# Approximation of Functions by Dunkl-Type Generalization of Szász-Durrmeyer Operators Based onp,q-Integers

Authors: Abdullah Alotaibi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, our main purpose is to define thep,q-variant of Szász-Durrmeyer type operators with the help of Dunkl generalization generated by an exponential function. We estimate moments and establish some direct results of the aforementioned operators. Moreover, we establish some approximation results in weighted spaces.


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## Body

## 1. Introduction and Preliminaries

The well-known Bernstein operators [1] and the q-Bernstein operators have become very important tools in the study of approximation theory and several branches of applied sciences and engineering [2, 3]. A good approach to introduce the p,q-analogues in approximation theory is given by Mursaleen et al. [4] by an idea of newly introduced integers known asp,q-integers and which is αp,q=pα−qα/p−q,α=0,1,2,⋯,q∈0,pandp∈q,1. In p,q-calculus, there are generally two types of exponential functions which are defined as follows:
(1)ep,qy=∑ℏ=0∞pℏℏ−1/2yℏℏp,q!,Ep,qy=∑ℏ=0∞qℏℏ−1/2yℏℏp,q!.In 1950, Szász [5] defined positive linear operators on 0,∞, and the Dunkl modification of these operators were given by Sucu [6] who was motivated by the work of Cheikh et al. [7]. The new generalization of these Szász operators [6] in quantum calculus (via q-analogue) was introduced in [8] by Içöz and Çekim. Very recent work on the quantum Dunkl analogue in postquantum calculus studied in [9] for a set of all continuous functionsfdefined on0,∞denote it asf∈C0,∞; for parameterλ>−1/2, they designed the following operators:
(2)Dα,p,qf;y=1eλ,p,qαp,qy∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2fpℏ+2λθℏ−qℏ+2λθℏpℏ−1pα−qα.Lemma 1.
Forft=1,t,t2, we have
(3)Dα,p,q1;y=1,Dα,p,qt;y=y,y2+q2λαp,q1−2λp,qeλ,p,qq/pαp,qyeλ,p,qαp,qyy≦Dα,p,qt2;y≦y2+1αp,q1+2λp,qy.Moreover, for everyλ>−1/2 and 0<q<p≦1, where exponential functions and recursion relations are given by
(4)eλ,p,qy=∑ℏ=0∞pℏℏ−1/2yℏγλ,p,qℏ,(5)γλ,p,qℏ=∏i=0ℏ+1/2−1p2λ−1i+1+1p2ip2λ+1−q2iq2λ+1∏j=0ℏ/2−1p2λ−1j+1p2jp2−q2jq2p−qℏ,(6)γλ,p,qℏ+1=p2λ−1ℓ+1+1p2λθℏ+1+ℏ+1−q2λθℏ+1+ℏ+1p−qγλ,p,qℏ,(7)θℏ=0,forℏ=0,2,4,⋯,1,forℏ=,1,3,5,⋯.Forℏ=0,1,2,⋯n, the number ℏ/2 denotes the greatest integer functions.Theq-analogues of Szász operators on the Dunkl type have been studied by several authors in [10–12] and for postquantum calculus in [9, 13–15]. We also refer some useful research articles on these topic (see [16–33]). Some convergence properties of operators through summability techniques can be examined in [34–39].


## 2. New Operators and Estimations of Moments

Here, with the motivational work of [9, 28], we design a different version of thep,q-Szász-Durrmeyer operators compared to the previous one, and we define it by ((9)). To obtain a generalized version of the approximation in Dunkl form generally, we take positive sequences p=pα and q=qα for every 0<qα<1 and qα<pα≦1, and also satisfy the following results:
(8)limα⟶∞pα⟶1,limα⟶∞qα⟶1,limα⟶∞pαα⟶m,limα⟶∞qαα⟶n,where the numbers m and n belong to 0,1.Definition 2.
Let0<q<p≦1,λ>−1/2 and θℏ be defined by (7). Then, for every f∈C0,∞ and y∈0,∞, we have
(9)Sα,p,qλf;y=∑ℏ=0∞Rα,p,qy1ℏ+2λθℏp,q!∫0∞Sα,p,qtfpℏ+2λθℏtpℏ−1dp,qt,where
(10)Rα,p,qy=αp,qeλ,p,qαp,qyαp,qyℏγλ,p,qℏpℏℏ−1/2,Sα,p,qt=pℏ+2λθℏℏ+2λθℏ−1/2αp,qtℏ+2λθℏEp,q−qαp,qt.
α∈ℕ and ζ>α. Moreover, for all α∈ℕ, the gamma functions in the postquantum calculus are defined as follows:
(11)Γp,qα=∫0∞pα−1α−2/2tα−1Ep,q−qtdp,qt,and
(12)Γp,qα+1=p⊖qp,qαp−qα=αp,qα−1p,qΓp,qα−1=αp,q!.Note that(13)ℏ+1+2λθℏp,q=qℏ+2λθℏp,q+pℏ+2λθℏ,(14)ℏ+2+2λθℏp,q=q2ℏ+2λθℏp,q+p+qpℏ+2λθℏ.For more detailed properties of thep,q-analogue of the beta and gamma functions, see [40, 41].Lemma 3.
For the operators in (9), we have Sα,p,qλ1;y=1:
(15)Sα,p,qλf;y≦y+1αp,q,forft=t,y2+1αp,q1+2p,q+1+2λp,qy+2p,qαp,q2,forft=t2,and
(16)Sα,p,qλf;y≧qy+p1+2λαp,q,forft=t,qαp,qα−2p,qq2+2λ1−2λp,qeλ,p,qq/pαp,qyeλ,p,qαp,qy+q+2p,qp2λy+q3y2,forft=t2,Proof.
We prove this Lemma by using the results obtained in (11), (12), (13), and (14). Therefore, for ft=1, we easily see that
(17)Sα,p,qλ1;y=∑ℏ=0∞Rα,p,qy1ℏ+2λθℏp,q!∫0∞Sα,p,qtdp,qt=∑ℏ=0∞Rα,p,qy1αp,qℏ+2λθℏp,q!Γp,qℏ+2λθℏ+1=1.
Takeft=t. Then, we have
(18)Sα,p,qλt;y=∑ℏ=0∞Rα,p,qypℏ+2λθℏpℏ−1αp,qℏ+2λθℏp,q!×∫0∞pℏ+2λθℏℏ+2λθℏ−1/2αp,qtℏ+2λθℏ+1Ep,q−qαp,qtdp,qt=1αp,q2∑ℏ=0∞Rα,p,qy1pℏ−1ℏ+2λθℏp,q!×∫0∞pℏ+2λθℏℏ+2λθℏ+1/2tℏ+2λθℏ+1Ep,q−qtdp,qt=1αp,q2∑ℏ=0∞Rα,p,qy1pℏ−1ℏ+2λθℏp,q!Γp,qℏ+2λθℏ+2=1αp,q2∑ℏ=0∞Rα,p,qy1pℏ−1ℏ+2λθℏ+1p,q=1αp,q2∑ℏ=0∞Rα,p,qyqℏ+2λθℏp,q+pℏ+2λθℏp1−ℏ=qeλ,p,qαp,qy∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2pℏ+2λθℏ−qℏ+2λθℏpℏ−1pα−qα+1αp,qeλ,p,qαp,qy∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2p2λθℏ+1=qDα,p,qt;y+pαp,qeλ,p,qαp,qy∑ℏ=0∞αp,qy2ℏγλ,p,q2ℏp2ℏ2ℏ−1/2p2λθ2ℏ+pαp,qeλ,p,qαp,qy∑ℏ=0∞αp,qy2ℏ+1γλ,p,q2ℏ+1p2ℏ2ℏ+1/2p2λθ2ℏ+1≧qy+p2λ+1αp,qeλ,p,qαp,qy∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2=qy+p2λ+1αp,q.
Similarly,(19)Sα,p,qλt;y≦y+1αp,q.
Ifft=t2, then
(20)Sα,p,qλt2;y=∑ℏ=0∞Rα,p,qypℏ+2λθℏpℏ−1αp,q2ℏ+2λθℏp,q!×∫0∞pℏ+2λθℏℏ+2λθℏ−1/2αp,qtℏ+2λθℏ+2Ep,q−qαp,qtdp,qt=1pαp,q3∑ℏ=0∞Rα,p,qy1p2ℏ−2ℏ+2λθℏp,q!Γp,qℏ+2λθℏ+3=1pαp,q3∑ℏ=0∞Rα,p,qy1p2ℏ−21+ℏ+2λθℏp,q2+ℏ+2λθℏp,q=q3pαp,q3∑ℏ=0∞Rα,p,qyp2−2ℏℏ+2λθℏp,q2+qp+2qpαp,q3∑ℏ=0∞Rα,p,qyp2−ℏ+2λθℏℏ+2λθℏp,q+p+qpαp,q3∑ℏ=0∞Rα,p,qyp2+4λθℏ=q3peλ,p,qαp,qy∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2pℏ+2λθℏ−qℏ+2λθℏpℏ−1pα−qα2+qp+2qpαp,q∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2p1+2λθℏpℏ+2λθℏ−qℏ+2λθℏpℏ−1pα−qα+p+qpαp,q2∑ℏ=0∞αp,qyℏγλ,p,qℏpℏℏ−1/2p2+2λθℏ.
We apply the resultsθℏ defined by (7) and separate it into even and odd terms, i.e., take ℏ=2m and ℏ=2m+1 for all m=0,1,2,⋯, and applying (2) and Lemma 1, we easily see that
(21)Sα,p,qλt2;y≧q3Dα,p,qt2;y+qp+2qp2λαp,qDα,p,qt;y+p+qp2λ+1αp,q2Dα,p,q1;y,and
(22)Sα,p,qλt2;y≦Dα,p,qt2;y+1+2p,qαp,qDα,p,qt;y+2p,qαp,q2Dα,p,q1;y.
These conclusions complete the proof of Lemma3.Lemma 4.
LetΔl=t−yl, for l=1,2; then, we have
(23)Sα,p,qλΔl;y≦1αp,q,forl=1,1αp,q2p,q+1+2λp,q−1y+2p,qαp,q2,forl=2.


## 3. Approximation in Weighted Spaces

To obtain the approximation in weighted Korovkin spaces, we take the weight functionσy=1+y2 and on 0,∞ consider Bσy, Cσy, and Cσyk such that
(24)Bσy=f:fy≦Mf1+y2,whereMfdepends on f,
(25)Cσy=f:f∈C0,∞∩Bσy,Cσyk=f:f∈Cσyandlimy→∞fy1+y2=k,where k is a constant, C0,∞ is the set of continuous functions on 0,∞, CB0,∞ is the set of all bounded and continuous functions on 0,∞ equipped with the norm fCB=supy∈0,∞fy, and on σy, a norm is given by fσy=supy∈0,∞fy/1+y2.Theorem 5.
Letqα∈0,1, pα∈qα,1, and Ωy=fy:y∈0,∞and0,∞/1+y2isconvergentasy⟶∞. Then, for each f∈Ωy∩C0,∞, the sequence Sα,pα,qαλα≧1 converges uniformly to f on each compact subset of 0,∞ if and only if limα⟶∞qα=1 and limα⟶∞pα=1.Proof.
Since the operatorsSα,pα,qαλ defined by (9) are positive and linear on 0,∞, if limα→∞qα⟶1 and limα→∞pα⟶1, then 1/αpα,qα⟶0. Therefore, from Korovkin’s theorem for every f∈C0,∞∩Ωy, the operators Sα,pα,qαλf;y converge uniformly to fy as α⟶∞ if and only if
(26)Sα,pα,qαλti;y⟶yi,i=1,2.
In another way for allf∈C0,∞∩Ωy, if we assume Sα,pα,qαλf;y converges uniformly to fy, when α approaches to ∞, then clearly qα⟶1 and pα⟶1. Suppose in the case where the sequences qα and pα do not converge to 1 and αpα,qα⟶1/p−q as α⟶∞. Thus, from Lemma 3, we have
(27)Sα,pα,qαλt;y⟶y+p−q↛y,and
(28)Sα,pα,qαλt2;y⟶y2+p−q2p,q+2λ+1p,q+1y+p−q22p,q↛y2,which leads to contradiction, and hence, qα⟶1 and pα⟶1 as α⟶∞.Theorem 6.
Let the sequences of positive numbers0<qα<1 and qα<pα≦1 satisfy qα⟶1 and pα⟶1 as α approaches to ∞. Then, for every f∈Cσyk on 0,∞, we have
(29)limα→∞Sα,pα,qαλf;y−fσy=0.Proof.
Takeft=ti for i=0,1,2. Since by Theorem 5, Sα,pα,qαλti;y converges to yi uniformly for i=0,1,2, from Lemma 3, we conclude that
(30)limα→∞Sα,pα,qαλ1;y−1σy=0.
Ifft=t,
(31)Sα,pα,qαλt;y−yσy=supy≧0Sα,pα,qαλt;y−y1+y2≦1αpα,qαsupy≧011+y.
Then, we have(32)limα→∞Sα,pα,qαλt;y−yσy=0.
Similarly, if we takei=2, we have
(33)Sα,pα,qαλt2;y−y2σy=supy≧0Sα,pα,qαλt2;y−y21+y2≦1αpα,qα2pα,qα+2λ+1pα,qα+1supy≧0y1+y2+2pα,qααpα,qα2supy≧011+y2,which implies that
(34)limα→∞Sα,pα,qαλt2;y−y2σy=0.
These explanations complete the proof of Theorem6.Let the modulus of the continuity off for any δ>0 and ρ>0 be defined as follows:
(35)ωρf;δ=supt−yδsupy,t∈0,ρft−fy,where it is obvious that limδ→0+ωρf;δ=0 and for f∈C0,∞(36)ft−fy≦1+t−yδωρf;δ.Theorem 7.
Take the numbersq=qα,p=pα with the positive sequences qα∈0,1, pα∈qα,1 satisfying qα⟶1 and pα⟶1 as α⟶∞. Let ωρf;δ be defined on the interval 0,ρ+1⊂0,∞ for ρ>0. Then, for every f∈Cσyk on 0,∞, we have
(37)Sα,p,qλf;y−fy≦6Mf1+ρ2δαy+2ωρ+1f;δαy,where Mf is a constant depending only on f.Proof.
Lety∈0,ρ and t≦ρ+1 for ρ>0. Then, clearly one has
(38)ft−fy≦6Cf1+ρ2t−y2.Also, when y∈0,ρ and t>ρ+1 for ρ>0, then for a given δ>0(39)ft−fy≦ωρ+1f;t−y≦t−yδ+1ωρ+1f;δ.
From (38) and (39), we easily see that
(40)ft−fy≦6Cf1+ρ2t−y2+1+t−yδωρ+1f;δ,which implies that
(41)Sα,pα,qαλft−fy;y≦6Cf1+ρ2Sα,pα,qαλΔ2;y+Sα,pα,qαλ1+t−yδ;yωρ+1f;δ.
The Cauchy-Schwartz inequality gives us(42)Sα,pα,qαλt−y;y≦Sα,pα,qαλt−y2;y1/2.
From an easy calculation, this leads us to(43)Sα,pα,qαλf;y−fy≦Sα,pα,qαλft−fy;y.
Therefore, in view of (41)–(43), clearly we get
(44)Sα,pα,qαλf;y−fy≦6Cf1+ρ2Sα,pα,qαλΔ2;y+1+1δSα,pα,qαλΔ2;y1/2ωρ+1f;δ.
Finally, if we takeδ=δαy1/2=Sα,p,qλΔ2;y, then we use a denumerable to get the result.


## 4. Pointwise Approximation

In an approximation process for measuring the smoothness of a continuous function, we need Peetre’sK-functional [42] defined as follows.Definition 8.
Letf∈C0,∞, and for a givenδ>0of theK-functional, we have
(45)K2f;δ=infy∈0,∞δψ″CB0,∞+f−ψCB0,∞:ψ∈CB20,∞.Now, from [43], there exists a positive constant C such that
(46)K2f;δ≦Cmin1,δfCB0,∞+ω2f;δ,where the modulus of continuity of order two is given by
(47)ω2f;δ=sup0<h<δsupy∈0,∞fy+2h−2fy+h+fy.Moreover, the classical modulus of continuity is given by(48)ωf;δ=sup0<h<δsupy∈0,∞fy+h−fy.Theorem 9.
Supposeqα and pα are the sequences of positive numbers satisfying qα∈0,1,pα∈qα,1 such that qα⟶1,pα⟶1 as α⟶∞. Let us define an auxiliary operator such that Tα,p,qλf;y=Sα,p,qλf;y+fy−fαp,qy+1/αp,q. Then, for every ψ∈CB20,∞, we have
(49)Tα,pα,qαλψ;y−ψy≦Θαyψ″,where Θαy=δαy+αpα,qαy+1/αpα,qα−y2, and δαy is defined in Theorem 7.Proof.
Letψ∈CB20,∞. We easily get Tα,pα,qαλ1;y=1 and
(50)Tα,pα,qαλt;y=Sα,pα,qαλt;y+y−αpα,qαy+1αpα,qα=y,where we easily know that
(51)Sα,pα,qαλf;y≦f.
Therefore,(52)Tα,pα,qαλf;y≦Sα,pα,qαλf;y+fy−fαpα,qαy+1αpα,qα≦3f.
In view of the Taylor series expansion, we have(53)ψt=ψy+t−yψ′y+∫ytt−xψ″xdx.
On operatingTα,pα,qαλ, we conclude that
(54)Tα,pα,qαλψ;y−ψy=ψ′yTα,pα,qαλt−y;y+Tα,pα,qαλ∫ytt−xψ″xdx;y=Tα,pα,qαλ∫ytt−xψ″xdx;y=Sα,pα,qαλ∫ytt−xψ″xdx;y−∫yαpα,qαy+1/αpα,qααpα,qαy+1αpα,qα−αψ″xdxTα,pα,qαλψ;y−ψy≦Sα,pα,qαλ∫ytt−xψ″xdx;y+∫yαpα,qαy+1/αpα,qααpα,qαy+1αpα,qα−xψ″xdx.
Since we know that(55)∫ytt−xψ″xdx≦t−y2ψ″,∫yαpα,qαy+1/αpα,qααpα,qαy+1αpα,qα−xψ″xdx≦αpα,qαy+1αpα,qα−y2ψ″,we get
(56)Tα,pα,qαℏ,λψ;y−ψy≦Sα,pα,qαℏ,λt−y2;y+αpα,qαy+1αpα,qα−y2ψ″.
Hence, the above discussion completes the proof.Theorem 10.
LetSα,pα,qαλ be defined by (9); then, for every ψ∈CB20,∞, there exists an absolute constant C>0 such that
(57)Sα,pα,qαλψ;y−fy≦Cω2f;Θαy2+min1,Θαy4f+ωf;1αpα,qα,where Θαy is defined by Theorem 9.Proof.
In the view of the result asserted by Theorem9, we prove this theorem. For all f∈CB0,∞ and ψ∈CB20,∞, we have
(58)Sα,pα,qαλψ;y−fy=Tα,pα,qαλψ;y−fy+fαpα,qαy+1αp,q−fy≦Tα,pα,qαλf−ψ;y+Tα,pα,qαλψ;y−ψy+ψy−fy+fαpα,qαy+1αpα,qα−fy≦4f−ψ+Θαyψ″+ωf;αpα,qααpα,qα−1y+1αpα,qα.
By taking the infimum over allψ∈CB20,∞ and using (45), we get
(59)Sα,pα,qαλψ;y−fy≦4K2f;Θαy4+ωf;αpα,qααpα,qα−1y+1αpα,qα≦Cω2f;Θαy2+min1;Θαy4f+ωf;1αpα,qα.
We consider the Lipschitz-type maximal function by [44] and obtain the local approximation such as for f∈C0,∞, 0<κ≦1, and t,y∈0,∞. We recall that
(60)ωκf;y=supt≠y,t∈0,∞ft−fyt−yκ.Theorem 11.
For allf∈CB0,∞ and κ∈0,1, we have
(61)Sα,pα,qαλf;y−fy≦ωκf;yδαyκ/2,where theωκf;yLipschitz maximal function is defined by (60), andδαyis defined by Theorem 7.Proof.
We prove Theorem11 by applying (60) and the well-known Hölder inequality:
(62)Sα,pα,qαλf;y−fy≦Sα,pα,qαλft−fy;y≦ωκf;y∣Sα,pα,qαλt−yκ;y≦ωκf;ySα,pα,qαλ1;y2−κ/2Sα,pα,qαλt−y2;yκ/2=ωκf;ySα,pα,qαλΔ2;yκ/2.
The desired results are proven.We next denote(63)CB20,∞=ψ:ψ∈CB0,∞andψ′,ψ″∈CB0,∞,ψCB2ℝ+=ψCB0,∞+ψ′CB0,∞+ψ″CB0,∞,ψCB0,∞=supy∈0,∞ψy.Theorem 12.
Letψ∈CB20,∞. Then, Sα,pα,qαλ defined by (9) satisfies
(64)Sα,pα,qαλψ;y−ψy≦δαy+δαy2ψCB20,∞.Proof.
From the Taylor series expansion of order two, we have(65)ψt=ψy+ψ′yt−y+ψ″φt−y22,for φ∈y,t. Let
(66)S=supy0,∞ψ′y=ψ′CB0,∞≦ψCB20,∞,T=supy0,∞ψ″y=ψ″CB0,∞≦ψCB20,∞.
Then, we have(67)ψt−ψy≦St−y+12Tt−y2≦t−y+12t−y2ψCB20,∞.
Therefore, we have(68)Sα,pα,qαλψ;y−ψy≦Sα,pα,qαλt−y;y+12Sα,pα,qαλt−y2;yψCB20,∞≦Sα,pα,qαλΔ2;y1/2+12Sα,pα,qαλΔ2;yψCB20,∞.
This completes the proof of Theorem12.



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