A Dunkl type generalization of Szász operators via post quantum calculus

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R E S E A R C H

Open Access

A Dunkl type generalization of Szász

operators via post-quantum calculus

Abdullah Alotaibi

1

_{, Md. Nasiruzzaman}

2

_{and M. Mursaleen}

1,3*

*_{Correspondence:}

[email protected] 1_{Operator Theory and Applications}

Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

3_{Department of Mathematics,}

Aligarh Muslim University, Aligarh, India

Full list of author information is available at the end of the article

Abstract

The object of this paper to construct Dunkl type Szász operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. We also study the bivariate version of these operators.

MSC: Primary 41A25; 41A36; secondary 33C45

Keywords: (p,q)-integers; (p,q)-analogues of the exponential function; Dunkl analogue; Szász operator; modulus of continuity

1 Introduction and preliminaries

The ﬁrst well-known positive linear operator introduced by S.N. Bernstein [9] in 1912 plays an important role in approximation theory, and the ﬁrstq-analogue of the well-known Bernstein polynomials was introduced by Lupaş [16] in 1987 who applied the idea ofq-integers. In 1997, Phillips considered anotherq-analogue of the classical Bernstein polynomials [26]. Later on, many authors introducedq-generalization of various opera-tors and investigated several approximation properties. In 1950, forx≥0 andf ∈C[0,∞), a positive linear operator was introduced by Szász [30].

A (p,q)-integer [n]p,qis deﬁned as [n]p,q:=p

n_–_qn

p–q ,n= 0, 1, 2, . . . , 0 <q<p≤1. Recently,

Mursaleen et al. [19] applied the (p,q)-calculus in the ﬁeld approximation theory and in-troduced the ﬁrst (p,q)-analogue of Bernstein operators as

Bp_m,q_,(f;x) = 1

pn(n2–1)

n

k=0

n k

p,q

pk(k2–1)_xk

n–k–1

s=0

ps–qsxf

[k]p,q

pk–n_[n] p,q

, x∈[0, 1]. (1.1)

They have also investigated several approximation properties by deﬁning diﬀerent pos-itive linear operators in an approximation process based on a (p,q)-analogue (see [1–8,

14,18,23–25]). Recently they have also studied the Szász-type operators via Dunkl gener-alizations (see [17,20–22,28]). After that many authors introducedq-generalizations of various operators and published their work on Dunkl type generalization (see [10,13,29,

31]).

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There are two types of a (p,q)-analogue of the exponential function (see [15]),

ep,q(x) = ∞

n=0

pn(n2–1) x

n

[n]p,q!

and Ep,q(x) = ∞

n=0

qn(n2–1) x

n

[n]p,q!

,

which satisfy the equalityep,q(x)Ep,q(–x) = 1, and forp= 1,ep,q(x) andEp,q(x) reduce to

q-exponential functions. Forx≥0,f ∈C[0,∞),μ≥0,n∈N, Sucu [29] deﬁned a Dunkl analogue of Szász operators via a generalization of the exponential function given by [27]. Ben Cheikh et al. [10] introduced theq-Dunkl classicalq-Hermite type polynomials. They gave deﬁnitions ofq-Dunkl analogues of exponential functions, recursion relations and notations forμ> –1₂and 0 <q< 1. Forμ>₂1,x≥0, 0 <q< 1 andf ∈C[0,∞), Gürhan Içöz gave a Dunkl generalization of Szász operators viaq-calculus [12] as

Dn,q(f;x) =

1 eμ,q([n]qx)

∞

k=0

([n]qx)k γμ,q(k)

f

1 –q2μθk+k

1 –qn . (1.2)

Here we deﬁne (p,q)-Dunkl analogues of exponential functions, recursion relations and notations forμ>1₂and 0 <q<p≤1, respectively, by

eμ,p,q(x) = ∞

n=0

pn(n2–1) x

n

γμ,p,q(n)

, x∈[0,∞),n∈N(natural number). (1.3)

We deﬁne an explicit formula forγμ,p,q(n) as follows:

γμ,p,q(n)

=

[n+1₂ ]–1

i=0 p2μ(–1)

i+1₊₁

((p2₎i_p2μ+1_{– (}_q2₎i_q2μ+1₎[n2]–1 j=0 p2μ(–1)

j₊₁

((p2₎j_p2_{– (}_q2₎j_q2₎

(p–q)n , (1.4)

where [n+1₂ ] and [n₂] denote the greatest integer functions forn∈N∪ {0}, and we have

(x–y)n_p_,_q=

⎧ ⎨ ⎩

n–1

i=0(pix–qiy) ifn∈N,

1 ifn= 0.

Also

γμ,p,q(n+ 1) =

p2μ(–1)n+1+1_(p2μθn+1+n+1_–_q2μθn+1+n+1₎

(p–q) γμ,p,q(n), n∈N,

θn= ⎧ ⎨ ⎩

0 ifn= 2m,m∈N,

1 ifn= 2m+ 1,m∈N.

(1.5)

We can derive some of the special cases ofγμ,p,q(n) as follows:

γμ,p,q(0) = 1

from (p,q)-binomial expansion,

γμ,p,q(1) =p–2μ+1

p2μ+1_–_q2μ+1

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γμ,p,q(2) =p2

p2μ+1_–_q2μ+1

p–q

p2_–_q2

p–q ,

γμ,p,q(3) =p3–2μ

p2μ+1_–_q2μ+1

p–q

p2_–_q2

p–q

p2μ+3_–_q2μ+3

p–q ,

γμ,p,q(4) =p4

p2μ+1_–_q2μ+1

p–q

p2_–_q2

p–q

p2μ+3_–_q2μ+3

p–q

p4_–_q4

p–q ,

γμ,p,q(5) =p5–2μ

p2μ+1_–_q2μ+1

p–q

p2_–_q2

p–q

p2μ+3_–_q2μ+3

p–q

×

p4_–_q4

p–q

p2μ+5_–_q2μ+5

p–q .

2 Auxiliary results

For anyx∈[0,∞),f ∈C[0,∞)n∈N, 0 <q<p≤1, andμ>1₂, we deﬁne

Dn,p,q(f;x) =

1 eμ,p,q([n]p,qx)

∞

k=0

([n]p,qx)k γμ,p,q(k)

pk(k2–1)_f

p2μθk+k_–_q2μθk+k

pk–1_(pn_–_qn₎ . (2.1)

If p= 1 in (2.1), then the Dunkl generalization of Szász operators via (q)-calculus (1.2) becomes a particular case of (p,q)-operators deﬁned by us. Thus we can say that our op-erators can be considered a generalization of opop-erators (1.2).

Lemma 2.1 Let Dn,p,q(·,·)be the operators given by(2.1).Then we have the following

re-sults:

(1) Dn,p,q(e0;x) = 1.

(2) Dn,p,q(e1;x) =x.

(3) x2+_[_nq2_]μ

p,q[1 – 2μ]p,q

eμ,p,q(pq[n]p,qx)

eμ,p,q([n]p,qx) x≤Dn,p,q(e2;x)≤x 2₊ 1

[n]p,q[1 + 2μ]p,qx, where ej(t) =tj,j= 0, 1, 2, . . . .

Proof (1)Dn,p,q(1;x) =_e_μ_,_p_,_q_([1_n_]_p_,_q_x₎

_∞

k=0 ([n]p,qx)k

γμ,p,q(k)p k(k–1)

2 _{= 1 (from (}_1.3_)). (2)

Dn,p,q(e1;x) =

∞

k=0

pk(k2–1)

p2μθk+k_–_q2μθk+k pk–1_(pn_–_qn₎

= 1 [n]p,q

∞

k=1

([n]p,qx)k γμ,p,q(k– 1)

p(k–1)(2k–2)

= 1 [n]p,q

∞

k=0

([n]p,qx)k+1 γμ,p,q(k)

pk(k2–1)

= x

eμ,p,q([n]p,qx) ∞

k=0

pk(k2–1)

=x.

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Dn,p,q(e2;x) =

∞

k=0

pk(k2–1)

p2μθk+k_–_q2μθk+k pk–1_(pn_–_qn₎

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= 1 [n]2

p,q

∞

k=1

pk2–52k+4

p2μθk+k_–_q2μθk+k (p–q)

= 1 [n]2

p,q

∞

k=1

p(k–1)(2k–4)

p2μθk+k_–_q2μθk+k (p–q) ,

hence

Dn,p,q(e2;x)

= 1 [n]2

p,q

∞

k=0

([n]p,qx)k+1 γμ,p,q(k)

pk(k2–3)

p2μθk+1+k+1_–_q2μθk+1+k+1

(p–q) . (2.2)

From a simple calculation we know that

[2μθk+1+k+ 1]p,q=p2μ(–1)

k₊₁

[2μθk+k]p,q+q2μθk+k

2μ(–1)k+ 1_p_,_q. (2.3)

Fork= 2k, Eq. (2.3) implies

[2μθ2k+1+ 2k+ 1]p,q=p2μ+1

p2μθ2k+2k_–_q2μθ2k+2k

p–q +q

2μθ2k+2k_{[1 + 2}_μ_]

p,q, (2.4)

and fork= 2k+ 1, we have

[2μθ2k+2+ 2k+ 2]p,q=p–2μ+1

p2μθ2k+1+2k+1_–_q2μθ2k+1+2k+1 p–q

+q2μθ2k+1+2k+1_{[1 – 2}_μ_]

p,q. (2.5)

Now by separating (2.2), into the even and odd terms and using (2.4)–(2.5), we have

Dn,p,q(e2;x)

= 1 [n]2

p,q

∞

k=0

p2μ(–1)k₊₁([n]p,qx)k+1 γμ,p,q(k)

pk(k2–3)

(p–q) k=2k,2k+1

+ 1 [n]2

p,q

∞

k=0

([n]p,qx)2k+1 γμ,p,q(2k)

pk(2k–3)q2μθ2k+2k_{[1 + 2}_μ_]

p,q

+ 1 [n]2

p,q

∞

k=0

([n]p,qx)2k+2 γμ,p,q(2k+ 1)

p(k–1)(2k+1)q2μθ2k+1+2k+1_{[1 – 2}_μ_]

p,q

≥ x2

k=0

p2μ(–1)k+1([n]p,qx)k γμ,p,q(k)

pk(k2–1)

+ 1 q[n]2

p,q

[1 – 2μ]p,q ∞

k=0

(q[n]p,qx)2k+1 γμ,q(2k)

pk(2k–3)

+q

2μ–1

[n]2

p,q

[1 – 2μ]p,q ∞

k=0

(q[n]p,qx)2k+2 γμ,p,q(2k+ 1)

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Here we have used the inequality [1 – 2μ]p,q≤[1 + 2μ]p,q, and, for 0 <q<p≤1 andμ>1₂,

a simple calculation led top2μ_≤_1,_p–2μ_≥_{1. Therefore,}

Dn,p,q(e2;x)≥x2+

q2μ [n]p,q

x eμ,p,q([n]p,qx)

[1 – 2μ]p,q ∞

k=0

(q_p[n]p,qx)k γμ,p,q(k)

pk(k2–1)

≥x2+ q

2μ [n]p,q

[1 – 2μ]p,q

eμ,p,q(q_p[n]p,qx)

eμ,p,q([n]p,qx)

x.

On the other hand, we have

Dn,p,q(e2;x)

≤x2+ 1 [n]p,q

[1 + 2μ]p,q ∞

k=0

(q[n]p,qx)2k γμ,q(2k)

pk(2k–3)

+ q

2μ [n]p,q

[1 + 2μ]p,q ∞

k=0

(q[n]p,qx)2k+1 γμ,p,q(2k+ 1)

p(k–1)(2k+1)

≤x2+ 1 [n]p,q

[1 + 2μ]p,q ∞

k=0

(_pq[n]p,qx)2k γμ,q(2k)

pk(2k–1)

+ q

2μ [n]p,q

[1 + 2μ]p,q ∞

k=0

(q_p[n]p,qx)2k+1 γμ,p,q(2k+ 1)

pk(2k+1)

≤x2+ 1 [n]p,q

[1 + 2μ]p,q ∞

k=0

pk(k2–1)

≤x2+ 1 [n]p,q

[1 + 2μ]p,qx.

Lemma 2.2 Let the operators Dn,p,q(·,·)be given by(2.1).Then we have the following

re-sults:

(1) Dn,p,q(e1–x;x) = 0.

(2) _[q_n_]2μ

p,q[1 – 2μ]p,q

eμ,p,q(pq[n]p,qx)

eμ,p,q([n]p,qx) x≤Dn,p,q((e1–x)

2_;_x)_≤ 1

[n]p,q[1 + 2μ]p,qx. 3 Main results

In this section we obtain a Korovkin’s type approximation theorem and compute conver-gence of the considered operators by using the modulus of continuity and also rate of convergence for functions belonging to the Lipschitz class presented here.

LetCB(R+) be the set of all bounded and continuous functions onR+= [0,∞), which is

a linear normed space with

fCB=sup

x≥0

f(x).

And also let

H:=

f :x∈[0,∞), f(x)

1 +x2 is convergent asx→ ∞

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In order to obtain the convergence results for the operatorsDn,p,q(·,·), we takeq=qn,p=

pnwhereqn∈(0, 1) andpn∈(qn, 1] satisfy

lim

n pn→1, limn qn→1 and limn p n

n→a, lim_n qnn→b. (3.1)

Theorem 3.1 Let p=pn,q=qnsatisfy(3.1),with0 <qn<pn≤1.Let Dn,pn,qn(·,·)be the operators given by(2.1).Then for any function f ∈X[0,∞)∩H,

lim

n→∞Dn,pn,qn(f;x) =f(x)

uniformly on each compact subset of[0,∞).

Proof The proof is based on the well-known Korovkin’s theorem regarding the conver-gence of a sequence of linear and positive operators, so it is enough to prove the conditions

lim

n→∞Dn,pn,qn(ej;x) =xj, j= 0, 1, 2,

uniformly on [0, 1]. Clearly, from (3.1) and_[_n_]1

pn,qn →0, whenn→ ∞, we have

lim

n→∞Dn,pn,qn(e1;x) =x, nlim→∞Dn,pn,qn(e2;x) =x

2_,

which completes the proof.

We recall the weighted spaces of the functions onR+_{, which are deﬁned as follows:}

PρR+=f :f(x)≤Mfρ(x)

,

QρR+=f :f ∈PρR+∩C[0,∞), Qk_ρR+=

f :f ∈QρR+and lim

x→∞

f(x)

ρ(x)=k(kis a constant)

,

whereρ(x) = 1 +x2is a weight function andMf is a constant depending only onf. And

Qρ(R+) is a normed space with the normfρ=sup_x_≥0|ρf((xx))|.

Theorem 3.2 Let p=pn,q=qnsatisfy(3.1)with0 <qn<pn≤1and let Dn,pn,qn(·,·)be the operators given by(2.1).Then for any function f ∈Qk

ρ(R+)we have

lim

n→∞Dn,pn,qn(f;x) –fρ= 0.

Proof From Lemma2.1, the ﬁrst condition of (1) is fulﬁlled forτ= 0. Now forτ = 1, 2, it is easy to see that from (2)–(3) of Lemma2.1, by using (3.1),

lim

n→∞Dn,pn,qn

eτ

1;x

–xτ ρ= 0.

This completes the proof.

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Letf∈C[0,∞]. The modulus of continuity off, denoted byω(f,δ), gives the maximum oscillation off in any interval of length not exceedingδ> 0 and it is given by the relation

ω(f,δ) = sup |y–x|≤δ

f(y) –f(x), x,y∈[0,∞). (3.2)

It is known thatlimδ→0+ω(f,δ) = 0 forf ∈C[0,∞), and for anyδ> 0 one has

_f(y) –f(x)≤

|y–x|

δ + 1 ω(f,δ). (3.3)

Theorem 3.3 Let f ∈ ˜C[0,∞),x∈[0,∞).Then for0 <q<p≤1the operators Dn,p,q(·,·)

deﬁned by(2.1)satisfy

Dn,p,q(f;x) –f(x)≤

[1 + 2μ]p,qx

ω

f; 1 [n]p,q

,

whereC[0,˜ ∞)is the space of uniformly continuous functions onR+_and_ω_(f_,_δ₎_{is the}

mod-ulus of continuity of a function f ∈ ˜C[0,∞)deﬁned in(3.2).

Proof We prove the claim by using (3.2)–(3.3) and Cauchy–Schwarz inequality:

Dn,p,q(f;x) –f(x)

≤ 1

k=0

pk(k2–1)_f

pk–1_(pn_–_qn₎ –f(x)

≤ 1

k=0

pk(k2–1)

1 +1

δ

p2μθk+k_–_q2μθk+k pk–1_(pn_–_qn₎ –x

ω(f;δ)

=

1 +1

δ

∞

k=0

pk(k2–1)p

2μθk+k_–_q2μθk+k pk–1_(pn_–_qn₎ –x

ω(f;δ)

≤

1 +1

δ

∞

k=0

pk(k2–1)

p2μθ_k+k_–_q2μθ_k+k

pk–1_(pn_–_qn₎ –x

212

×Dn,p,q(e0;x)

1 2

ω(f;δ)

=

1 +1

δ

Dn,p,q(e1–x)2;x

1 2

ω(f;δ)

≤

1 +1

δ

1 [n]p,q

[1 + 2μ]p,qx

ω(f;δ),

where, choosingδ=δn,p,q=

1

[n]p,q, we get our result. Now we give the rate of convergence of the operatorsDn,p,q(f;x) deﬁned in (2.1) in terms

of the elements of the usual Lipschitz classLip_M(ν).

Letf ∈C[0,∞),M> 0 and 0 <ν≤1. We recall that the classLip_M(ν) is given by

Lip_M(ν) =f :f(ζ1) –f(ζ2)≤M|ζ1–ζ2|ν,ζ1,ζ2∈[0,∞)

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Theorem 3.4 Let Dn,p,q(·,·)be the operator deﬁned in(2.1).Then for each f ∈LipM(ν),M>

0, 0 <ν≤1,satisfying(3.4)we have

Dn,p,q(f;x) –f(x)≤M

1 [n]p,q

[1 + 2μ]p,qx

ν

2 .

Proof We prove the claim by using (3.4) and Hölder inequality:

Dn,p,q(f;x) –f(x)≤Dn,p,q

f(e1) –f(x);x

≤Dn,p,qf(e1) –f(x);x

≤ |MDn,p,q

|e1–x|ν;x

.

Therefore,

≤M 1 eμ,p,q([n]p,qx)

∞

k=0

pk(k2–1)p

2μθk+k_–_q2μθk+k pk–1_(pn_–_qn₎ –x

ν

≤M 1 eμ,p,q([n]p,qx)

∞

k=0

([n]p,qx)kp

k(k–1) 2

γμ,p,q(k)

2–ν

2

×

([n]p,qx)kp

k(k–1) 2

γμ,p,q(k)

ν

2

ν

≤M

1 (eμ,p,q([n]p,qx))

∞

k=0

([n]p,qx)kp

k(k–1) 2

γμ,p,q(k) 2–ν

2

×

1 (eμ,p,q([n]p,qx))

∞

k=0

([n]p,qx)kp

k(k–1) 2

γμ,p,q(k)

2

ν

2

≤MDn,p,q(e1–x)2;x

ν

2_.

This completes the proof.

We denote by CB[0,∞) the space of all bounded and continuous functions on R+=

[0,∞) and

C_B2R+=g∈CB

R+_:_g_,_g_∈_C B

R+_, _(3.5)

with the norm

g_C2

B(R+)=gCB(R

+₎+g

CB(R+)+g

CB(R+), (3.6)

also

gCB(R+)=sup

x∈R+

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Theorem 3.5 Let Dn,p,q(·,·)be the operator deﬁned in(2.1).Then for any g∈C2B(R+)we

have

Dn,p,q(f;x) –f(x)≤

[1 + 2μ]p,qx

2[n]p,q

g_C2 B(R+). Proof Letg∈C2

B(R+). Then by using the generalized mean value theorem in the Taylor

series expansion we have

g(e1) =g(x) +g(x)(e1–x) +g(ψ)

(e1–x)2

2 , ψ∈(x,e1). By applying the linearity property ofDn,p,q, we have

Dn,p,q(g,x) –g(x) =g(x)Dn,p,q

(e1–x);x

+g

₍_ψ₎

2 Dn,p,q

(e1–x)2;x

,

which implies

Dn,p,q(g;x) –g(x)≤

1 [n]p,q

[1 + 2μ]p,qx

gCB(R+)

2 .

From (3.6) we havegCB[0,∞)≤ gC2_B[0,∞)and so

Dn,p,q(g;x) –g(x)≤

1 [n]p,q

[1 + 2μ]p,qx g_C2

B(R+) 2 .

This completes the proof due to (1) of Lemma2.2.

The Peetre’sK-functional is deﬁned by

K2(f,δ) = inf

C2_B(R+₎

f –gCB(R+)+δg_C2 B(R+)

:g∈W2, (3.8)

where

W2₌_g_∈_C

B

R+_:_g_,_g_∈_C

B

R+_. _(3.9)

Then there exits a positive constantC> 0 such thatK2(f,δ)≤Cω2(f,δ

1

2_),_δ_{> 0, where the} second order modulus of continuity is given by

ω2

f,δ12₌ _sup

0<h<δ12

sup

x∈R+

f(x+ 2h) – 2f(x+h) +f(x). (3.10)

Theorem 3.6 Let Dn,p,q(·,·)be the operator deﬁned in(2.1)and CB[0,∞)the space of all

bounded and continuous functions onR+_._{Then for x}_∈_R+_,_f _∈_C

B(R+)we have Dn,p,q(f;x) –f(x)≤2M

ω2

f;

[1 + 2μ]p,qx

4[n]p,q

+min

1,[1 + 2μ]p,qx 4[n]p,q

fCB(R+)

,

where M is a positive constant andω2(f;δ)is the second order modulus of continuity of the

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Proof We prove this claim by using Theorem (3.5):

≤Dn,p,q(f–g;x)+Dn,p,q(g;x) –g(x)+f(x) –g(x) ≤2f–gCB(R+)+

[1 + 2μ]p,qx

2[n]p,q

g_C2 B(R+). From (3.6) we clearly havegCB[0,∞)≤ gC2

B[0,∞). Therefore,

Dn,p,q(f;x) –f(x)≤2

f –gCB(R+)+

[1 + 2μ]p,qx

4[n]p,q

g_C2 B(R+) . By taking the inﬁmum over allg∈C2

B(R+) and using (3.8), we get Dn,p,q(f;x) –f(x)≤2K2

f;[1 + 2μ]p,qx 4[n]p,q

.

Now for an absolute constantC> 0 given in [11] we have the relation

K2(f;δ)≤C

ω2(f;

√

δ) +min(1,δ)f.

4 Bivariate operators and rate of convergence

In this section, we construct a bivariate extension of the operators (2.1).

LetR2₊= [0,∞)×[0,∞),f :C(R2₊)→Rand 0 <qn1,qn2 <pn1,pn2≤1. We deﬁne the bivariate extension of the Dunkl (p,q)-Szász operators (2.1) as follows:

D(pn1,pn2),(qn1,qn2)

n1,n2 (f;x,y)

= 1

eμ1,pn₁,qn₁([n1]pn₁,qn₁x)

1

eμ2,pn₂,qn₂([n2]pn₂,qn₂y)

× ∞

k1=0

∞

k2=0

([n1]pn1,qn1x)

k1

γμ1,pn₁,qn₁(k1)

([n2]pn2,qn2y)

k2

γμ2,pn₂,qn₂(k2)

×p k1(k1–1)

2

n1 p

k2(k2–1) 2

n2 f

p2μ1θk1+k1

n1 –q

2μ1θk₁+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1) ,p

2μ2θk₂+k2

n2 –q

2μ2θk₂+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2)

, (4.1)

where eμ1,pn1,qn1([n1]pn1,qn1x) =

∞

k1=0

([n1]pn1 ,qn1x) k₁ γμ1,pn1 ,qn1(k1)p

k₁₍k_1–1) 2

n1 and eμ2,pn2,qn2([n2]pn2,qn2y) =

_∞

k2=0

([n2]pn2 ,qn2y) k₂ γμ2,pn2 ,qn2(k2) p

k₂₍k_2–1) 2

n2 .

Lemma 4.1 Let ei,j:R2+→[0,∞),ei,j= (uv)ij,i,j= 0, 1, 2, . . . ,be the two-dimensional test

functions.Then the q-bivariate operators deﬁned in(4.1)satisfy the following: (1) D(pn1,pn2),(qn1,qn2)

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(2) D(pn1,pn2),(qn1,qn2)

n1,n2 (e1,0;x,y) =x, (3) D(pn1,pn2),(qn1,qn2)

n1,n2 (e0,1;x,y) =y, (4) D(pn1,pn2),(qn1,qn2)

n1,n2 (e2,0;x,y)≤x2+_[_n₁_]_pn1

1 ,qn1

[1 + 2μ1]pn₁,qn₁x, (5) D(pn1,pn2),(qn1,qn2)

n1,n2 (e0,2;x,y)≤y2+

1 [n2]pn2 ,qn2

[1 + 2μ2]pn2,qn2y.

Lemma 4.2 The q-bivariate operators deﬁned in(4.1)satisfy the following:

(1) D(pn1,pn2),(qn1,qn2)

n1,n2 (e1,0–x;x,y) = 0, (2) D(pn1,pn2),(qn1,qn2)

n1,n2 (e0,1–y;x,y) = 0, (3) D(pn1,pn2),(qn1,qn2)

n1,n2 ((e1,0–x)

2_;_x,_y)_≤ 1

[n1]pn1 ,qn1[1 + 2

μ1]pn1,qn1x, (4) D(pn1,pn2),(qn1,qn2)

n1,n2 ((e0,1–y)2;x,y)≤_[_n₂_]_pn1

2 ,qn2

[1 + 2μ2]pn₂,qn₂y.

The rate of convergence of operatorsD(pn1,pn2),(qn1,qn2)

n1,n2 (f;x,y) deﬁned in (4.1) by means of modulus of continuity of some bivariate modulus of smoothness functions is now in-troduced.

To obtain the convergence results for the operatorsD(pn1,pn2),(qn1,qn2)

n1,n2 (f;x,y), we takeq= qn1,qn2andp=pn1,pn2where 0 <qn1<pn1≤1 and 0 <qn2<pn2≤1 are such that

lim

n1,n2

pn1,pn2→1 and lim

n1,n2

qn1,qn2→1. (4.2)

The modulus of continuity for the bivariate case is deﬁned as follows: Forf ∈Hω(R2

+),

ω(f;δ1,δ2)

=sup

u,x≥0

f(u,v) –f(x,y);|u–x| ≤δ1,|v–y| ≤δ2, (u,v), (x,y)∈R2+

, (4.3)

whereHω(R+_{) is the space of all real-valued continuous functions. Then for all}_f _∈_Hω(_R +),

ω(f;δ1,δ2) satisﬁes the following:

(i) limδ1,δ2→0ω(f;δ1,δ2)→0,

(ii) |f(u,v) –f(x,y)| ≤ω(f;δ1,δ2)(|uδ–1x|+ 1)(

|v–y|

δ2 + 1).

Theorem 4.3 Let pn=pn1,pn2and qn=qn1,qn2satisfy(4.2)and consider(x,y)∈[0,∞), 0 <

qn1,qn2<pn1,pn2 ≤1.Suppose D

(pn₁,pn₂),(qn₁,qn₂)

n1,n2 (f;x,y)are the operators deﬁned by(4.1). Then for any function f ∈ ˜C([0,∞)×[0,∞)),we have

n1,n2 (f;x,y) –f(x,y)

≤1 +[1 + 2μ1]pn₁,qn₁x

1 +[1 + 2μ2]pn₂,qn₂y

×ω

f; 1 [n1]pn1,qn1

, 1 [n2]pn2,qn2

,

whereC[0,˜ ∞)is the space of uniformly continuous functions onR+_and_ω_(f_,_δ

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Proof We prove the claim by using the results for the modulus of continuity and Cauchy– Schwarz inequality:

n1,n2 (f;x,y) –f(x,y)

≤ 1

eμ1,pn1,qn1([n1]pn1,qn1x)

1

eμ2,pn2,qn2([n2]pn2,qn2y)

× ∞

k1=0

∞

k2=0

([n1]pn₁,qn₁x)k1

γμ1,pn1,qn1(k1)

([n2]pn₂,qn₂y)k2

γμ2,pn2,qn2(k2) p

k1(k1–1) 2

n1 p

k2(k2–1) 2

n2

×f

p2μ1θk1+k1

n1 –q

2μ1θk1+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1) ,p

2μ2θk2+k2

n2 –q

2μ2θk2+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2)

–f(x,y)

≤ 1

eμ1,pn₁,qn₁([n1]pn₁,qn₁x)

1

× ∞

k1=0

∞

k2=0

([n1]pn₁,qn₁x)k1

γμ1,pn1,qn1(k1)

([n2]pn₂,qn₂y)k2

γμ2,pn2,qn2(k2) p

k1(k1–1) 2

n1 p

k2(k2–1) 2

n2

×

1 + 1

δn1

p2

μ1θk₁+k1

n1 –q

2μ1θk₁+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1)

–x

×

1 + 1

δn2

p2

μ2θk2+k2

n2 –q

2μ2θk2+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2)

–y ω(f;δn1,δn2)

≤

1 + 1

δn1

1

∞

k1=0

([n1]pn₁,qn₁x)k1

γμ1,pn1,qn1(k1) p

k1(k1–1) 2

n1

×

p2μ1θk1+k1

n1 –q

2μ1θk₁+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1) –x

212

×

1 + 1

δn2

1

∞

k2=0

([n2]pn2,qn2y)

k2

p k₂₍k_2–1)

2

n2

×

p2μ2θk2+k2

n2 –q

2μ2θk2+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2) –y

212

×ω(f;δn1,δn2)

=

1 + 1

δn1

n1,n2 (e1,0–x)

2_;_x,_y12

×

1 + 1

δn2

n1,n2 (e0,1–y)2;x,y

1 2

×ω(f;δn1,δn2)

≤

1 + 1

δn1

1 [n1]pn1,qn1

[1 + 2μ1]pn₁,qn₁x

×

1 + 1

δn2

1 [n2]pn₂,qn₂

[1 + 2μ2]pn2,qn2y ω(f;δ1,δ2).

Choosingδ1=δn1=

1 [n1]qn1 and

δ2=δn2=

1

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Now we give the rate of convergence of the operatorsD(pn1,pn2),(qn1,qn2)

n1,n2 (f;x,y) deﬁned in (4.1) in terms of the elements of the usual Lipschitz classLip_M(ν1,ν2).

Letf ∈C([0,∞)×[0,∞)),M> 0 and 0 <ν1,ν2≤1. We recall that the classLipM(ν1,ν2)

is deﬁned by

Lip_M(ν1,ν2) =

f :f(u,v) –f(x,y)≤M|u–x|ν1|_v_–_y|ν2_{, (u,}_{v), (x,}_y)∈_[0,∞₎2_{. (4.4)}

Theorem 4.4 Let D(pn1,pn2),(qn1,qn2)

n1,n2 (f;x,y)be the operator deﬁned in(4.1).Then for each f ∈Lip_M(ν1,ν2),M> 0, 0 <ν1,ν2≤1,satisfying(4.4),we have

n1,n2 (f;x,y) –f(x,y)≤M

λn1(x)

ν1 2_λ

n2(y)

ν2 2_,

whereλn1(x) =D

n1,n2 ((e1,0–x)2;x,y),andλn2(y) =D

n1,n2 ((e0,1–y)2;x,y). Proof We prove the claim by using (4.4) and Hölder inequality:

n1,n2 (f;x,y) –f(x,y)

≤D(pn1,pn2),(qn1,qn2)

n1,n2

f(u,v) –f(x,y);x,y

n1,n2 f(u,v) –f(x,y);x,y

n1,n2

|e1,0–x|ν1;x,yD

(pn1,pn2),(qn1,qn2)

n1,n2

|e0,1–y|ν2;x,y

.

Therefore,

n1,n2 (f;x,y) –f(x,y)

≤M 1

∞

k1=0

([n1]pn₁,qn₁x)k1

γμ1,pn1,qn1(k1) p

k1(k1–1) 2

n1

p2μ1θk1+k1

n1 –q

2μ1θk1+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1) –x

ν1

× 1

eμ₂,pn₂,qn₂([n2]pn₂,qn₂y)

∞

k2=0

([n2]pn2,qn2y)

k2

p k₂₍k_2–1)

2

n2

p2

μ2θk2+k2

n2 –q

2μ2θk2+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2) –y

ν2

≤M

1

(eμ₁,pn₁,qn₁([n1]pn₁,qn₁x))

∞

k1=0

([n1]pn1,qn1x)

k1_p k₁₍k_1–1)

2

n1

γμ1,pn₁,qn₁(k1)

2–ν1 2

×

1

(eμ1,pn1,qn1([n1]pn1,qn1x))

× ∞

k1=0

([n1]pn₁,qn₁x)k1p k1(k1–1)

2

n1

γμ1,pn1,qn1(k1)

p2

μ1θk1+k1

n1 –q

2μ1θk1+k1

n1 pk1–1

n1 (p

n1

n1–q

n1

n1) –x 2 ν₁ 2 × 1

(eμ2,pn2,qn2([n2]pn2,qn2y))

∞

k2=0

([n2]pn₂,qn₂y)k2p k2(k2–1)

2

n2

γμ2,pn2,qn2(k2)

2–ν2 2

×

1

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× ∞

k2=0

([n2]pn2,qn2y)

k2_p k₂₍k_2–1)

2

n2

p2

μ2θk₂+k2

n2 –q

2μ2θk₂+k2

n2 pk2–1

n2 (p

n2

n2–q

n2

n2) –y

2

ν2 2

n1,n2 (e1,0–x)

2_;_x,_yν21_D(pn1,pn2),(qn1,qn2)

n1,n2 (e0,1–y)

2_;_x,_yν22_.

5 Conclusion

In this paper we have constructed a (p,q)-analogue of Dunkl type Szász operators. We obtained some approximation results for these operators and showed the convergence of the operators by using the modulus of continuity. Furthermore, we obtained the rate of convergence of these operators for functions belonging to the Lipschitz class. We have also studied the bivariate version of these operators.

Funding

This project was funded by the Deanship of Scientiﬁc Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (RG-18-130-37). The authors, therefore, acknowledge with thanks DSR for technical and ﬁnancial support.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors of the manuscript have read it and agreed on its content and are accountable for all aspects of the accuracy and integrity of the manuscript.

Author details

1_{Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz}

University, Jeddah, Saudi Arabia.2_{Department of Computer Science (SEST), Jamia Hamdard, New Delhi, India.} 3_{Department of Mathematics, Aligarh Muslim University, Aligarh, India.}

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Received: 9 July 2018 Accepted: 9 October 2018 References

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