# Approximation by Szász-Jakimovski-Leviatan-Type Operators via Aid of Appell Polynomials

Authors: Md. Nasiruzzaman; A. F. Aljohani
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The main purpose of the present article is to construct a newly Szász-Jakimovski-Leviatan-type positive linear operators in the Dunkl analogue by the aid of Appell polynomials. In order to investigate the approximation properties of these operators, first we estimate the moments and obtain the basic results. Further, we study the approximation by the use of modulus of continuity in the spaces of the Lipschitz functions, Peetres K-functional, and weighted modulus of continuity. Moreover, we studyA-statistical convergence of operators and approximation properties of the bivariate case.


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## Body

## 1. Introduction

In 1969, Jakimovski and Leviatan introduced a sequence of positive linear operatorsLnn≥1 [1], by using Appell polynomials [2] Fvevy=∑s=0∞Psyvs and defined as
(1)Lnh;y=e−nyF1∑s=0∞Psnyhsn,where F1≠0, Fv=∑s=0∞csvs, and Psy=∑i=0sciys−i/s−i!s∈ℕ. For all n∈ℕ and ci/F1≥0, the positive linear operators Ln are defined on 0,1 given by Wood in [3]. If we take h∈E0,∞, then an analogue of Szász operators was proved by Jakimovski and Leviatan, where E0,∞ denotes the set of functions on 0,∞} such that ∣hy∣≤aeκy, where a, κ are positive constants. They established limn→∞Lnh;y⟶hy is uniformly on each compact subset of 0,1 (see [1, 4]). Precisely, for F1=1 in (1), the well-known classical Szász operators [5] were obtained defined in 1950 such that
(2)Snh;y=e−ny∑s=0∞nyss!hsn.Recently, Szász-Mirakyan operators have been obtained by researchers via the Dunkl generalization in approximation process; for instance, we refer the readers to [6–12]. For more details, related results relevant to the present article in different functional spaces are seen in [13–19] and [20–23]. Sucu [24] introduced Szász-Mirakyan operators by using the new exponential function given in [25] as
(3)eλy=∑s=0∞ysγλs.γλ2p=22pp!Γp+λ+1/2Γλ+1/2,γλ2p+1=22p+1p!Γp+λ+3/2Γλ+1/2.Forp=0,1,2,⋯ a recursion of γλ is given as
(4)γλp+1p+1+2λθp+1=γλp,θp=0ifp=2s,s∈ℕ,1ifp=2s+1,s∈ℕ.These types of generalizations gave rise to exponential function and generalization of Hermite-type polynomials, expressed in the form of the confluent hypergeometric function (see [25]).


## 2. Construction of Operators and Estimation of Moments

For everyh∈Cϑ0,∞=h∈C0,∞:hs=Osϑass⟶∞, and all y∈0,∞, ϑ>n, n∈ℕ, F1≠0, λ≥0, we define
(5)Jn,λ∗h;y=1F1eλny∑s=0∞Psnyhs+2λθsn.Lemma 1.
For ally∈0,∞, Psy≥0, λ≥0, and F1≠0, if we define
(6)Fαeλαy=∑s=0∞Psyαs.
Then for alln∈ℕ, we have
(7)F1eλny=∑s=0∞Psny,∑s=0∞sPsny=F′1+nyF1eλny,∑s=0∞s2Psny=F″1+2ny+1F′1+nyny+1F1eλny,∑s=0∞s3Psny=F‴1+3ny+1F″1+3n2y2+6ny+2F′1+nyn2y2+3ny+2F1eλny,∑s=0∞s4Psny=F′v1+4ny+6F‴1+6n2y2+18ny+11F″1+4n3y3+18n2y2+22ny+6F′1+nyn3y3+6n2y2+11ny+6F1eλny.Lemma 2.
Letλ∈0,∞, F1≠0 and take ϕr=sr for r=0,1,2,3,4.
Then, for operatorsJn,λ∗·;· by (5), we have the following estimates:
(8)Jn,λ∗ϕ0;y=1,Jn,λ∗ϕ1;y=y+1nF′1F1+2λ,Jn,λ∗ϕ2;y=y2+1n2F′1F1+4λ+1y+1n2F″1F1+1+4λF′1F1+4λ2,Jn,λ∗ϕ3;y=y3+3nF′1F1+2λ+1y2+1n23F″1F1+61+2λF′1F1+2+6λy+1n33F‴1F1+31+2λF″1F1+21+3λ+6λ2F′1F1+8λ3,Jn,λ∗ϕ4;y=y4+1nF′1F1+8λ+6y3+1n26F″1F1+81+3λF′1F1+11+24λ+24λ2y2+1n36F‴1F1+81+3λF″1F1+211+24λ+24λ2F′1F1+6+16λ+24λ2+32λ2y+1n4F′v1F1+23+4λF‴1F1+11+24λ+24λ2F″1F1+6+16λ+24λ2+32λ3F′1F1+16λ4.Proof.
(1)
Takeh=ϕ0, then(9)Jn,λ∗ϕ0;y=1F1eλny∑s=0∞Psny=1.(2)
Forh=ϕ1(10)Jn,λ∗ϕ1;y=1F1eλny∑s=0∞Psnys+2λθsn=1nF1eλny∑s=0∞sPsny+2λnF1eλny∑s=2k+1∞θsPsnyfork=0,1,2,3,⋯=1nF1eλnyF′1+nyF1eλny+2λn.(3)
Forh=ϕ2(11)Jn,λ∗ϕ2;y=1F1eλny∑s=0∞Psnys+2λθsn2=1n2F1eλny∑s=0∞s2Psny+4λn2F1eλny∑s=2k+1∞sPsnyθsfork=0,1,2,3,⋯+4λ2n2F1eλny∑s=2k+1∞Psnyθs2fork=0,1,2,3,⋯=1F1eλnyF″1+2ny+1F′1+nyny+1F1eλny+4λ2n2F1eλnyF′1+nyF1eλny+4λ2n2.
Similarly, we can prove easily (4) and (5).Lemma 3.
Letψj=ϕ1−yjforj=1,2,3,be the central moments, then
(12)Jn,λ∗ψ1;y=1nF′1F1+2λ,Jn,λ∗ψ2;y=yn++1n2F″1F1+1+4λF′1F1+4λ2,Jn,λ∗ψ4;y=1n4F′1F1+8λy3+1n2−10F′1F1+3y2+1n3−4F″1F1+27+12λF′1F1+23+8λ+12λ2y+1n4F′v1F1+23+4λF‴1F1+11+24λ+24λ2F″1F1+6+16λ+24λ2+32λ3F′1F1+16λ4+1n−4F′1F1−8λ.


## 3. Global Approximation

In the present section, we follow Gadžiev [11] and recall the weighted spaces of the functions on0,∞, as well as additional conditions under which the analogous theorem of P.P. Korovkin holds for such a kind of functions. Take y⟶ϕy be continuous and strictly increasing function with σy=1+ϕ2y and limy→∞σy=∞. Let Bσ0,∞ be a set of functions defined on [0,∞), verifying the results
(13)Bσ0,∞=hy:hy≤Khσy,where Kh is a constant and depending only on function h and Bσ0,∞ is space of all continuous as well as bounded functions on 0,∞. Let the set of all continuous functions on 0,∞ will be denoted by Cσ0,∞ and Bσ0,∞⊂Cσ0,∞ equipped with the norm hσ=supy∈0,∞h/σy.Let us denote(14)Cσm0,∞=h∈Cσ:limy→∞hyσy=k,kispositiveconstant.It is well known that (see [26]) the sequence of linear positive operators Lnn≥1 maps Cσ0,∞ into Bσ0,∞ if and only if
(15)∣Lnσ;y∣≤Cσy,where C is a positive constant.Definition 4.
For allh∈C0,∞, the modulus of continuity for a uniformly continuous function h defined by
(16)ωh;δ=sup∣s1−s2∣≤δhs1−hs2,s1,s2∈0,∞.
For everyδ>0 and uniformly continuous function h∈C0,∞, we suppose
(17)hs1−hs2≤1+s1−s2δ2ωh;δ.Theorem 5.
For allh∈0,∞∩h:y≥0,hy/σyisconvergentasy⟶∞, operators Jn,λ∗ defined in (5) satisfy Jn,λ∗⇒h on each compact subset of 0,∞, with ⇒ stands for uniform convergence.Proof.
From the well-known Korovkin’s theorem (see [27]), for all r=0,1,2, it is sufficient to see that
(18)Jn,λ∗ϕr;y⟶yr.
In the view of Lemma2, it is obvious that Jn,λ∗ϕr;y⟶yr as n⟶∞, r=0,1,2, which completes Theorem 5.Theorem 6.
LetJn,λ∗:Cσm0,∞⟶Bσ0,∞. Then for every h∈Cσm0,∞, we have
(19)limn→∞Jn,λ∗h;y−hσ=0.Proof.
We prove this theorem by applying Korovkin’s theorem so it is sufficient to show that(20)limn→∞Jn,λ∗ϕj;y−yjσ=0,forj=0,1,2.
From Lemma2, we easily see that
(21)Jn,λ∗ϕo;y−y0σ=supy∈0,∞Jn,λ∗1;y−1σy=0forj=0.
Similarly, for(22)Jn,λ∗ϕ1;y−yσ=1nF′1F1+2λsupy∈0,∞1σy.which imply that Jn,λ∗ϕ1;y−yσ⟶0 as n⟶∞. For j=2(23)Jn,λ∗ϕ2;y−y2σ=supy∈0,∞Jn,λ∗ϕ2;y−y2σy=1n2F′1F1+4λ+1supy∈0,∞yσy+1n2F″1F1+1+4λF′1F1+4λ2supy∈0,∞1σy,which clearly shows that Jn,λ∗ϕ2;y−y2σ⟶0, whenever n⟶∞.Theorem 7.
For allh∈CB0,∞, operators given by (5) satisfy
(24)Jn,λ∗h;y−hy≤2ωh;δny,where δny=Jn,λ∗ψ2;y and CB0,∞ stand for space of all continuous and bounded functions defined on 0,∞.Proof.
We prove Theorem7 by using the well-known Cauchy-Schwarz inequality and modulus of continuity. Thus, we see that
(25)Jn,λ∗h;y−hy≤1F1eλny∑s=0∞Psny;hs+2λθsn−hy≤1F1eλny∑s=0∞Psny1+1δs+2λθsn−yωh;δ=1+1F1eλny∑s=0∞Psnys+2λθsn−yωh;δ≤1+1δ1F1eλny∑s=0∞Psnys+2λθsn−y212Jn,λ∗ϕ0;y12ωh;δ=1+1δJn,λ∗ϕ1−y2;y12ωh;δ.
If we takeδ=δn=Jn,λ∗ψ2;y, we get the required result asserted by Theorem 7.


## 4. Some Direct Results ofJn,λ∗

The present section gives some direct approximation results in the space ofK-functional and in the Lipschitz spaces. We suppose the following.Definition 8.
For everyδ>0 and h∈C0,∞, we define
(26)K2h;δ=infh−ψCB0,∞+δψ′′CB0,∞:ψ,ψ′∈CB20,∞,where CB20,∞ is defined by
(27)CBk0,∞=h:h∈CB0,∞,k∈ℕ;suchthatlimy→∞hyσy=kh<∞.
Now, there exists an absolute constantC>0 such that
(28)K2h;δ<Cω2h;δ+min1,δhCB0,∞,where ω2h;δ is the second-order modulus of continuity given by
(29)ω2h;δ=sup0<η<δsupyϵ0,∞hy+2η−2hy+η+hy.
Moreover, the modulus of continuity of order one is(30)ωh;δ=sup0<η<δsupy∈0,∞hy+η−hy.Theorem 9.
Leth∈CB20,∞, we define an auxiliary operators Kn,λ∗ such that
(31)Kn,λ∗h;y=Jn,λ∗h;y+hy−hy+1nF′1F1+2λ.
Then, for everyψ∈CB20,∞, operators Kn,λ∗ satisfy
(32)Kn,λ∗ψ;y−ψy≤Θnyψ″,where Θny=δny2+1/n2F′1/F1+2λ2 andδny are defined in Theorem 7.Proof.
Takeψ∈CB20,∞; then, we easily conclude that Kn,λ∗ϕ0;y=1 and
(33)Kn,λ∗ϕ1;y=Jn,λ∗ϕ1;y+y−y+1nF′1F1+2λ=y.
We also know easily(34)Jn,λ∗h;y≤h.Therefore,
(35)Kn,λ∗h;y≤Jn,λ∗h;y+hy+hy+1nF′1F1+2λ≤3h.
From the Taylor series we see(36)ψs=ψy+s−yψ′y+∫yss−μψ″μdμ.
ApplyingKn,λ∗, we have
(37)Kn,λ∗ψ;y−ψy=ψ′yKn,λ∗ϕ1−y;y+Kn,λ∗∫yϕ1ϕ1−μψ″μdμ;y=Kn,λ∗∫yϕ1ϕ1−μψ″μdμ;y=Jn,λ∗∫yϕ1ϕ1−μψ″μdμ;y−∫yy+1nF′1F1+2λy+1nF′1F1+2λ−μψ″μdμ,Kn,λ∗ψ;y−ψy≤Jn,λ∗∫yϕ1ϕ1−μψ″μdμ;y+∫yy+1nF′1F1+2λy+1nF′1F1+2λ−μψ″μdμ.
Since we know(38)∫yss−μψ″μdμ≤s−y2ψ″,∫yy+1nF′1F1+2λy+1nF′1F1+2λ−μψ″μdμ≤1nF′1F1+2λ2ψ″.
Therefore, we get(39)Kn,λ∗ψ;y−ψy≤Jn,λ∗ϕ1−y2;y+1n2F′1F1+2λ2ψ″.
This gives the complete proof.Theorem 10.
Leth∈CB0,∞ and any ψ∈CB20,∞. Then, there exists a constant C>0 such that
(40)Jn,λ∗h;y−hy≤Cω2h;Θny2+min1,Θny4hCB0,∞+ωh;1nF′1F1+2λ,where Θny is defined by Theorem 9.Proof.
We prove the result asserted by Theorem10 in the light of Theorem 9. Therefore, for all h∈CB0,∞ and ψ∈CB20,∞, we get
(41)Jn,λ∗h;y−hy=Kn,λ∗h;y−hy+hy+1nF′1F1+2λ−hy≤Kn,λ∗h−ψ;y+Kn,λ∗ψ;y−ψy+ψy−hy+hy+1nF′1F1+2λ−hy≤4h−ψ+Θnyψ″+ωh;1nF′1F1+2λ.
Taking infimum over allψ∈CB20,∞ and using (26), we get
(42)Jn,λ∗h;y−hy≤4K2h;Θny4+ωh;1nF′1F1+2λ≤Cω2h;Θny4+min1;Θny4hCB0,∞+ωh;1nF′1F1+2λ.
Here, we obtain some local approximation results ofJn,λ∗ in the Lipschitz spaces. For all the Lipschitz maximal function h∈C0,∞,0<ϑ≤1 and s,y∈0,∞, we recall that
(43)ωϑh;y=sups≠y,s∈0.∞hs−hys−yϑ.Theorem 11.
Let0<ϑ≤1, then for all h∈CB0,∞, operators Jn,λ∗ satisfy
(44)Jn,λ∗h;y−hy≤ωϑh;yδnyϑ,where ωϑh;y is the Lipschitz maximal function defined by (43) and δny by Theorem 7.Proof.
To prove Theorem11, we use the well-known Hölder inequality by applying (43)
(45)Jn,λ∗h;y−hy≤Jn,λ∗hs−hy;y≤ωϑh;yJn,λ∗s−yϑ;y≤ωϑh;yJn,λ∗ϕ0;y2−ϑ2Jn,λ∗ϕ1−y2;yϑ2=ωϑh;yJn,λ∗ψ2;yϑ2.
The proof is complete.From [28] for an arbitrary h∈Cσm0,∞, the weighted modulus of continuity is introduced such that
(46)Ωh;δ=supy∈0,∞,η≤δhy+η−hy1+η21+y2.The two main properties of this modulus of continuity are limδ→0Ωh;δ=0 and
(47)hs−hy≤21+s−yδ1+δ21+y21+s−y2Ωh;δ,where s,y∈0,∞.Theorem 12.
Let the operatorsJn,λ∗ be defined by (5); then for every h∈Cσm0,∞, there exists a constant C>0 such that
(48)supy∈0,O1nJn,λ∗h;y−hyσy≤C1+O1nΩh;O1n,where σy=1+y2 and C=2+C1+2C2 with C1>0,C2>0.Proof.
In light of (46), (47), and Cauchy-Schwarz inequality, we prove this theorem. Thus, we see
(49)Jn,λ∗h;y−hy≤21+δ21+y2Ωh;δ1+Jn,λ∗ϕ1−y2;y+Jn,λ∗1+ϕ1−y2ϕ1−yδ;y(50)Jn,λ∗1+ϕ1−y2ϕ1−yδ;y≤1+2Jn,λ∗ϕ1−y4;y12Jn,λ∗ϕ1−y4δ2;y12.
From Lemma3, we easily conclude that for any positive C1 and C1(51)Jn,λ∗ϕ1−y2;y=O1ny+12≤C1y+12asn⟶∞,(52)Jn,λ∗ϕ1−y4;y=O1ny+14≤C2y+14asn⟶∞.
Therefore,(53)Jn,λ∗ϕ1−y2δ2;y12=1δO1n1+y.(54)Jn,λ∗ϕ1−y4;y12≤C21+y2.
Hence, in light of (49), (50), (51), (52), (53) and (54) and choosing δO1/n, if we take the supremum y∈0,O1/n, we get the result.


## 5.A-Statistical Convergence

Here, we obtain theA-statistical convergence for the operators Jn,λ∗ by (5). From [29], we recall the needed notations and notions for A-statistical convergence. Take G=Dnk be a nonnegative infinite summability matrix. For a given sequence y=yk, the A-transform of y is denoted by Gy:Gyn where the series converges for each n and defined by
(55)Gyn=∑k=1∞ykDnk.The matrixG is said to be regular if limGyn=L whenever limx=L and y=yn are said to be a A-statistically convergent to L, i.e., stG−limy=L if for each ∈>0,limn∑k:yk−L≥∈Dnk=0. For the recent work on statistical convergence and statistical approximation, we refer to [30–37].Theorem 13.
Let operatorsJn,λ∗ be defined by 1 and a nonnegative regular summability matrix be G=Dnk; then, for every h∈Cσm0,∞(56)stG−limnJn,λ∗h;y−hσ=0.Proof.
It is enough to show that(57)stG−limnJn,λ∗ϕj;y−yjσ=0,forj=0,1,2.
From Lemma2, we conclude that
(58)Jn,λ∗ϕ2;y−yσ=1nF′1F1+2λsupy∈0,∞1σy≤1nF′1F1+2λ.which implies that
(59)stG−limn1nF′1F1+2λ=0.
Similarly forj=2(60)Jn,λ∗ϕ2;y−y2σ=1n2F′1F1+4λ+1supy∈0,∞yσy+1n2F″1F1+1+4λF′1F1+4λsupy∈0,∞1σy≤121n2F′1F1+4λ+1+1nF′1F1+2λ.which shows that
(61)stG−limn121n2F′1F1+4λ+1=stG−limn1nF′1F1+2λ=0.
For a givenε>0, we define the sets such that
(62)U1≔n:Jn,λ∗ϕ2;y−y2≥∈,U2≔n:121n2F′1F1+4λ+1≥∈2,U3≔n:1n2F′1F1+2λ≥∈2.
Therefore, we conclude thatU1⊆U2∪U3, and ∑k1∈U1Dnk1≤∑k1∈U2Dnk1+∑k1∈U3Dnk1. Hence, (61) implies that
(63)stG−limnJn,λ∗ϕ2;y−y2σ=0.
This is denumerable to complete the proof.


## 6. Bivariate Operators and Their Moments Estimation

Letℝ+2=ℝ+×ℝ+ with ℝ+=0,∞ and ℝ+2=y,y1:0≤y<∞,0≤y1<∞. Suppose Cℝ+2 is a set of all continuous functions on ℝ+2, endowed with the norm given by gCℝ+2=supy,y1∈ℝ+2gy,y1. Then, for all g∈Cℝ+2 and n,m∈ℕ, we define
(64)In,m∗g;y,y1=1F1eλnyG1eηmy1∑s,t=0∞PsnyPtmy1gs+2λθsn,t+2ηθtm,where λ,η≥0 and F1,G1≠0.Fori,j=0,1,2,3,4, if we take s=s+2λθs/n, t=t+2ηθt/m, and consider central moments as
(65)gsi,tj=s−yit−y1j.then from Lemma 2 and Lemma 3, we easily conclude Lemma 14 as follows:Lemma 14.
For ally,y1∈ℝ+2 and sufficiently large n,m∈ℕ, we have
(66)In,m∗s−y2;y,y1=O1ny+12≤M1y+12asn,m⟶∞,In,m∗t−y12;y,y1=O1my1+12≤M2y1+12asn,m⟶∞,In,m∗s−y4;y,y1=O1ny+14≤M3y+14asn,m⟶∞,In,m∗t−y14;y,y1=O1my1+14≤M4y1+14asn,m⟶∞.Lemma 15.
If we let(67)Kn,λ∗g;y,y1=1F1eλny∑s=0∞Psnygs+2λθsn,y1,Lm,η∗g;y,y1=1G1eηmy1∑t=0∞Ptmy1gy,t+2ηθtm.
Then, it follows that(68)In,m∗g;y,y1=Kn,λ∗Lm,η∗g;y,y1=Lm,η∗Kn,λ∗g;y,y1.Proof.
We easily see that(69)Kn,λ∗Lm,η∗g;y,y1=Kn,λ∗1G1eηmy1∑t=0∞Ptmy1gy,t+2ηθtm=1G1eηmy1∑t=0∞Kn,λ∗gy,t+2ηθtmPtmy1=1F1eλny1G1eηmy1∑s,t=0∞PsnyPtmy1gs+2λθsn,t+2ηθtm=In,n∗g;y,y1.
Similarly, we can seeLm,η∗Kn,λ∗g;y,y1=Lm,η∗g;y,y1.
Let the weighted functionρbeρy,y1=1+y2+y12. Take Bρℝ+2=g:gy,y1≤Mgρy,y1Mg>0. We denote the set of k-times continuously differentiable functions on ℝ+2=y,y1∈ℝ2:y,y1∈0,∞ by Ckℝ+2. We also denote the class of functions such that
(70)Cρℝ+2=g:g∈Bρ∩Cρℝ+2,Cρkℝ+2=g:g∈Cρℝ+2;suchthatlimy,y1→∞gy,y1ρy,y1=kg<∞,Cρ0ℝ+2=g:g∈Cρkℝ+2;suchthatlimy,y1→∞gy,y1ρy,y1=0.
Let the norm onBρ be defined as gρ=supy,y1∈ℝ+2gy,y1/ρy,y1.
For allg∈Cρ0ℝ+2 and δ1,δ2>0, the weighted modulus of continuity is given as
(71)ωρg;δ1,δ2=supy,y1∈0,∞sup0≤α≤δ1,0≤β≤δ2gy+α,y1+β−gy,y1ρy,y1ρα,β,and for any r1,r2>0 satisfying the inequality
(72)ωρg;r1δ1,r2δ2≤41+r11+r21+δ121+δ22ωρg;δ1,δ2,it also follows that
(73)gs,t−gy,y1≤ρy,y1ρs−y,t−y1ωρg;s−y,t−y1≤1+y2+y121+s−y21+t−y12ωρg;s−y,t−y1.Theorem 16.
For allg∈Cρ0ℝ+2 and sufficiently large n,m∈ℕ(74)In,m∗g;y,y1−gy,y11+y2+y12≤ξy,y11+On−11+Om−1ωρg;On−12,Om−12,where ξy,y1=1+y+1+M1y+12+M3y+131+y1+1+M2y1+12+M4y1+13, and M1, M2, M3, M4>0.Proof.
In view of the above explanation for allδn,δm>0, we see that
(75)gs,t−gy,y1≤41+y2+y121+s−y21+t−y12×1+s−yδn1+t−y1δm1+δn21+δm2ωρg;δn,δm=41+y2+y121+δn21+δm2×1+s−yδn+s−y2+s−yδn+s−y2×1+t−y1δm+t−y12+t−y1δmt−y12ωρg;δn,δm.
On applying the operatorsIn,m∗, we get
(76)In,m∗g;y,y1−gy,y1≤In,m∗gs,t−gy,y1;y,y141+y2+y12×In,m∗1+s−yδn+s−y2+s−yδns−y2;y,y1×In,m∗1+t−y1δm+t−y12+t−y1δmt−y12;y,y1×1+δn21+δm2ωρg;δn,δm=41+y2+y121+δn21+δm2ωρg;δn,δm×1+1δnIn,m∗s−y;y,y1+In,m∗s−y2;y,y1+1δnIn,m∗s−ys−y2;y,y1×1+1δmIn,m∗t−y1;y,y1In,m∗t−y12;y,y1+1δmIn,m∗t−y1t−y12;y,y1.
From Cauchy-Schwarz inequality, we see(77)In,m∗g;y,y1−gy,y1≤41+y2+y121+δn21+δm2ωρg;δn,δm×1+1δnIn,m∗s−y2;y,y1+In,m∗s−y2;y,y1+1δnIn,m∗s−y2;y,y1In,m∗s−y4;y,y1×1+1δmIn,m∗t−y12;y,y1+In,m∗t−y12;y,y1+1+1δmIn,m∗t−y12;y,y1+In,m∗t−y14;y,y1.
From Lemma14 we get
(78)In,m∗g;y,y1−gy,y1≤41+y2+y121+δn21+δm2ωρg;δn,δm×1+1δnO1ny+1+M1y+12+1δnO1nM3y+13×1+1δmO1my1+1+M2y1+12+1δmO1mM4y1+13.
By choosingδn=On−1/2 and δm=Om−1/2, we arrived to our desired results.


## 7. Conclusion

The motivation of this present article is to provide the generalized error estimation of convergence rather than the classical Dunkl-Szász-Mirakyan operators. Here, we have defined Szász-Jakimovski-Leviatan operators by using the Appel polynomials with the aid of a new parameterλ∈0,∞. These types of approximation are able to give the generalized results and error estimation in comparison to earlier study demonstrations. We have obtained the approximations via the well-known weighted Korovkin’s spaces and investigated approximations in Peetre’s K-functional and Lipschitz spaces with the aid of modulus of continuity. Further, we have also obtained the approximation in A-statistical convergence. Lastly, we have studied the approximation properties for the bivariate case.



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