimate of the operators defined in (.), using theLipschitz-type maximal function of order η introduced by Lenze [] as
ωη(f , x) = supt �=x,t∈[,∞)
|f (t) – f (x)||t – x|η , x ∈ [,∞) and η ∈ (, ]. (.)
Theorem Let f ∈ CB[,∞) and < η ≤ . Then, for all x ∈ [,∞) we have
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ ωη(f , x)γ η/n,q (x).
Proof From (.), we have
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ ωη(f , x)B(α,β)n,q
(|t – x|η; x).
Applying Hölder’s inequality with p = η
and q = –
p , we get
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ ωη(f , x)B(α,β)n,q
((t – x); x
) η ≤ ωη(f , x)γ η/
n,q (x).
Thus, the proof is completed. �
4 Statistical convergenceLet A = (ank) be a non-negative infinite summability matrix. For a given sequence x := (x)n,the A-transform of x denoted by Ax : (Ax)n is defined as
(Ax)n =∞∑
k=
ankxk
provided the series converges for each n. A is said to be regular if limn(Ax)n = L wheneverlimn(x)n = L. Then x = (x)n is said to be A-statistically convergent to L i.e. stA- limn(x)n = L
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 13 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
if for every ε > , limn∑
k:|xk–L|≥ε ank = . If we replace A by C then A is a Cesaro matrix oforder one and A-statistical convergence is reduced to the statistical convergence. Similarly,if A = I , the identity matrix, then A-statistical convergence is called ordinary convergence.Kolk [] proved that statistical convergence is better than ordinary convergence. In thisdirection, significant contributions have been made by (cf. [, , –] etc.).
Let qn ∈ (, ) be a sequence such that
stA- limn
qn = , stA- limn
qnn = a (a < ) and stA- lim
n
[n]qn
= . (.)
Theorem Let A = (ank) be a non-negative regular summability matrix and (qn) be asequence satisfying (.). Then, for any compact set K ⊂ [,∞) and for each function f ∈C(K ), we have
stA- limn
∥∥B(α,β)n,qn (f ; ·) – f
∥∥ = .
Proof Let x = maxx∈K x. From Lemma , stA- limn ‖B(α,β)n,qn (e; ·) – e‖ = . Again, by
Lemma , we have
supx∈K
∣∣B(α,β)n,qn (e; x) – e(x)
∣∣ ≤∣∣∣∣ [n]
qn
([n]qn + β)[n – ]qn–
∣∣∣∣x +α
[n]qn + β.
For ε > , let us define the following sets:
G :={
k :∥∥B(α,β)
k,qk(e; ·) – e
∥∥ ≥ ε}
,
G :={
k :∣∣∣∣ [k]
qk
([k]qk + β)[k – ]qk
– ∣∣∣∣ ≥ ε
},
G :={
k :α
[k]qk + β≥ ε
},
which implies that G ⊆ G ∪ G and hence for all n ∈N, we obtain
∑k∈G
ank ≤∑k∈G
ank +∑k∈G
ank .
Hence, taking the limit as n → ∞, we have stA- limn ‖B(α,β)n,qn (e; ·) – e‖ = .
Similarly, by using Lemma , we have
supx∈K
∣∣B(α,β)n,qn (e; x) – e(x)
∣∣ ≤∣∣∣∣ qn[n]
qn + α[n – ]qn
[n – ]qn [n – ]qn ([n]qn + β) – ∣∣∣∣x
+[]qn [n]
qn
qn[n – ]qn [n – ]qn ([n]qn + β) x +α
([n]qn + β) .
Now, let us define the following sets:
M :={
k :∥∥B(α,β)
k,qk(e; ·) – e
∥∥ ≥ ε}
,
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 14 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
M :={
k :∣∣∣∣ qk[k]
qk+ α[k – ]qk
[k – ]qk [k – ]qk ([k]qk + β) – ∣∣∣∣ ≥ ε
},
M :={
k :[]qk [k]
qk
qk[k – ]qk [k – ]qk ([k]qk + β) ≥ ε
},
M :={
k :α
([k]qk + β) ≥ ε
}.
Then we obtain M ⊆ M ∪ M ∪ M, which implies that
∑k∈M
ank ≤∑
k∈M
ank +∑
k∈M
ank +∑
k∈M
ank .
Thus, as n → ∞ we get stA- limn ‖B(α,β)n,qn (e; ·) – e‖ = . This completes the proof. �
Theorem Let A = (ank) be a non-negative regular summability matrix and (qn) be asequence in (, ) satisfying (.). Let the operators B(α,β)
n,qn , n ∈N, be defined as in (.). Then,for each function f ∈ C[,∞), we have
stA- limn
∥∥B(α,β)n,qn (f ; ·) – f
∥∥ρ
= ,
where ρ(x) = + x+λ, λ > .
Proof From [, p., Theorem ], it is sufficient to prove that stA- limn ‖B(α,β)n,qn (ei; ·)–ei‖ =
, where ei(x) = xi, i = , , .From Lemma , stA- limn ‖B(α,β)
n,qn (e; ·) – e‖ = holds.Again using Lemma , we have
∥∥B(α,β)n,qn (e; ·) – e
∥∥ ≤ sup
x∈[,∞)
{x
( + x)
∣∣∣∣ [n]qn
([n]qn + β)[n – ]qn–
∣∣∣∣ +
( + x)α
([n]qn + β)
}
≤∣∣∣∣ [n]
qn
([n]qn + β)[n – ]qn–
∣∣∣∣ +α
([n]qn + β). (.)
For each ε > , let us define the following sets:
G :={
k :∥∥B(α,β)
k,qk(e; ·) – e
∥∥ ≥ ε}
,
G :={
k :∣∣∣∣ [k]
qk
([k]qk + β)[k – ]qk
– ∣∣∣∣ ≥ ε
},
G :={
k :α
[k]qk + β≥ ε
},
which yields G ⊆ G ∪ G in view of (.) and therefore for all n ∈N, we have
∑k∈G
ank ≤∑k∈G
ank +∑k∈G
ank .
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 15 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
Hence, on taking the limit as n → ∞, stA- limn ‖B(α,β)n,qn (e; ·) – e‖ = . Proceeding similarly,
∥∥B(α,β)n,qn (e; ·) – e
∥∥ ≤
∣∣∣∣ qn[n]qn + α[n – ]qn
[n – ]qn [n – ]qn ([n]qn + β) – ∣∣∣∣
+[]qn [n]
qn
qn[n – ]qn [n – ]qn ([n]qn + β) +α
([n]qn + β) .
Now, let us define the following sets:
M :={
k :∥∥B(α,β)
k,qk(e; ·) – e
∥∥ ≥ ε}
,
M :={
k :∣∣∣∣ qk[k]
qk+ α[k – ]qk
[k – ]qk [k – ]qk ([k]qk + β) – ∣∣∣∣ ≥ ε
},
M :={
k :[]qk [k]
qk
qk[k – ]qk [k – ]qk ([k]qk + β) ≥ ε
},
M :={
k :α
([k]qk + β) ≥ ε
}.
Then we obtain M ⊆ M ∪ M ∪ M, which implies that
∑k∈M
ank ≤∑
k∈M
ank +∑
k∈M
ank +∑
k∈M
ank .
Hence, taking the limit as n → ∞ we get stA- limn ‖B(α,β)n,qn (e; ·) – e‖ = . This completes
the proof of the theorem. �
5 Better estimatesIt is well known that the classical Bernstein polynomials preserve constant as well as linearfunctions. To make the convergence faster, King [] proposed an approach to modifythe Bernstein polynomials, so that the sequence preserves test functions e and e, whereei(t) = ti, i = , , . As the operator B(α,β)
n,q (f ; x) defined in (.) reproduces only constantfunctions, this motivated us to propose the modification of this operator, so that it canpreserve constant as well as linear functions.
The modification of the operators given in (.) is defined as
B(α,β)n,q (f ; x) =
∞∑ν=
qn,ν(rq
n(x))qν–
∫ ∞/A
bn,ν–(q, t)f
(qν[n]qt + α
[n]q + β
)dqt
+ eq(–[n]qrq
n(x))f(
α
[n]q + β
),
where rqn(x) = ([n]q+β)[n–]qx–α[n–]q
[n]q
for x ∈ In = [ α[n]q+β
,∞) and n > .
Lemma For each x ∈ In, by simple computations, we have() B(α,β)
n,q (; x) = ;() B(α,β)
n,q (t; x) = x;
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() B(α,β)n,q (t; x) = [n–]q(q[n]
q+α[n–]q)[n–]q[n]
qx + []q[n]
q–qα[n–]q(q[n]q+α[n–]q)
q([n]q+β)[n–]q[n]q
x +α[n]
q(q[n–]q+[n–]q)+α[n–]q[n–]q([n]q+β)[n]
q[n–]q.
Consequently, for each x ∈ In, we have the following equalities:
B(α,β)n,q (t – x; x) = ,
B(α,β)n,q
((t – x); x
)=
[n]q(q[n – ]q – [n – ]q) + α[n – ]q[n – ]q
[n – ]q[n]q
x
+[]q[n]
q – qα[n – ]q(q[n]q – α[n – ]q)
q([n]q + β)[n – ]q[n]q
x
+α[n]
q(q[n – ]q + [n – ]q) + α[n – ]q[n – ]q
([n]q + β)[n]q[n – ]q
=: ξn(x), (.)
say.
Theorem Let f ∈ CB[,∞) and x ∈ In. Then there exists a positive constant C such that
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ Cω(f ,
√ξn(x)
),
where ξn(x) is given by (.).
Proof Let g ∈ W , x ∈ In and t ∈ [,∞). Using Taylor’s expansion we have
g(t) = g(x) + (t – x)g ′(x) +∫ t
x(t – u)g ′′(u) du.
Applying B(α,β)n,q on both sides and using Lemma , we get
B(α,β)n,q (g; x) – g(x) = B(α,β)
n,q((t – x); x
)g ′(x) + B(α,β)
n,q
(∫ t
x(t – u)g ′′(u) du; x
).
Obviously, we have | ∫ tx (t – u)g ′′(u) du| ≤ (t – x)‖g ′′‖. Therefore
∣∣B(α,β)n,q (g; x) – g(x)
∣∣ ≤ B(α,β)n,q
((t – x); x
)∥∥g ′′∥∥ = ξn(x)∥∥g ′′∥∥.
Since |B(α,β)n,q (f ; x)| ≤ ‖f ‖, we get
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ ∣∣B(α,β)n,q (f – g; x)
∣∣ +∣∣(f – g)(x)
∣∣ +∣∣B(α,β)
n,q (g; x) – g(x)∣∣
≤ ‖f – g‖ + ξn(x)∥∥g ′′∥∥.
Finally, taking the infimum over all g ∈ W and using (.)-(.) we obtain
∣∣B(α,β)n,q (f ; x) – f (x)
∣∣ ≤ Cω(f ,
√ξn(x)
),
which proves the theorem. �
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441 Page 17 of 18http://www.journalofinequalitiesandapplications.com/content/2014/1/441
Competing interestsThe authors declare that they have no competing interests.
Authors’ contributionsAll authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details1Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India. 2Department ofMathematics, Faculty of Science and Arts, Abant Izzet Baysal University, Bolu, 14280, Turkey.
AcknowledgementsThe authors are extremely grateful to the reviewers for a careful reading of the manuscript and making valuablesuggestions leading to a better presentation of the paper. The last author is thankful to the ‘Council of Scientific andIndustrial Research’, India, for financial support to carry out the above research work.
Received: 18 June 2014 Accepted: 1 October 2014 Published: 03 Nov 2014
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10.1186/1029-242X-2014-441Cite this article as: Agrawal et al.: Szász-Baskakov type operators based on q-integers. Journal of Inequalities andApplications 2014, 2014:441
