Approximation by the q-Szلsz-Mirakjan Operators

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 754217, 16 pagesdoi:10.1155/2012/754217

Research ArticleApproximation by the q-Szasz-Mirakjan Operators

N. I. Mahmudov

Department of Mathematics, Eastern Mediterranean University, Gazimagusa,North Cyprus, Mersin 10, Turkey

Correspondence should be addressed to N. I. Mahmudov, [email protected]

Received 6 September 2012; Accepted 4 December 2012

Academic Editor: Behnam Hashemi

Copyright q 2012 N. I. Mahmudov. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper deals with approximating properties of the q-generalization of the Szasz-Mirakjanoperators in the case q > 1. Quantitative estimates of the convergence in the polynomial-weightedspaces and the Voronovskaja’s theorem are given. In particular, it is proved that the rate ofapproximation by the q-Szasz-Mirakjan operators (q > 1 ) is of order q−n versus 1/n for the classicalSzasz-Mirakjan operators.

1. Introduction

The approximation of functions by using linear positive operators introduced via q-Calculusis currently under intensive research. The pioneer work has been made by Lupas [1] andPhillips [2] who proposed generalizations of Bernstein polynomials based on the q-integers.The q-Bernstein polynomials quickly gained the popularity, see [3–11]. Other importantclasses of discrete operators have been investigated by using q-Calculus in the case 0 < q < 1,for example, q-Meyer-Konig operators [12–14], q-Bleimann, Butzer and Hahn operators [15–17], q-Szasz-Mirakjan operators [18–21], and q-Baskakov operators [22, 23].

In the present paper, we introduce a q-generalization of the Szasz operators in the caseq > 1. Notice that different q-generalizations of Szasz-Mirakjan operators were introducedand studied by Aral and Gupta [18, 19], by Radu [20], and by Mahmudov [21] in the case0 < q < 1. Since we define q-Szasz-Mirakjan operators for q > 1, the rate of approximationby the q-Szasz-Mirakjan operators (q > 1) is of order q−n, which is essentially better than 1/n(rate of approximation for the classical Szasz-Mirakjan operators). Thus our q-Szasz-Mirakjanoperators have better approximation properties than the classical Szasz-Mirakjan operatorsand the other q-Szasz-Mirakjan operators.

2 Abstract and Applied Analysis

The paper is organized as follows. In Section 2, we give standard notations that will beused throughout the paper, introduce q-Szasz-Mirakjan operators, and evaluate the momentsof Mn,q. In Section 3 we study convergence properties of the q-Szasz-Mirakjan operators inthe polynomial-weighted spaces. In Section 4, we give the quantitative Voronovskaja-typeasymptotic formula.

2. Construction of Mn,q and Estimation of Moments

Throughout the paper we employ the standard notations of q-calculus, see [24, 25].q-integer and q-factorial are defined by

[n]q :=

⎧⎪⎨

⎪⎩

1 − qn

1 − q, if q ∈ R+ \ {1},

n, if q = 1,for n ∈ N, [0] = 0,

[n]q! := [1]q[2]q . . . [n]q, for n ∈ N, [0]! = 1.

(2.1)

For integers 0 ≤ k ≤ n q-binomial is defined by

[nk

]

q

:=[n]q!

[k]q![n − k]q!. (2.2)

The q-derivative of a function f(x), denoted by Dqf , is defined by

(Dqf)(x) :=

f(qx) − f(x)

(q − 1

)x

, x /= 0,(Dqf)(0) := lim

x→ 0

(Dqf)(x). (2.3)

The formula for the q-derivative of a product and quotient are

Dq(u(x)v(x)) = Dq(u(x))v(x) + u(qx)Dq(v(x)). (2.4)

Also, it is known that

Dqxn = [n]qx

n−1, DqE(ax) = aE(qax). (2.5)

If |q| > 1, or 0 < |q| < 1 and |z| < 1/(1 − q), the q-exponential function eq(x) was defined byJackson

eq(z) :=∞∑

k=0

zk

[k]q!. (2.6)

Abstract and Applied Analysis 3

If |q| > 1, eq(z) is an entire function and

eq(z) =∞∏

j=0

(

1 +(q − 1

) z

qj+1

)

,∣∣q∣∣ > 1. (2.7)

There is another q-exponential function which is entire when 0 < |q| < 1 and which convergeswhen |z| < 1/|1 − q| if |q| > 1. To obtain it we must invert the base in (2.6), that is, q → 1/q:

Eq(z) := e1/q(z) =∞∑

k=0

qk(k−1)/2zk

[k]q!. (2.8)

We immediately obtain from (2.7) that

Eq(z) =∞∏

j=0

(1 +(1 − q

)zqj), 0 <

∣∣q∣∣ < 1. (2.9)

The q-difference equations corresponding to eq(z) and Eq(z) are

Dqeq(az) = aeq(qz), DqEq(az) = aEq

(qaz),

D1/qeq(z) = D1/qE1/q(z) = E1/q

(q−1z)= eq(q−1z), q /= 0.

(2.10)

Let Cp be the set of all real valued functions f , continuous on [0,∞), such that wpf isuniformly continuous and bounded on [0,∞) endowed with the norm

∥∥f∥∥p := sup

x∈[0,∞)wp(x)

∣∣f(x)

∣∣. (2.11)

Here

w0(x) := 1, wp(x) := (1 + xp)−1, if p ∈ N. (2.12)

The corresponding Lipschitz classes are given for 0 < α ≤ 2 by

Δ2hf(x) := f(x + 2h) − 2f(x + h) + f(x),

ω2p

(f ; δ):= sup

0<h≤δ

∥∥∥Δ2

hf∥∥∥p, Lip2

pα :={f ∈ Cp : ω2

p

(f ; δ)= 0(δα), δ → 0+

}.

(2.13)

Now we introduce the q-parametric Szasz-Mirakjan operator.


Definition 2.1. Let q > 1 and n ∈ N. For f : [0,∞) → R one defines the Szasz-Mirakjanoperator based on the q-integers

Mn,q

(f ;x):=

∞∑

k=0

f

([k]q[n]q

)1

qk(k−1)/2[n]kqx

k

[k]q!eq(−[n]qq−kx

). (2.14)

Similarly as a classical Szasz-Mirakjan operator Sn, the operator Mn,q is linear andpositive. Furthermore, in the case of q → 1+ we obtain classical Szasz-Mirakjan operators.

Moments Mn,q(tm;x) are of particular importance in the theory of approximationby positive operators. From (2.14) one easily derives the following recurrence formula andexplicit formulas for moments Mn,q(tm;x), m = 0, 1, 2, 3, 4.

Lemma 2.2. Let q > 1. The following recurrence formula holds

Mn,q

(tm+1;x

)=

m∑

j=0

(mj

)xqj

[n]m−jq

Mn,q

(tj ; q−1x

). (2.15)

Proof. The recurrence formula (2.15) easily follows from the definition ofMn,q and q[k]q+1 =[k + 1]q as show below:

Mn,q

(tm+1;x

)

=∞∑

k=0

[k]m+1q

[n]m+1q

1qk(k−1)/2

[n]kqxk


)

=∞∑

k=1

[k]mq[n]mq

1qk(k−1)/2

[n]k−1q xk

[k − 1]q!eq(−[n]qq−kx

)

=∞∑

k=0

(q[k]q + 1

)m

[n]mq

1qk(k+1)/2

[n]kqxk+1

[k]q!eq(−[n]qq−kq−1x

)

=∞∑

k=0

1[n]mq

m∑

j=0

(mj

)

qj[k]jq1

qk(k+1)/2

[n]kqxk+1

[k]q!eq(−[n]qq−kq−1x

)

=m∑

j=0

(mj

)xqj

[n]m−jq

∞∑

k=0

[k]jq

[n]jq

1qk(k−1)/2

[n]kqxk

[k]q!qkeq(−[n]qq−kq−1x

)

=m∑

j=0

(mj

)xqj

[n]m−jq

Mn,q

(tj ; q−1x

).

(2.16)


Lemma 2.3. The following identities hold for all q > 1, x ∈ [0,∞), n ∈ N, and k ≥ 0:

xDqsnk(q;x)= [n]q

([k]q[n]q

− x

)

snk(q;x),

Mn,q

(tm+1;x

)=

x

[n]qDqMn,q(tm;x) + xMn,q(tm;x),

(2.17)

where snk(q;x) := (1/qk(k−1)/2)([n]kqxk/[k]q!)eq(−[n]qq−kx).

Proof. The first identitiy follows from the following simple calculations

xDqsnk(q;x)

= [k]q1

qk(k−1)/2[n]kqx

k

[k]q!eq(−[n]q−kx

)− xq−k[n]q

1qk(k−1)/2

[n]kqqkxk


)

= [k]qsnk(q;x) − x[n]qsnk

(q;x)= [n]q

([k]q[n]q

− x

)

snk(q;x).

(2.18)

The second one follows from the first:

xDqMn,q(tm;x) = [n]q∞∑

k=0

([k]q[n]q

)m( [k]q[n]q

− x

)

snk(q;x)

= [n]q∞∑

k=0

([k]q[n]q

)m+1

snk(q;x) − [n]qx

∞∑

k=0

([k]q[n]q

)m

snk(q;x)

= [n]qMn,q

(tm+1;x

)− [n]qxMn,q(tm;x).

(2.19)

Lemma 2.4. Let q > 1. One has

Mn,q(1;x) = 1, Mn,q(t;x) = x, Mn,q

(t2;x)= x2 +

1[n]q

x,

Mn,q

(t3;x)= x3 +

2 + q

[n]qx2 +

1

[n]2qx,

Mn,q

(t4;x)= x4 +

(3 + 2q + q2

) x3

[n]q+(3 + 3q + q2

) x2

[n]2q+

1

[n]3qx.

(2.20)

Proof. For a fixed x ∈ R+, by the q-Taylor theorem [24], we obtain

ϕn(t) =∞∑

k=0

(t − x)k1/q[k]1/q!

Dk1/qϕn(x). (2.21)


Choosing t = 0 and taking into account

(−x)k1/q = (−1)kxkq−k(k−1)/2, Dk1/qeq

(−[n]qx

)= (−1)kq−k(k−1)/2[n]kqeq

(−[n]qq−kx

)(2.22)

we get for ϕn(x) = eq(−[n]qx) that

1 = ϕn(0) =∞∑

k=0

(−1)kxk

qk(k−1)/2[k]1/q!Dk

1/qϕn(x)

=∞∑

k=0

(−1)kxk

[k]q!(−1)kq−k(k−1)/2[n]kqeq

(−[n]qq−kx

)

=∞∑

k=0

[n]kqxk

[k]q!qk(k−1)/2eq(−[n]qq−kz

).

(2.23)

In other words Mn,q(1;x) = 1.Calculation of Mn,q(ti;x), i = 1, 2, 3, 4, based on the recurrence formula (2.17)

(or (2.15)). We only calculate Mn,q(t3;x) and Mn,q(t4;x):

Mn,q

(t3;x)=

x

[n]qDqMn,q

(t2;x)+ xMn,q

(t2;x)

=x

[n]q

(

[2]qx +1

[n]q

)

+ x

(

x2 +1

[n]qx

)

=1

[n]2qx +

2 + q

[n]qx2 + x3,

Mn,q

(t4;x)=

x

[n]qDqMn,q

(t3;x)+ xMn,q

(t3;x)

=x

[n]q

⎛

⎝1

[n]2q+2 + q

[n]q[2]qx + [3]qx

2

⎞

⎠ + x

⎛

⎝1

[n]2qx +

2 + q

[n]qx2 + x3

⎞

⎠

=1

[n]3qx +(3 + 3q + q2

) x2

[n]2q+(3 + 2q + q2

) x3

[n]q+ x4.

(2.24)


Lemma 2.5. Assume that q > 1. For every x ∈ [0,∞) there hold

Mn,q

((t − x)2;x

)=

x

[n]q, (2.25)

Mn,q

((t − x)3;x

)=

1

[n]2qx +(q − 1

) x2

[n]q, (2.26)

Mn,q

((t − x)4;x

)=

1

[n]3qx +(q2 + 3q − 1

) x2

[n]2q+(q − 1

)2 x3

[n]q. (2.27)

Proof. First of all we give an explicit formula forMn,q((t − x)4;x).

Mn,q

((t − x)3;x

)= Mn,q

(t3;x)− 3xMn,q

(t2;x)+ 3x2Mn,q(t;x) − x3

= x3 +2 + q

[n]qx2 +

1

[n]2qx − 3x

(

x2 +x

[n]q

)

+ 3x3 − x3

=1

[n]2qx +(q − 1

) x2

[n]q,

Mn,q

((t − x)4;x

)= Mn,q

(t4;x)− 4xMn,q

(t3;x)+ 6x2Mn,q

(t2;x)− 4x3Mn,q(t;x) + x4

=1

[n]3qx +(3 + 3q + q2

) x2

[n]2q+(3 + 2q + q2

) x3

[n]q+ x4

− 4x

⎛

⎝1

[n]2qx +

2 + q

[n]qx2 + x3

⎞

⎠ + 6x2

(

x2 +x

[n]q

)

− 4x4 + x4

=1

[n]3qx +(−1 + 3q + q2

) x2

[n]2q+(q − 1

)2 x3

[n]q.

(2.28)

Now we prove explicit formula for the momentsMn,q(tm;x), which is a q-analogue ofa result of Becker, see [26, Lemma 3].

Lemma 2.6. For q > 1, m ∈ N there holds

Mn,q(tm;x) =m∑

j=1

Sq

(m, j) xj

[n]m−jq

, (2.29)


where

Sq

(m + 1, j

)=[j]Sq

(m, j)+ Sq

(m, j − 1

), m ≥ 0, j ≥ 1,

Sq(0, 0) = 1, Sq(m, 0) = 0, m > 0, Sq

(m, j)= 0, m < j.

(2.30)

In particularMn,q(tm;x) is a polynomial of degreem without a constant term.

Proof. Because ofMn,q(t;x) = x,Mn,q(t2;x) = x2+x/[n]q, the representation (2.29) holds truefor m = 1, 2 with Sq(2, 1) = 1, Sq(1, 1) = 1.

Now assume (2.29) to be valued form then by Lemma 2.3 we have

Mn,q

(tm+1;x

)=

x

[n]qDqMn,q(tm;x) + xMn,q(tm;x)

=x

[n]q

m∑

j=1

[j]

qSq

(m, j) xj−1

[n]m−jq

+ xm∑

j=1

Sq

(m, j) xj

[n]m−jq

=m∑

j=1

[j]

qSq

(m, j) xj

[n]m−j+1q

+m∑

j=1

Sq

(m, j) xj+1

[n]m−jq

=x

[n]mSq(m, 1) + xm+1

Sq(m,m)

+m∑

j=2

([j]

qSq

(m, j)+ Sq

(m, j − 1

)) xj

[n]m−j+1q

.

(2.31)

Remark 2.7. Notice that Sq(m, j) are Stirling numbers of the second kind introduced byGoodman et al. in [8]. For q = 1 the formulae (2.30) become recurrence formulas satisfiedby Stirling numbers of the second type.

3. Mn,q in Polynomial-Weighted Spaces

Lemma 3.1. Let p ∈ N ∪ {0} and q ∈ (1,∞) be fixed. Then there exists a positive constant K1(q, p)such that

∥∥Mn,q

(1/wp;x

)∥∥p≤ K1

(q, p), n ∈ N. (3.1)

Moreover for every f ∈ Cp one has

∥∥Mn,q

(f)∥∥p≤ K1

(q, p)∥∥f∥∥p, n ∈ N. (3.2)

Thus Mn,q is a linear positive operator from Cp into Cp for any p ∈ N ∪ {0}.


Proof. The inequality (3.1) is obvious for p = 0. Let p ≥ 1. Then by (2.29)we have

wp(x)Mn,q

(1/wp;x

)= wp(x) +wp(x)

p∑

j=1

Sq

(p, j) xj

[n]p−jq

≤ K1(q, p), (3.3)

K1(q, p) is a positive constant depending on p and q. From this follows (3.1). On the otherhand

∥∥Mn,q

(f)∥∥p≤ ∥∥f∥∥p

∥∥∥∥∥Mn,q

(1wp

)∥∥∥∥∥p

, (3.4)

for every f ∈ Cp. By applying (3.1), we obtain (3.2).

Lemma 3.2. Let p ∈ N ∪ {0} and q ∈ (1,∞) be fixed. Then there exists a positive constant K2(q, p)such that

∥∥∥∥∥Mn,q

((t − ·)2wp(t)

; ·)∥∥∥∥∥p

≤ K2(q, p)

[n]q, n ∈ N. (3.5)

Proof. The formula (2.25) imply (3.5) for p = 0. We have

Mn,q

((t − x)2

wp(t);x

)

= Mn,q

((t − x)2;x

)+Mn,q

((t − x)2tp;x

), (3.6)

for p, n ∈ N. If p = 1 then we get

Mn,q

((t − x)2(1 + t);x

)= Mn,q

((t − x)2;x

)+Mn,q

((t − x)2t;x

)

= Mn,q

((t − x)3;x

)+ (1 + x)Mn,q

((t − x)2;x

),

(3.7)

which by Lemma 2.5 yields (3.5) for p = 1.Let p ≥ 2. By applying (2.29), we get

wp(x)Mn,q

((t − x)2tp;x

)

= wp(x)(Mn,q

(tp+2;x

)− 2xMn,q

(tp+1;x

)+ x2Mn,q(tp;x)

)

= wp(x)

⎛

⎝xp+2 +p+1∑

j=1

Sq

(p + 2, j

) xj

[n]p+2−jq

− 2xp+2 − 2p∑

j=1

Sq

(p + 1, j

) xj+1

[n]p+1−jq

+xp+2 +p−1∑

j=1

Sq

(p, j) xj+2

[n]p−jq

⎞

⎠


= wp(x)

⎛

⎝p∑

j=2

(Sq

(p + 2, j

) − 2Sq

(p + 1, j

)+ Sq

(p, j)) xj+1

[n]p+1−jq

+ Sq

(p + 2, 1

) x

[n]p+1q

+(Sq

(p + 2, 2

) − 2Sq

(p + 2, 1

)) x2

[n]pq

⎞

⎠

= wp(x)x

[n]qPp

(q;x),

(3.8)

where Pp(q;x) is a polynomial of degree p. Therefore one has

wp(x)Mn,q

((t − x)2tp;x

)≤ K2

(q, p) x

[n]q. (3.9)

Our first main result in this section is a local approximation property of Mn,q statedbelow.

Theorem 3.3. There exists an absolute constant C > 0 such that

wp(x)∣∣Mn,q

(g;x) − g(x)

∣∣ ≤ K3

(q, p)∥∥g ′′∥∥ x

[n]q, (3.10)

where g ∈ C2p, q > 1 and x ∈ [0,∞).

Proof. Using the Taylor formula

g(t) = g(x) + g ′(x)(t − x) +∫ t

x

∫s

x

g ′′(u)duds, g ∈ C2p, (3.11)

we obtain that

wp(x)∣∣Mn,q

(g;x) − g(x)

∣∣ = wp(x)

∣∣∣∣∣Mn,q

(∫ t

x

∫ s

x

g ′′(u)duds;x

)∣∣∣∣∣

≤ wp(x)Mn,q

(∣∣∣∣∣

∫ t

x

∫ s

x

g ′′(u)duds

∣∣∣∣∣;x

)

≤ wp(x)Mn,q

(∥∥g ′′∥∥

p

∣∣∣∣∣

∫ t

x

∫s

x

(1 + um)duds

∣∣∣∣∣;x

)

≤ wp(x)12∥∥g ′′∥∥

pMn,q

((t − x)2

(1/wp(x) + 1/wp(t)

);x)


≤ 12∥∥g ′′∥∥

p

(Mn,q

((t − x)2;x

)+wp(x)Mn,q

((t − x)2wp(t);x

))

≤ K3(q, x)∥∥g ′′∥∥

p

x

[n]q.

(3.12)

Now we consider the modified Steklov means

fh(x) :=4h2

∫∫h/2

0

[2f(x + s + t) − f(x + 2(s + t))

]dsdt. (3.13)

fh(x) has the following properties:

f(x) − fh(x) =4h2

∫∫h/2

0Δ2

s+tf(x)dsdt, f ′′h(x) = h−2

(8Δ2

h/2f(x) −Δ2hf(x)

)(3.14)

and therefore

∥∥f − fh

∥∥p ≤ ω2

p

(f ;h),

∥∥f ′′

h

∥∥p≤ 1

9h2ω2

p

(f ;h). (3.15)

We have the following direct approximation theorem.

Theorem 3.4. For every p ∈ N ∪ {0}, f ∈ Cp and x ∈ [0,∞), q > 1, one has

wp(x)∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ Mpω

2p

(

f ;√

x

[n]q

)

= Mpω2p

⎛

⎝f ;

√√√√

(q − 1

)x

(qn − 1

)

⎞

⎠. (3.16)

Particularly, if Lip2pα for some α ∈ (0, 2], then

wp(x)∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ Mp

(x

[n]q

)α/2

. (3.17)

Proof. For f ∈ Cp and h > 0

∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ ∣∣Mn,q

((f − fh

);x) − (f − fh

)(x)∣∣ +∣∣Mn,q

(fh;x) − fh(x)

∣∣ (3.18)

and therefore

wp(x)∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ ∥∥f − fh

∥∥p

(

wp(x)Mn,q

(1

wp(t);x

)

+ 1

)

+K3(q, p)∥∥f ′′

h

∥∥p

x

[n]q.

(3.19)


Since wp(x)Mn,q(1/wp(t);x) ≤ K1(q, p), we get that

wp(x)∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ M

(q, p)ω2

p

(f ;h)[

1 +x

[n]qh2

]

. (3.20)

Thus, choosing h =√x/[n]q, the proof is completed.

Corollary 3.5. If p ∈ N ∪ {0}, f ∈ Cp, q > 1 and x ∈ [0,∞), then

limn→∞

Mn,q

(f ;x)= f(x). (3.21)

This converegnce is uniform on every [a, b], 0 ≤ a < b.

Remark 3.6. Theorem 3.4 shows the rate of approximation by the q-Szasz-Mirakjan operators(q > 1) is of order q−n versus 1/n for the classical Szasz-Mirakjan operators.

4. Convergence of q-Szasz-Mirakjan Operators

An interesting problem is to determine the class of all continuous functions f such thatMn,q(f) converges to f uniformly on the whole interval [0,∞) as n → ∞. This problemwas investigated by Totik [27, Theorem 1] and de la Cal and Carcamo [28, Theorem 1]. Thefollowing result is a q-analogue of Theorem 1 [28].

Theorem 4.1. Assume that f : [0,∞) → R is bounded or uniformly continuous. Let

f∗(z) = f(z2), z ∈ [0,∞). (4.1)

One has, for all t > 0 and x ≥ 0,

∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ 2ω

(

f∗;

√1

[n]q

)

. (4.2)

Therefore, Mn,q(f ;x) converges to f uniformly on [0,∞) as n → ∞, whenever f∗ is uniformlycontinuous.

Proof. By the definition of f∗ we have

Mn,q

(f ;x)= Mn,q

(f∗(√·);x). (4.3)


Thus we can write

∣∣Mn,q

(f ;x) − f(x)

∣∣ =∣∣Mn,q

(f∗(√·);x) − f∗(√x

)∣∣

=

∣∣∣∣∣∣

∞∑

k=0

⎛

⎝f∗

⎛

⎝

√√√√

[k]q[n]q

⎞

⎠ − f∗(√x)

⎞

⎠sn,k(q;x)

∣∣∣∣∣∣

≤∞∑

k=0

∣∣∣∣∣∣

⎛

⎝f∗

⎛

⎝

√√√√

[k]q[n]q

⎞

⎠ − f∗(√x)

⎞

⎠

∣∣∣∣∣∣sn,k(q;x)

≤∞∑

k=0

ω

⎛

⎝f∗;

∣∣∣∣∣∣

√√√√

[k]q[n]q

− √x

∣∣∣∣∣∣

⎞

⎠sn,k(q;x)

≤∞∑

k=0

ω

⎛

⎜⎝f∗;

∣∣∣√[k]q/[n]q −

√x∣∣∣

Mn,q

(∣∣√· − √

x∣∣;x)Mn,q

(∣∣√· − √

x∣∣;x)

⎞

⎟⎠sn,k

(q;x).

(4.4)

Finally, from the inequality

ω(f∗;αδ

) ≤ (1 + α)ω(f∗; δ), α, δ ≥ 0, (4.5)

we obtain

∣∣Mn,q

(f ;x) − f(x)

∣∣ ≤ ω

(f∗;Mn,q

(∣∣√· − √

x∣∣;x)) ∞∑

k=0

⎛

⎜⎝1 +

∣∣∣√[k]q/[n]q −

√x∣∣∣

Mn,q

(∣∣√· − √

x∣∣;x)

⎞

⎟⎠sn,k

(q;x)

= 2ω(f∗;Mn,q

(∣∣√· − √

x∣∣;x)).

(4.6)

In order to complete the proof we need to show that we have for all t > 0 and x > 0,

Mn,q

(∣∣√· − √

x∣∣;x) ≤√

1[n]q

. (4.7)


Indeed we obtain from the Cauchy-Schwarz inequality:

Mn,q

(∣∣√· − √

x∣∣;x)=

∞∑

k=0

∣∣∣∣∣∣

√√√√

[k]q[n]q

− √x

∣∣∣∣∣∣sn,k(q;x)

=∞∑

k=0

∣∣∣[k]q/[n]q − x

∣∣∣

√[k]q/[n]q +

√xsn,k(q;x) ≤ 1√

x

∞∑

k=0

∣∣∣∣∣

[k]q[n]q

− x

∣∣∣∣∣sn,k(q;x)

≤ 1√x

√√√√

∞∑

k=0

∣∣∣∣∣

[k]q[n]q

− x

∣∣∣∣∣

2

sn,k(q;x)=

1√x

√

Mn,q

((· − x)2;x

)

=1√x

√1

[n]qx =

√1

[n]q

(4.8)

showing (4.2), and completing the proof.

Next we prove Voronovskaja type result for q-Szasz-Mirakjan operators.

Theorem 4.2. Assume that q ∈ (1,∞). For any f ∈ C2p the following equality holds

limn→∞

[n]q(Mn,q

(f ;x) − f(x)

)=

12f ′′(x)x, (4.9)

for every x ∈ [0,∞).

Proof. Let x ∈ [0,∞) be fixed. By the Taylor formula we may write

f(t) = f(x) + f ′(x)(t − x) +12f ′′(x)(t − x)2 + r(t;x)(t − x)2, (4.10)

where r(t;x) is the Peano form of the remainder, r(·;x) ∈ Cp, and limt→xr(t;x) = 0. ApplyingMn,q to (4.10) we obtain

[n](Mn,q

(f ;x) − f(x)

)= f ′(x)[n]qMn,q(t − x;x)

+12f ′′(x)[n]qMn,q

((t − x)2;x

)+ [n]qMn,q

(r(t;x)(t − x)2;x

).

(4.11)

By the Cauchy-Schwartz inequality, we have

Mn,q

(r(t;x)(t − x)2;x

)≤√

Mn,q(r2(t;x);x)√

Mn,q

((t − x)4;x

). (4.12)


Observe that r2(x;x) = 0. Then it follows from Corollary 3.5 that

limn→∞

Mn,q

(r2(t;x);x

)= r2(x;x) = 0. (4.13)

Now from (4.12), (4.13), and Lemma 2.5 we get immediately

limn→∞

[n]qMn,q

(r(t;x)(t − x)2;x

)= 0. (4.14)

The proof is completed.

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