Recently, Karsli [8] defined the following MKZD operators for functions defined on [0, bn], named Chlodovsky-type MKZD operators as Ln(f;x

65, 2 (2013), 187–196 June 2013

research paper

ON CONVERGENCE OF q-CHLODOVSKY-TYPE MKZD OPERATORS

Harun Karsli and Vijay Gupta

Abstract. In the present paper, we define a new kind of MKZD operators for functions defined on [0, bn], named q-Chlodovsky-type MKZD operators, and give some approximation properties.

1. Introduction

For a function defined on the interval [0,1], the Meyer-K¨onig and Zeller oper- atorsMn(f, x) [10] are defined as

Mn(f;x) = X∞

k=0

mn,k(x)f µ k

n+k

¶

(1.1) where mn,k = ¡_n+k−1

k

¢x^k(1 −x)ⁿ. In 1989 Guo [2] introduced the integrated Meyer-K¨onig and Zeller operators Mfn by the means of the operators (1.1), to approximate Lebesgue integrable functions on the interval [0,1]. Such operators have been defined as

Mfn(f;x) = X∞ k=0

e mn,k(x)

Z

Ik

f(t)dt (1.2)

whereIk = [_n+k^k ,_n+k+1^k+1 ] andmen,k(x) = (n+ 1)¡_n+k+1

k

¢x^k(1−x)ⁿ. Similar results may be also found in the papers [3, 4].

Recently, Karsli [8] defined the following MKZD operators for functions defined on [0, bn], named Chlodovsky-type MKZD operators as

Ln(f;x) = X∞ k=0

n+k bn mn,k

µx bn

¶ Z _bn

0

f(t)bn,k

µ t bn

¶

dt, 0≤x, t≤bn, (1.3)

2010 AMS Subject Classification: 41A25, 41A36

Keywords and phrases: q-Chlodovsky-type MKZD operators; modulus of continuity; Peetre- K functional; Lipschitz space.

187

where (bn) is a positive increasing sequence with the properties

n→∞lim bn =∞and lim

n→∞

bn

n = 0 andbn,k(t) =n¡_n+k

k

¢t^k(1−t)ⁿ⁻¹. We now deal with theq-analogue of Chlodovsky- type MKZD operatorsLn,q,defined as

Ln,q(f;x) = X∞ k=0

[n+k]_q bn mn,k,q

µx bn

¶ Z _bn

0

q^−kf(t)bn,k,q

µqt bn

¶

dqt, 0≤x≤bn, (1.4) where

mk,n,q(x) =

·n+k−1 k

¸

q

x^k

n−1Y

s=0

(1−q^sx) and

bn,k,q(t) = [n]q

·n+k k

¸

q

t^k

n−2Y

s=0

(1−q^st) (0≤t, x≤1),

provided theq-integral and the infinite series on the r.h.s. of (1.4) are well-defined.

It can be easily verified that in the caseq= 1 the operators defined by (1.4) reduce to the Chlodovsky-type MKZD operators defined by (1.3).

Actually theq-analogue of the linear positive operators was started in the last decade when Phillips [11] first introducedq-Bernstein polynomials, and later their Durrmeyer variants were studied and discussed in [5, 6]. Very recently Govil and Gupta [1] studied the approximation properties of q-MKZD operators. Here our aim is to study theq-analogue of summation-integral-type CMKZD operators. We shall prove that the operatorsLn,qf being defined in (1.4) converge to the limitf. Before getting onto the main subject, we first give definitions of q-integer, q-binomial coefficient and q-integral, which are required in this paper. For any fixed real numberq >0 and non-negative integerr theq-integerof the numberr is defined by

[r]q =

½(1−q^r)/(1−q), q6= 1

r, q= 1.

Theq-f actorialis defined by [r]q! =

½[r]q[r−1]q· · ·[1]q, r= 1,2,3, . . .

1, r= 0.

andq-binomial coefficient is defined as

·n r

¸

q

= [n]q! [r]q![n−r]q!, for integersn≥r≥0. Theq-integral is defined as (see [9])

Z _a

0

f(x)dqx= (1−q)a X∞ n=0

f(aqⁿ)qⁿ

provided the sum converges absolutely. Note that the series on the right-hand side is guaranteed to be absolutely convergent as the functionf is such that, for some M >0,α >−1,|f(x)|< M x^α in a right neighbourhood ofx= 0.

Definition 1.1. A functionf isq-integrable on [0,∞) if the series Z _∞

0

f(x)dqx= (1−q)X

n∈Z

f(qⁿ)qⁿ converges absolutely. We use the notation

(a−b)ⁿ_q =

n−1Y

j=0

(a−q^jb).

Theq-analogue of Beta function (see [7]) is defined as Bq(m, n) =

Z ₁

0

t^m−1(1−qt)ⁿ⁻¹_q dqt, m, n >0.

Also

Bq(m, n) =[m−1]![n−1]!

[m+n−1]! . 2. Auxiliary results

In this section we give certain results, which are necessary to prove our main theorem.

Lemma 2.1. Fors∈N, (Ln,qt^s) (x) =b^s_n

X∞

k=0

mn,k,q

µx bn

¶[n+k]_q! [k]q!

[k+s]_q!

[k+s+n]_q!. (2.1) Proof. We have

(Ln,qt^s) (x) = X∞ k=0

[n+k]_q bn mn,k,q

µx bn

¶ Z _bn

0

q^−kt^sbn,k,q

µqt bn

¶ dqt

= X∞ k=0

[n+k]_q bn

mn,k,q

µx bn

¶ Z _b

n

0

t^s

·n+k−1 k

¸

q

µ t bn

¶_kµ 1−qt

bn

¶_n−1

q

dqt.

Settingu=t/bn, we get (Ln,qt^s) (x) =

X∞

k=0

[n+k]_q bn mn,k,q

µx bn

¶ b^s+1_n

·n+k−1 k

¸

q

Z ₁

0

u^k+s(1−qu)ⁿ⁻¹_q dqu

= X∞

k=0

[n+k]_q bn mn,k,q

µx bn

¶ b^s+1_n

·n+k−1 k

¸

q

Bq(k+s+ 1, n)

=b^s_n X∞ k=0

[n+k]_qmn,k,q

µx bn

¶ [n+k−1]_q! [n−1]_q! [k]_q!

Γq(k+s+ 1) Γq(n) Γq(k+s+n+ 1)

=b^s_n X∞ k=0

m_n,k,q µx

bn

¶[n+k]_q! [k]q!

[k+s]_q! [k+s+n]_q!. Fors= 0,1 and 2 in (2.1), we get respectively

(Ln,q1) (x) = X∞

k=0

mn,k,q

µx bn

¶

= X∞

k=0

·n+k−1 k

¸

q

µx bn

¶k n−1Y

s=0

µ

1−q^sx bn

¶

= 1, (2.2) since

Q_n−1 1

s=0

³

1−q^{s x}_b

n

´ = X∞ k=0

·n+k−1 k

¸

q

µx bn

¶k

.

(Ln,qt)(x) =bn

X∞

k=0

mn,k,q

µx bn

¶[n+k]_q! [k]_q!

[k+ 1]_q! [n+k+ 1]_q!

=bn n−1Y

s=0

µ

1−q^sx b_n

¶X∞ k=0

[n+k−1]_q! [n−1]_q! [k]_q!

[k+ 1]_q [n+k+ 1]_q

µx b_n

¶_k

=bn n−1Y

s=0

µ

1−q^sx bn

¶X∞ k=1

[n+k−2]_q! [n−1]_q! [k−1]_q!

µx bn

¶k [k+ 1]_q [n+k+ 1]_q

[n+k−1]_q [k]_q

=bn n−1Y

s=0

µ

1−q^sx bn

¶X∞ k=1

[n+k−2]_q! [n−1]_q! [k−1]_q!

µx bn

¶_k

[k+ 1]_q [k]_q

[n+k−1]_q [n+k+ 1]_q

≥bn n−1Y

s=0

µ

1−q^sx bn

¶X∞ k=1

[n+k−2]_q! [n−1]_q! [k−1]_q!

µx bn

¶_k

[k+ 1]_q [k]_q

[n−1]_q [n+ 1]_q

= [n−1]_q [n+ 1]_qbn

n−1Y

s=0

µ

1−q^s x bn

¶X∞ k=1

[n+k−2]_q! [n−1]_q! [k−1]_q!

µx bn

¶_k

= [n−1]_q [n+ 1]_qbn

n−1Y

s=0

µ

1−q^s x bn

¶X∞ k=0

[n+k−1]_q! [n−1]_q! [k]_q!

µx bn

¶_k+1

= [n−1]_q [n+ 1]_q

x bn

bn

X∞ k=0

[n+k−1]_q! [n−1]_q! [k]_q!

µx bn

¶_{k n−1}Y

s=0

µ

1−q^s x bn

¶

= [n−1]_q [n+ 1]_qx

X∞

k=0

·n+k−1 k

¸

q

µx bn

¶k n−1Y

s=0

µ

1−q^sx bn

¶

= [n−1]_q

[n+ 1]_qx, (2.3)

and

(Ln,qt²)(x) =b²_n X∞ k=0

mn,k,q

µx bn

¶[n+k]_q! k!

[k+ 2]_q! [k+ 2 +n]_q!

=b²_n

n−1Y

s=0

µ

1−q^sx bn

¶X∞ k=0

[n+k−1]_q! [n−1]_q! [k]_q!

µx bn

¶_k

[k+ 2]_q[k+ 1]_q [k+ 2 +n]_q[k+ 1 +n]_q

=b²_n

n−1Y

s=0

µ

1−q^sx bn

¶X∞ k=0

[n+k−1]_q! [n−1]_q! [k]_q!

µx bn

¶_k

1 +q+q[k]_q+ 2q²[k]_q+q³[k]²_q [k+ 2 +n]_q[k+ 1 +n]_q

≤b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q!×

× X∞

k=0

[n+k−3]_q! [k]_q!

µx bn

¶_k³

1 +q+q[k]_q+ 2q²[k]_q+q³[k]²_q

´

= (1 +q)b²_n

n−1Y

s=0

µ

1−q^sx b_n

¶ 1

[n−1]_q[n−2]_q X∞

k=0

[n+k−3]_q! [n−3]_q! [k]_q!

µx b_n

¶_k

+¡

q+ 2q²¢ b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q

X∞ k=0

[n+k−2]_q! [n−2]_q! [k]_q!

µx bn

¶k+1

+q³b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q!

X∞

k=1

[n+k−3]_q! [k−1]_q!

µx bn

¶_k [k]_q

= (1 +q)b²_n 1

[n−1]_q[n−2]_q +¡

q+ 2q²¢ b²_n x

[n−1]_q +q³b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q!

X∞

k=1

[n+k−3]_q! [k−1]_q!

µx bn

¶_k³

1 +q[k−1]_q

´

= (1 +q)b²_n 1

[n−1]_q[n−2]_q +¡

q+ 2q²¢ b²_n 1

[n−1]_q x bn

+q³b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q!

X∞ k=0

[n+k−2]_q! [k]_q!

µx bn

¶_k+1

+q⁴b²_n

n−1Y

s=0

µ

1−q^sx bn

¶ 1 [n−1]_q!

X∞

k=0

[n+k−1]_q! [k]_q!

µx bn

¶_k+2

= (1 +q)b²_n [n−1]_q[n−2]_q +¡

q+ 2q²+q³¢ bn

[n−1]_qx+q⁴x². (2.4) From (2.2), (2.3) and (2.4), an easy computation gives

¡Ln,q(t−x)²¢

(x)≤ (1 +q)b²_n [n−1]_q[n−2]_q +

¡q+ 2q²+q³¢ bn

[n−1]_q x

+

"

q⁴−2[n−1]_q [n+ 1]_q + 1

#

x²:=An,q(x). (2.5) It is observed here that for 0< q <1,one has [n]_q → _1−q¹ as n→ ∞. This implies that (Ln,qt²)(x) and¡

Ln,q(t−x)²¢

(x) does not converge tox² and 0 respectively, as n → ∞. To obtain some convergence results for q-CMKZD operators defined in (1.4), we will consider a sequence (qn) of real numbers such that 0< qn < 1, limn→∞qn= 1, and

n→∞lim bn

[n]_qn = 0. (2.6)

3. Main results

Now we are ready to obtain some convergence results onq-CMKZD operators.

Theorem 3.1. Let (qn) be a sequence of real numbers such that0 < qn <1 andlimn→∞qn= 1. If f ∈C[0,∞), we have

|(Ln,qnf)(x)−f(x)| ≤2ω(f, q

An,qn(x)), (3.1)

where ω(f,·) is the usual modulus of continuity of f in the space of continuous functions.

Proof. Using (1.4) forq=qn, we have

|(Ln,qnf)(x)−f(x)|

=

¯¯

¯¯ X∞

k=0

[n+k]_qn bn mn,k,qn

µx bn

¶ Z _bn

0

q_n^−kf(t)bn,k,qn

µqnt bn

¶

dqnt−f(x)

¯¯

¯¯

≤ X∞

k=0

[n+k]_qn bn

mn,k,qn

µx bn

¶ Z _b

n

0

q_n^−k|f(t)−f(x)|bn,k,qn

µqnt bn

¶ dqnt

≤ X∞ k=0

[n+k]_qn bn

m_n,k,qn µx

bn

¶ Z _b

n

0

q_n^−k

µ|t−x|

δ + 1

¶

ω(f, δ)b_n,k,qn µqnt

bn

¶ d_qnt

=ω(f, δ) X∞ k=0

[n+k]_q

n

bn mn,k,qn

µx bn

¶ Z _bn

0

q_n^−kbn,k,qn

µq_nt bn

¶ dqnt +ω(f, δ)

δ X∞

k=0

[n+k]_q

n

bn mn,k,qn

µx bn

¶ Z _bn

0

q_n^−k|t−x|bn,k,qn

µqnt bn

¶ dqnt

≤ω(f, δ) +ω(f, δ) δ

©¡L_n,qn(t−x)²¢ (x)ª_1/2

≤ω(f, δ) +ω(f, δ)

δ {An,qn(x)}^1/2 Now, if we chooseδ²=A_n,qn(x),we get

|(Ln,qnf)(x)−f(x)| ≤2ω(f, q

An,qn(x)), and the proof of Theorem 3.1 is thus complete.

It is easy to see that, the right-hand side of formula (3.1) can diverge. Indeed, forx=^b₂ⁿ we cannot guaranteeδ→0 asn→ ∞.

From Lemma 2.1 and Theorem 3.1, we can immediately give the following Bohman-Korovkin-type theorem.

Theorem 3.2. Let (qn) be a sequence of real numbers such that0 < qn <1 and limn→∞qn = 1. Then, for f ∈ C[0,∞), the sequence Ln,qn(f, x) converges uniformly to f(x) on any closed finite subinterval [0, A], where A > 0 being a constant.

Definition 3.3. For f ∈ C[a, b] and t > 0, the Peetre-K Functional are defined by

K(f, δ) := inf

g∈C²[a,b]

n

kf−gk_C[a,b]+tkgk_C2[a,b]

o . Theorem 3.4. If g∈C²[0, A], then

|(Ln,qg)(x)−g(x)| ≤An,q(x)kgk_C2[0,A], whereA >0 is a constant.

Proof. By Taylor formula with integral reminder term, we write

g(t) =g(x) + (t−x)g⁰(x) + Z _t−x

0

(t−x−u)²g⁰⁰(x+u)du. (3.2) If we apply the operator (1.4) to (3.2), we get

|(Ln,qg)(x)−g(x)|

=

¯¯

¯¯g⁰(x)(Ln,q(t−x))(x) + µ

Ln,q

µZ _t−x

0

(t−x−u)²g⁰⁰(x+u)du

¶¶

(x)

¯¯

¯¯

≤ kg⁰kC[0,A]|(Ln,q(t−x))(x)|

+kg⁰⁰kC[0,A]

¯¯

¯¯ µ

Ln,q

µZ _t−x

0

(t−x−u)²du

¶¶

(x)

¯¯

¯¯.

Since Z _t−x

0

(t−x−u)²du= (t−x)²

2 ,

one gets from (2.5)

|(Ln,qg)(x)−g(x)| ≤ kg⁰k_C[0,A]{An,q(x)}^1/2+kg⁰⁰k_C[0,A]An,q(x).

Now noting that

kgk_C2[a,b] =kgk_C[a,b]+kg⁰k_C[a,b]+kg⁰⁰k_C[a,b], we get

|(L_n,qg)(x)−g(x)| ≤A_n,q(x)kgk_C2[0,A],

and this completes the proof of Theorem 3.4.

Now, we are ready to prove the following theorem.

Theorem 3.5. Let (qn) be a sequence of real numbers such that0 < qn <1 andlim_n→∞q_n= 1. If f ∈C[0,∞), then

k(Ln,qnf)−fk_C[0,A]≤2K(f, Bn,qn),

where Bn,qn is the maximum value of An,qn(x) on [0, A], A > 0 is a constant;

namely,

Bn,q = (1 +q)b²_n [n−1]_q[n−2]_q +

¡q+ 2q²+q³¢ bn

[n−1]_q A+

"

q⁴−2[n−1]_q [n+ 1]_q + 1

# A².

Proof. By the linearity property of (Ln,qn), we get

|(Ln,qnf)(x)−f(x)|

≤ |(Ln,qnf)(x)−(Ln,qng)(x)|+|(Ln,qng)(x)−g(x)|+|g(x)−f(x)|

≤ kf −gk_C[0,A]|(Ln,qn1)(x)|+kf−gk_C[0,A]+|(Ln,qng)(x)−g(x)|. From Theorem 3.4, one has

|(Ln,qnf)(x)−f(x)| ≤2kf−gk_C[0,A]+An,qn(x)kgk_C2[0,A], and hence

k(Ln,qnf)−fk_C[0,A]≤2kf −gk_C[0,A]+Bn,qnkgk_C2[0,A]. (3.3) If we take the infimum on the right-hand side of (3.3) over allg∈C²[0, A],we get

k(Ln,qnf)−fk_C[0,A]≤2K(f, Bn,qn).

This completes the proof.

Theorem 3.6. Let (qn) be a sequence of real numbers such that0 < qn <1 and limn→∞qn = 1. If f ∈ Lip^α_M[0,∞), then for any A >0 and x ∈ [0, A] the inequality

|(Ln,qnf)(x)−f(x)| ≤M{Bn,qn}^α2

holds with the constant M, which is independent of n and Bn,qn is as defined in Theorem 3.5.

Proof. For convenience we writeLn,qn(f;x) instead of (Ln,qnf)(x). Note that

|L_n,qn(f;x)−f(x)| ≤L_n,qn(|f(t)−f(x)|;x)

= X∞

k=0

[n+k]_qn bn mn,k,qn

µx bn

¶ Z _bn

0

q^−k_n |f(t)−f(x)|bn,k,qn

µqnt bn

¶ dqnt

≤M Z _bn

0

q_n^−k|t−x|^α X∞ k=0

[n+k]_qn bn

mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶ dqnt.

If we choosep1=_α² andp2= _2−α² , then _p¹

1 +_p¹

2 = 1. Therefore

|Ln,qn(f;x)−f(x)|

≤M Z _bn

0

½

|t−x|²q_n^−k X∞

k=0

[n+k]_q

n

bn mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶¾¹

p1

×

×

½ q_n^−k

X∞

k=0

[n+k]_qn bn mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶¾¹

p2

dqnt.

By H¨older inequality, we have

|Ln,qn(f;x)−f(x)|

≤M

½Z _b

n

0

q_n^−k|t−x|² X∞

k=0

[n+k]_qn bn

mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶ dqnt

¾¹

p1

×

×

½Z _bn

0

q_n^−k X∞ k=0

[n+k]_q

n

bn mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶ dqnt

¾¹

p2

=M

½Z _bn

0

q_n^−k|t−x|² X∞

k=0

[n+k]_qn bn

mn,k,qn

µx bn

¶ bn,k,qn

µqnt bn

¶ dqnt

¾^α

2

.

From (2.5) we obtain

|Ln,qn(f;x)−f(x)| ≤M{An,qn(x)}^α2 . This implies that forx∈[0, A]

|(Ln,qnf)(x)−f(x)| ≤M{Bn,qn}^α2 which in view of (2.5) and (2.6) tends to zero asn→ ∞.

Acknowledgement. The authors are thankful to the referees for their valu- able remarks and suggestions.

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(received 16.06.2011; in revised form 06.02.2012; available online 15.03.2012)

Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280 Golkoy Bolu

E-mail:karsli [email protected]

School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka New Delhi- 110075, India

E-mail:[email protected]