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Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
http://www.journalofinequalitiesandapplications.com/content/2014/1/441
RESEARCH
Open Access
Szász-Baskakov type operators based on
q-integers
Purshottam N Agrawal1 , Harun Karsli2 and Meenu Goyal1*
*
Correspondence:
[email protected]
1
Department of Mathematics,
Indian Institute of Technology
Roorkee, Roorkee, 247667, India
Full list of author information is
available at the end of the article
Abstract
In the present paper, we introduce the q-analog of the Stancu variant of
Szász-Baskakov operators. We establish the moments of the operators by using the
q-derivatives and then prove the basic convergence theorem. Next, the Voronovskaja
type theorem and some direct results for the above operators are discussed. Also, the
rate of convergence and weighted approximation by these operators in terms of
modulus of continuity are studied. Then we obtain a point-wise estimate using the
Lipschitz type maximal function. Lastly, we study the A-statistical convergence of
these operators and also, in order to obtain a better approximation, we study a King
type 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 Introduction
In recent years, there has been intensive research on the approximation of functions by
positive 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 Bernstein polynomials. Derriennic [] discussed modified Bernstein polynomials with Jacobi
weights. Gupta [] and Gupta and Heping [] proposed the q-analogs of usual and discretely defined Durrmeyer operators and studied their approximation properties. In [],
(α,β)
Stancu introduced the positive linear operators Pn : C[, ] → C[, ] by modifying the
Bernstein polynomials as
Pn(α,β) (f ; x) =
n
k=
where pn,k (x) =
n
k
pn,k (x)f
k+α
,
n+β
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 Stancu
type generalization of certain q-Baskakov operators and studied some local direct results
for these operators. Subsequently, several authors (cf. [–] etc.) have considered such a
modification 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 Commons
Attribution 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.
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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Page 2 of 18
To approximate Lebesgue integrable functions defined on [, ∞), Gupta et al. [] introduced the following positive linear operators:
Mn (f ; x) =
∞
∞
qn,ν (x)
bn,ν (t)f (t) dt,
(.)

ν=
ν
ν

x
where bn,ν (x) = B(ν+,n)
, qn,ν (x) = e–nx (nx)
, x ∈ [, ∞), and studied the asymptotic
ν!
(+x)n+ν+
approximation and error estimation in simultaneous approximation. Subsequently, for
f ∈ Cγ [, ∞) := {f ∈ C[, ∞) : |f (t)| ≤ Mf ( + t γ ) for some Mf > , γ > }, Gupta [] introduced the following operators:
Sn (f ; x) =
∞
qn,ν (x)
∞
bn,ν– (t)f (t) dt + e–nx f (),
(.)

ν=
by considering the value of the function at zero explicitly, and studied an estimate of error
in terms of the higher order modulus of continuity in simultaneous approximation for
a linear combination of the operators (.), introduced by May []. Later on, Gupta and
Noor [] discussed some direct results in a simultaneous approximation for the operators
(.).
In [], Aral introduced Szász-Mirakjan operators based on q-integers and established
some 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 and
Doˇgru [] constructed Szász-King type operators and investigated the statistical convergence and the rate of local and global convergence for functions with a polynomial
growth. Örkcü and Doˇgru [] proposed two different modifications of q-Szász-Mirakjan
Kantorovich operators and obtained the rate of convergence in terms of the modulus of
continuity. In [], they introduced Kantorovich type generalization of q-Szász-Mirakjan
operators and discussed their A-statistical approximation properties.
|f (x)|
Let the space Cγ [, ∞) be endowed with the norm f γ = supt∈[,∞) (+t
γ ) then for f ∈
Cγ [, ∞),  < q < ,  ≤ α ≤ β and each positive integer n, we introduce the following
Stancu type modification of the operators (.) based on q-integers:
B(α,β)
n,q (f ; x) =
∞
ν=
+ eq –[n]q x f
∞/A
qn,ν (q, x)qν–
bn,ν– (q, t)f

qν [n]q t + α
dq t
[n]q + β
α
,
[n]q + β
ν qν(ν–)/
eq (–[n]q x)[n]νq xν
and bn,ν (q, t) = (+t)tn+ν+
.
[ν]q !
Bq (ν+,n)
(α,β)
we denote Bn,q (f ; x) by Bn,q (f ; x).
–
(.)
where qn,ν (q, x) =
For α = β = ,
Clearly, if q →  and α = β = , the operators defined by (.) reduce to the operators
given by (.).
The purpose of the present paper is to study the basic convergence theorem, Voronovskaja type asymptotic formula, local approximation, rate of convergence, weighted approximation, point-wise estimation and A-statistical convergence of the operators (.). Fur-
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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Page 3 of 18
ther, to obtain a better approximation we also propose a modification of these operators
by using a King type approach.
We recall that the q-analog of beta function of second kind [, ] is defined by
∞/A
Bq (t, s) = K(A, t)

xt–
dq x,
( + x)t+s
q
(.)
s–
 t
s
i
+
where K(x, t) = x+
x ( + x )tq ( + x)–t
q , and (a + b)q =
i= (a + q b), s ∈ Z .
In particular, for any positive integers n, m, we have
K(x, n) = q
n(n–)

,
K(x, ) = 
(.)
and
Bq (m, n) =
q (m)q (n)
.
q (m + n)
2 Moment estimates
Lemma  For Bn,q (t m ; x), m = , , , one has
() Bn,q (; x) = ;
[n]q x
() Bn,q (t; x) = [n–]
, for n > ;
q
() Bn,q (t  ; x) =
q[n]q
x
[n–]q [n–]q
+
[]q [n]q
x,
q[n–]q [n–]q
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) =
∞
qn,ν (q, x)qν–
∞
bn,ν– (q, t) dq t + eq –[n]q x

ν=
=
∞/A
qn,ν (q, x) = .
ν=
Next, for f (t) = t, again applying (.) and (.), we get
Bn,q (t; x) =
∞
qn,ν (q, x)qν–
bn,ν– (q, t)qν t dq t

ν=
=
∞/A
[n]q
x.
[n – ]q
Proceeding similarly, we have
Bn,q t  ; x =
q[n]q
[]q [n]q
x+
x ,
q[n – ]q [n – ]q
[n – ]q [n – ]q
by using [ν + ]q = []q + q [ν]q .
(α,β)
Lemma  For the operators Bn,q (f ; x) as defined in (.), the following equalities hold:
(α,β)
() Bn,q (; x) = ;
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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[n]q
x + [n]αq +β , for n > ;
([n]q +β)[n–]q
[n]q  (q[n]q +α[n–]q ) 
[] [n]q
(α,β)
) { [n–]q [n–]q x + q[n–]qq [n–]
x} + ( [n]αq +β ) ,
Bn,q (t  ; x) = ( [n]q +β
q
(α,β)
() Bn,q (t; x) =
()
for n > .
Proof This lemma is an immediate consequence of Lemma . Hence the details of its proof
are omitted.
Lemma  For f ∈ CB [, ∞) (the space of all bounded and uniform continuous functions
on [, ∞) endowed with norm f = sup{|f (x)| : x ∈ [, ∞)}), one has
(α,β)
B
n,q (f ; x) ≤ f .
Proof In view of (.) and Lemma , the proof of this lemma easily follows.
Remark  For every q ∈ (, ), we have
([n]q – ([n]q + β)[n – ]q )x + α[n – ]q
B(α,β)
,
n,q (t – x); x =
([n]q + β)[n – ]q
n > ,
and

B(α,β)
n,q (t – x) ; x
[n]q
q[n]q
x
+
(α
–
)
= +
([n]q + β) [n – ]q [n – ]q
([n]q + β) [n – ]q
[]q [n]q
α
α
x+
+
–
, n>

q([n]q + β) [n – ]q [n – ]q [n]q + β
([n]q + β)
=: γn,q (x),
say.
3 Main results
(α,β)
Theorem  Let  < qn <  and A > . Then for each f ∈ Cγ [, ∞), the sequence {Bn,qn (f ; x)}
converges to f uniformly on [, A] if and only if limn→∞ qn = .
Proof First, we assume that limn→∞ qn = .
(α,β)
We have to show that {Bn,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 → ∞.
(α,β)
Therefore, the well-known property of the Korovkin theorem implies that {Bn,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 → ∞.
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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Thus,

[nk +s]qn
=
k
–qnk
–(qnk )nk +s
Page 5 of 18
(α,β)
→ ( – a) as k → ∞. Choosing n = nk , q = qnk in Bn,qn (t  ; x),
from Lemma , we have
 (a + α( – a))x
( – a )x
( – a) α 
+
+
= x
B(α,β)
n,qn t ; x →
( + ( – a)β)
a( + ( – a)β) ( + ( – a)β)
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 sequence
such that qn →  and qnn →  as n → ∞. Suppose that f (x) exists at a point x ∈ [, ∞),
then we have
lim [n]qn B(α,β)
n,qn (f ; x) – f (x) = (α – βx)f (x) + ( – α)x( – x)f (x).
n→∞
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) = .
(α,β)
Applying Bn,q (f , x) to the both sides of (.), we have

(α,β)
(α,β)

[n]qn B(α,β)
n,qn (f ; x) – f (x) = [n]qn f (x)Bn,qn (t – x); x + [n]qn f (x)Bn,qn (t – x) ; x

(α,β)

+ [n]qn Bn,q (t – x) r(t, x); x .
In view of Remark , we have
lim [n]qn B(α,β)
n,qn (t – x); x = α – βx
(.)

lim [n]qn B(α,β)
n,qn (t – x) ; x = x( – x)( – α).
(.)
n→∞
and
n→∞
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
(α,β) (α,β) 
 (t, x); x
;
x
≤
B
Bn,qn (t – x) ; x .
r(t,
x)(t
–
x)
r
B(α,β)
n,q
n
n,qn
(.)
We observe that r (x, x) =  and r (·, x) ∈ Cγ [, ∞). Then it follows from Theorem  that


lim B(α,β)
n,qn r (t, x); x = r (x, x) = ,
n→∞
(.)
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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(α,β)
in view of the fact that Bn,qn ((t – x) ; x) = O( [n] ). Now, from (.) and (.), we get
qn

lim B(α,β)
n,qn r(t, x)(t – x) ; x = ,
n→∞
(.)
and from (.), (.), and (.), we get the required result.
Theorem  (Voronovskaja type theorem) Let f ∈ Cγ [, ∞) and qn ∈ (, ) be a sequence
such that qn →  and qnn →  as n → ∞. If f (x) exists on [, ∞), then
lim [n]qn B(α,β)
n,qn (f ; x) – f (x) = (α – βx)f (x) + ( – α)x( – x)f (x)
n→∞
holds uniformly on [, A], where A > .
Proof Let x ∈ [, A]. The remainder part of the proof of this theorem is similar to that of
the proof of the previous theorem. So we omit it.
3.1 Local approximation
For CB [, ∞), let us consider the following K -functional:
K (f , δ) = inf f – g + δ g ,
g∈W 
(.)
where δ >  and W  = {g ∈ CB [, ∞) : g ∈ CB [, ∞)}. By [] there exists an absolute
constant C >  such that
√
K (f , δ) ≤ Cω (f , δ),
(.)
where
√
ω (f , δ) = sup√
sup f (x + h) – f (x + h) + f (x)
(.)
<h≤ δ x∈[,∞)
is the second order modulus of smoothness of f . By
ω(f , δ) = sup sup f (x + h) – f (x),
<h≤δ 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
n–
(α,β) B
≤ Cω f , δn (x) + ω f , ([n]q q – β[n – ]q )x + α[n – ]q ,
f
(t);
x
–
f
(x)
n,q
([n]q + β)[n – ]q
where C is an absolute constant and
([n]q qn– – β[n – ]q )x + α[n – ]q  /

;
x
+
.
(t
–
x)
δn (x) = B(α,β)
n,q
([n]q + β)[n – ]q
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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(α,β)
Proof For x ∈ [, ∞), we consider the auxiliary operators Bn,q defined by
(α,β)
[n]q x
Bn,q (f ; x) = B(α,β)
n,q (f ; x) – f
([n]q + β)[n – ]q
+
α
[n]q + β
+ f (x).
(.)
(α,β)
From Lemma , we observe that the operators Bn,q are linear and reproduce the linear
functions.
Hence
(α,β) Bn,q
(t – x); x = .
(.)
Let g ∈ W  . By Taylor’s theorem, we have
g(t) = g(x) + g (x)(t – x) +
t
(t – u)g (u) du,
t ∈ [, ∞).
x
(α,β)
Applying Bn,q to the both sides of the above equation and using (.), we have
(α,β)
(α,β)
Bn,q (g; x) = g(x) + Bn,q
t
(t – u)g (u) du; x .
x
Thus, by (.) we get
(α,β)
B
n,q (g; x) – g(x)
t
g (u) du; x
≤ B(α,β)
|t
–
u|
n,q
x
[n]q x
α
+
([n] + β)[n – ] + [n] + β – u g (u) du
q
q
q
x
 [n]q x
α

g –
x
;
x
+
+
(t
–
x)
≤ B(α,β)
n,q
([n]q + β)[n – ]q [n]q + β
≤ δn (x)g .
[n]
qx
α
([n]q +β)[n–]q + [n]q +β
(.)
On other hand, by (.) and Lemma , we have
(α,β)
(α,β)
B
n,q (f ; x) ≤ Bn,q (f ; x) + f ≤ f .
(.)
Using (.) and (.) in (.), we obtain
(α,β)
(α,β)
(α,β)
B
n,q (f ; x) – f (x) ≤ Bn,q (f – g; x) + (f – g)(x) + Bn,q (g; x) – (g)(x)
[n]q x
α
– f (x)
+
+ f
([n]q + β)[n – ]q [n]q + β
[n]q x + α[n – ]q

≤ f – g + δn (x) g + f
– f (x).
([n]q + β)[n – ]q
Hence, taking the infimum on the right hand side over all g ∈ W  and using (.), we get
the required result.
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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Page 8 of 18
Theorem  Let f ∈ C [, ∞), qn ∈ (, ) and ωa+ (f , δ) be its modulus of continuity on the
finite interval [, a + ] ⊂ [, ∞), where a > . Then, for every n > ,
(α,β)
≤ Mf  + a γn,q (x) + ωa+ f , γn,q (x) ,
B
(f
;
x)
–
f
(x)
n
n
n,qn
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| ωa+ (f , δ),
δ
δ > .
Thus, by applying the Cauchy-Schwarz inequality, we have
(α,β)
(α,β)

B
Bn,qn (t – x) ; x
n,qn (f ; x) – f (x) ≤ Mf  + a



+ ωa+ (f , δ)  + B(α,β)
(t
–
x)
;
x
δ n,qn
= 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 approximation
(x)|
Let C∗ [, ∞) := {f ∈ C [, ∞) : limx→∞ |f+x
 < ∞}. Throughout the section, we assume
that {qn } is a sequence in (, ) such that qn → .
Theorem  For each f ∈ C∗ [, ∞), we have
lim B(α,β)
n,qn (f ) – f  = .
n→∞
Proof Making use of the Korovkin type theorem on weighted approximation [], we see
that it is sufficient to verify the following three conditions:
k k
lim B(α,β)
n,qn t ; x – x  = ,
n→∞
k = , , .
(α,β)
Since Bn,qn (; x) = , the condition in (.) holds for k = .
By Lemma , we have for n > 
(α,β)
B
n,qn (t; x) – x 
[n]qn qnn– – β[n – ]qn α
x

sup
sup
≤ +
([n]qn + β)[n – ]qn x∈[,∞)  + x [n]qn + β x∈[,∞)  + x
[n]qn qnn– – β[n – ]qn α
+
,
≤
([n]qn + β)[n – ]qn [n]qn + β
which implies that the condition in (.) holds for k = .
(.)
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Similarly, we can write for n > 
(α,β)   B
≤
;
x
–
x
t
n,qn

qn [n]qn
[n – ]qn [n – ]qn
+
+
α
– 
[n – ]qn
[]qn [n]qn
α
+
,
qn [n – ]qn [n – ]qn ([n]qn + β)
(α,β)
which implies that limn→∞ Bn,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
(α,β)
|Bn,qn (f ; x) – f (x)|
= .
n→∞ x∈[,∞)
( + x )+α
lim sup
Proof Let x ∈ [, ∞) be arbitrary but fixed. Then
(α,β)
(α,β)
(α,β)
|Bn,qn (f ; x) – f (x)|
|Bn,qn (f ; x) – f (x)|
|Bn,qn (f ; x) – f (x)|
≤ sup
+ sup

+α

+α
( + x )
( + x )
( + x )+α
x≤x
x>x
x∈[,∞)
sup
(α,β)
|Bn,qn ( + t  ; x)|
≤ B(α,β)
n,qn (f ) – f C[,x ] + f  sup
( + x )+α
x>x
+ sup
x>x
|f (x)|
.
( + x )+α
(.)
|f (x)|
f 
Since |f (x)| ≤ f  ( + x ), we have supx>x (+x
 )+α ≤ (+x )α .

Let >  be arbitrary. We can choose x to be so large that
f 
< .
( + x )α 
(.)
In view of Theorem , we obtain
(α,β)
|Bn,qn ( + t  ; x)|
 + x
f 
f 
=
f  =
≤
< .
n→∞
( + x )+α
( + x )+α
( + x )α ( + x )α 
f  lim
(.)
Using Theorem , we can see that the first term of the inequality (.), implies that
(α,β)
B
n,qn (f ) – f C[,x ] < ,

as n → ∞.
Combining (.)-(.), we get the desired result.
For f ∈ C∗ [, ∞), the weighted modulus of continuity is defined as
 (f , δ) =
|f (x + h) – f (x)|
.
 + (x + h)
x≥,<h≤δ
sup
(.)
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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
(α,β)
+λ
B
 (f , δn ),
n,qn (f ; x) – f (x) ≤ K  + x
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|  (f , δ)
δ
|t – x|
 (f , δ)
≤  + (x + t)  +
δ

:= φx (t)  + ψx (t)  (f , δ),
δ
where φx (t) =  + (x + t) and ψx (t) = |t – x|. Then we obtain
(α,β)
 (α,β)
(α,β)
B
B
(φx ψx ; x)  (f , δn ).
n,qn (f ; x) – f (x) ≤ Bn,qn (φx ; x) +
δn n,qn
Now, applying the Cauchy-Schwarz inequality to the second term on the right hand side,
we get
(α,β)
B
n,qn (f ; x) – f (x)

(α,β)  (α,β)  (α,β)
≤ Bn,qn (φx ; x) +
Bn,qn φx ; x Bn,qn ψx ; x  (f , δn ).
δn
(.)
From Lemma ,

qn [n]qn + α[n – ]qn [n]qn

x

(α,β)

Bn,qn  + t ; x =
+



+x
+x
[n – ]qn [n – ]qn ([n]qn + β)  + x
+
[]qn [n]qn
qn [n – ]qn [n – ]qn ([n]qn
≤  + C ,
where C is a positive constant.
+ β)
for sufficiently large n,
α
x

+


+x
([n]qn + β)  + x
(.)
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(α,β)
From (.), there exists a positive constant K such that Bn,qn (φx ; x) ≤ K ( + x ), for
sufficiently large n.
(α,β)
Proceeding similarly, +x  Bn,qn ( + t  ; x) ≤  + C , for sufficiently large n, where C is a
positive constant.
(α,β)
Bn,qn (φx ; x) ≤ K ( + x ), where x ∈
So there exists a positive constant K , such that
[, ∞) and n is large enough. Also, we get
B(α,β)
n,qn
ψx ; x
=
qn [n]qn + α[n – ]qn [n]qn
[n]qn
x
+–
[n – ]qn [n – ]qn ([n]qn + β)
([n]qn + β)[n – ]qn

[n]qn
α
α
x
+
+
–
qn [n – ]qn [n – ]qn ([n]qn + β) [n]qn + β
([n]qn + β)
≤ α n x  + β n x + γn .
Hence, from (.), we have
(α,β)
 

B
K + K αn x + βn x + γn  (f , δn ).
n,qn (f ; x) – f (x) ≤  + x
δn
If we take δn = max{αn , βn , γn }, then we get
√
(α,β)

B
K + K x + x +   (f , δn )
n,qn (f ; x) – f (x) ≤  + x
≤ K  + x+λ  (f , δn ), for sufficiently large n and x ∈ [, ∞).
Hence, the proof is completed.
3.3 Point-wise estimates
In this section, we establish some point-wise estimates of the rate of convergence of the
(α,β)
operators Bn,q . First, we give the relationship between the local smoothness of f and local
approximation.
We know that a function f ∈ C[, ∞) is in LipM η on E, η ∈ (, ], E be any bounded
subset 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
(α,β)
η/
η
B
n,q (f ; x) – f (x) ≤ M γn,q (x) + d (x, E) ,
x ∈ [, ∞),
where M is a constant depending on η and f , and d(x, E) is the distance between x and E
defined as
d(x, E) = inf |t – x| : t ∈ E .
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Proof Let E be the closure of E in [, ∞). Then there exists at least one point x ∈ E such
that
d(x, E) = |x – x |.
(α,β)
By our hypothesis and the monotonicity of Bn,q , we get
(α,β)
(α,β) f (x) – f (x ); x
B
f (t) – f (x ); x + B(α,β)
n,q (f ; x) – f (x) ≤ Bn,q
n,q
η
η
≤ M B(α,β)
n,q |t – x | ; x + |x – x |
η
η
≤ M B(α,β)
n,q |t – x| ; x + |x – x | .
Now, applying Hölder’s inequality with p =

η
and

q
=  – p , we obtain
(α,β)
(α,β) η/

B
+ dη (x, E) ,
n,q (f ; x) – f (x) ≤ M Bn,q |t – x| ; x
from which the desired result immediate.
Next, we obtain the local direct estimate of the operators defined in (.), using the
Lipschitz-type maximal function of order η introduced by Lenze [] as
ωη (f , x) =
|f (t) – f (x)|
,
|t – x|η
t=x,t∈[,∞)
sup
x ∈ [, ∞) and η ∈ (, ].
(.)
Theorem  Let f ∈ CB [, ∞) and  < η ≤ . Then, for all x ∈ [, ∞) we have
(α,β)
ωη (f , x)γ η/ (x).
B
n,q (f ; x) – f (x) ≤ n,q
Proof From (.), we have
(α,β)
ωη (f , x)B(α,β) |t – x|η ; x .
B
n,q (f ; x) – f (x) ≤ n,q
Applying Hölder’s inequality with p =

η
and

q
=  – p , we get
(α,β)
η
η/
ωη (f , x)B(α,β) (t – x) ; x  ≤ B
ωη (f , x)γn,q
(x).
n,q (f ; x) – f (x) ≤ n,q
Thus, the proof is completed.
4 Statistical convergence
Let 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 =
∞
ank xk
k=
provided the series converges for each n. A is said to be regular if limn (Ax)n = L whenever
limn (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
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if for every > , limn k:|xk –L|≥ ank = . If we replace A by C then A is a Cesaro matrix of
order 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 this
direction, significant contributions have been made by (cf. [, , –] etc.).
Let qn ∈ (, ) be a sequence such that
stA - lim qnn = a (a < )
stA - lim qn = ,
n
n
and stA - lim
n

= .
[n]qn
(.)
Theorem  Let A = (ank ) be a non-negative regular summability matrix and (qn ) be a
sequence satisfying (.). Then, for any compact set K ⊂ [, ∞) and for each function f ∈
C(K), we have
stA - limB(α,β)
n,qn (f ; ·) – f = .
n
(α,β)
Proof Let x = maxx∈K x. From Lemma , stA - limn Bn,qn (e ; ·) – e = . Again, by
Lemma , we have
(α,β)
sup Bn,qn (e ; x) – e (x) ≤ x∈K
[n]qn
([n]qn + β)[n – ]qn
– x +
α
.
[n]qn + β
For > , let us define the following sets:
(α,β)
G := k : Bk,qk (e ; ·) – e ≥ ,
[k]qk
,
G := k : –  ≥
([k]qk + β)[k – ]qk

α
,
≥
G := k :
[k]qk + β 
which implies that G ⊆ G ∪ G and hence for all n ∈ N, we obtain
k∈G
ank ≤
k∈G
ank +
ank .
k∈G
(α,β)
Hence, taking the limit as n → ∞, we have stA - limn Bn,qn (e ; ·) – e = .
Similarly, by using Lemma , we have
supB(α,β)
≤
(e
;
x)
–
e
(x)


n,qn
x∈K
qn [n]qn + α[n – ]qn
[n – ]qn [n – ]qn ([n]qn + β)
+
[]qn [n]qn
qn [n – ]qn [n – ]qn ([n]qn
Now, let us define the following sets:
(α,β)
M := k : Bk,qk (e ; ·) – e ≥ ,
– x
x
+ β)
+
α
.
([n]qn + β)
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qk [k]qk + α[k – ]qk
≥
M := k : –

 ,
[k – ]qk [k – ]qk ([k]qk + β)
[]qk [k]qk
,
M := k :
≥
qk [k – ]qk [k – ]qk ([k]qk + β) 
α
.
≥
M := k :
([k]qk + β) 
Then we obtain M ⊆ M ∪ M ∪ M , which implies that
ank ≤
k∈M
ank +
k∈M
ank +
k∈M
ank .
k∈M
(α,β)
Thus, as n → ∞ we get stA - limn Bn,qn (e ; ·) – e = . This completes the proof.
Theorem  Let A = (ank ) be a non-negative regular summability matrix and (qn ) be a
(α,β)
sequence in (, ) satisfying (.). Let the operators Bn,qn , n ∈ N, be defined as in (.). Then,
for each function f ∈ C [, ∞), we have
stA - limB(α,β)
n,qn (f ; ·) – f ρ = ,
n
where ρ(x) =  + x+λ , λ > .
(α,β)
Proof From [, p., Theorem ], it is sufficient to prove that stA - limn Bn,qn (ei ; ·)–ei  =
, where ei (x) = xi , i = , , .
(α,β)
From Lemma , stA - limn Bn,qn (e ; ·) – e  =  holds.
Again using Lemma , we have
[n]qn

x α
+
–

( + x ) ([n] + β)
 qn
x∈[,∞) ( + x ) ([n]qn + β)[n – ]qn
[n]qn
α
.
(.)
≤ –  +
([n]qn + β)[n – ]qn
([n]qn + β)
(α,β)
B
n,qn (e ; ·) – e  ≤ sup
For each > , let us define the following sets:
(α,β)
G := k : Bk,qk (e ; ·) – e ≥ ,
[k]qk
,
G := k : –  ≥
([k]qk + β)[k – ]qk

α
,
≥
G := k :
[k]qk + β 
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 .
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(α,β)
Hence, on taking the limit as n → ∞, stA - limn Bn,qn (e ; ·) – e  = . Proceeding similarly,
(α,β)
B
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:
(α,β)
M := k : Bk,qk (e ; ·) – e ≥ ,
qk [k]qk + α[k – ]qk
,
M := k : –  ≥
[k – ]qk [k – ]qk ([k]qk + β)

[]qk [k]qk
,
M := k :
≥
qk [k – ]qk [k – ]qk ([k]qk + β) 
α
.
≥
M := k :
([k]qk + β) 
Then we obtain M ⊆ M ∪ M ∪ M , which implies that
ank ≤
k∈M
ank +
k∈M
ank +
k∈M
ank .
k∈M
(α,β)
Hence, taking the limit as n → ∞ we get stA - limn Bn,qn (e ; ·) – e  = . This completes
the proof of the theorem.
5 Better estimates
It is well known that the classical Bernstein polynomials preserve constant as well as linear
functions. To make the convergence faster, King [] proposed an approach to modify
the Bernstein polynomials, so that the sequence preserves test functions e and e , where
(α,β)
ei (t) = t i , i = , , . As the operator Bn,q (f ; x) defined in (.) reproduces only constant
functions, this motivated us to propose the modification of this operator, so that it can
preserve constant as well as linear functions.
The modification of the operators given in (.) is defined as
B˜ (α,β)
n,q (f ; x) =
∞
qn,ν rnq (x) qν–
ν=
+ eq –[n]q rnq (x)
q
where rn (x) =
([n]q +β)[n–]q x–α[n–]q
[n]q
f
∞/A
bn,ν– (q, t)f

qν [n]q t + α
dq t
[n]q + β
α
,
[n]q + β
for x ∈ In = [ [n]αq +β , ∞) and n > .
Lemma  For each x ∈ In , by simple computations, we have
(α,β)
() B˜ n,q (; x) = ;
(α,β)
() B˜ n,q (t; x) = x;
Agrawal et al. Journal of Inequalities and Applications 2014, 2014:441
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[n–]q (q[n]q +α[n–]q ) 
x
[n–]q [n]q
α  [n]q (q[n–]q +[n–]q )+α  [n–]q [n–]q
(α,β)
() B˜ n,q (t  ; x) =
+
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[]q [n]q –qα[n–]q (q[n]q +α[n–]q )
q([n]q +β)[n–]q [n]q
x+
.
([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) = ,
[n]q (q[n – ]q – [n – ]q ) + α[n – ]q [n – ]q 

;
x
=
x
B˜ (α,β)
(t
–
x)
n,q
[n – ]q [n]q
+
+
[]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
(α,β)
B˜
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
(t – u)g (u) du.
x
(α,β)
Applying B˜ n,q on both sides and using Lemma , we get
˜ (α,β)
˜ (α,β)
B˜ (α,β)
n,q (g; x) – g(x) = Bn,q (t – x); x g (x) + Bn,q
Obviously, we have |
t
x (t
t
(t – u)g (u) du; x .
x
– u)g (u) du| ≤ (t – x) g . Therefore
(α,β)

˜ (α,β)
= ξn (x)g .
B˜
n,q (g; x) – g(x) ≤ Bn,q (t – x) ; x g
(α,β)
Since |B˜ n,q (f ; x)| ≤ f , we get
(α,β)
(α,β)
(α,β)
˜
˜
B˜
n,q (f ; x) – f (x) ≤ Bn,q (f – g; x) + (f – g)(x) + Bn,q (g; x) – g(x)
≤ f – g + ξn (x)g .
Finally, taking the infimum over all g ∈ W  and using (.)-(.) we obtain
(α,β)
B˜
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
http://www.journalofinequalitiesandapplications.com/content/2014/1/441
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India. 2 Department of
Mathematics, Faculty of Science and Arts, Abant Izzet Baysal University, Bolu, 14280, Turkey.
Acknowledgements
The authors are extremely grateful to the reviewers for a careful reading of the manuscript and making valuable
suggestions leading to a better presentation of the paper. The last author is thankful to the ‘Council of Scientific and
Industrial 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-2014-441
Cite this article as: Agrawal et al.: Szász-Baskakov type operators based on q-integers. Journal of Inequalities and
Applications 2014, 2014:441
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