# Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

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

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## Abstract

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.


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## Body

## 1. Introduction

In the year 1950, a famous mathematician Szász [1] invented the positive linear operators for the continuous function f on 0,∞ and that were extensively searched rather than Bernstein operators [2]. For z∈0,∞ and f∈C0,∞, Szász introduced the operators as follows:
(1)Srf;z=e−rz∑k=0∞rzkk!fkr,where C0,∞ is the space of continuous functions on 0,∞. In recent years, Szász-Mirakjan operators were introduced by Sucu [3] by proposing an exponential function on Dunkl generalization by including a nonnegative number η≥0, such that
(2)Sr∗f;z=1eηrz∑κ=0∞rzκγηκfκ+2ηθκr,where eηz=∑κ=0∞zκ/γηκ and a recursion formula for s=0,1,2,⋯.
(3)γηκ+1κ+1+2ηθκ+1=γηκ,θκ=0ifκ=2r,r∈ℕ∪0,1ifκ=2r+1,r∈ℕ∪0.In 1969, Jakimovski and Leviatan introduced the sequence of Szász-Mirakjan-type positive linear operators by the use of Appell polynomials [4], Lueuz=∑κ=0∞Hκzuκ such that
(4)Jrh;z=e−rzL1∑κ=0∞Hκrzfκr,where L1≠0, Lu=∑κ=0∞bκuκ, Hκz=∑j=0κbjzκ−j/κ−j!κ∈ℕ. Note that, if L1=1 in (4), the Szász-Mirakjan operator (1) is obtained. Most recently, in [5], Nasiruzzaman and Aljohani have introduced the Szász-Mirakjan-Jakimovski-Leviatan-type operators involving the Dunkl generalization for the function f∈C0,∞ by
(5)Pr,η∗f;z=1L1eηrz∑κ=0∞Hκrzfκ+2ηθκr.Lemma 1. [5].
For the test functionγj=tj, if j=0,1,2,3,4, the operators Pr,η∗ have Pr,η∗γ0;z=1,Pr,η∗γ1;z=z+1/rL′1/L1+2η, and the following identities:
(6)Pr,η∗γ2;z=z2+1r2L′1L1+4η+1z+1r2L′′1L1+1+4ηL′1L1+4η2,Pr,η∗γ3;z=z3+3rL′1L1+2η+1z2+1r23L′′1L1+61+2ηL′1L1+2+6η+12η2z+1r33L′′′1L1+31+2ηL′′1L1+21+3η+6η2L′1L1+8η3,Pr,η∗γ4;z=z4+1r4L′1L1+8η+6z3+1r26L′′1L1+81+3ηL′1L1+11+24η+24η2z2+1r34L′′′1L1+81+3ηL′′1L1+211+24η+24η2L′1L1+6+16η+24η+32η3z+1r4L′v1L1+23+4ηL′′′1L1+11+24η+24η2L′′1L1+6+16η+24η2+32η3L′1L1+16η4.There are several research articles mentioned regarding the Szász-Mirakjan-type operators, for instance, [6–13]. For some further related concepts and approximation, we refer to see [9, 10, 14–20].


## 2. Kantorovich Operators Involving Appell Polynomials and Their Moments

In this section, we construct the generalized operators of recent investigation [5] including the Kantorovich polynomial. For this purpose, we let f∈CΦ0,∞=f∈C0,∞:ft=OtΦ as t→∞; then, for all z∈0,∞, Φ>r, r∈ℕ, L1≠0, and η≥0, we define the operators as follows:
(7)Rr,η∗f;z=rL1eηrz∑κ=0∞Hκrz∫κ+2ηθκ/rκ+1+2ηθκ/rftdt.Lemma 2.
Forj=0,1,2,3,4, let the test function be γj=tj. Then, operators Rr,η∗·;· have the following identities:
(8)Rr,η∗γ0;z=1,Rr,η∗γ1;z=z+12r2L′1L1+4η+1,Rr,η∗γ2;z=z2+1r2L′1L1+4η+2z+13r23L′′1L1+61+2ηL′1L1+12η2+6η+1,Rr,η∗γ3;z=z3+32r2L′1L1+4η+3z2+32r22L′′1L1+23+4ηL′1L1+8η2+8η+3z+14r312L′′′1L1+63+4ηL′′1L1+68η2+8η+3L′1L1+32η3+8η2+8η+1,Rr,η∗γ4;z=z4+1r4L′1L1+8η+8z3+1r26L′′1L1+14+24ηL′1L1+24η2+36η+9z2+1r34L′′′1L1+14+24ηL′′1L1+38+72η+48η2L′1L1+32η3+48η2+36η+13z+1r4L′v1L1+43+2ηL′′′1L1+19+36η+24η2L′′1L1+13+36η+48η2+32η3L′1L1+16η4+16η3+8η2+2η+1.Proof.
To prove this Lemma, we take into account [5] Lemma 1. Thus, for all j=0,1,2,3,4 and γj=tj, we can conclude that
(9)∫κ+2ηθκ/rκ+1+2ηθκ/rtjdt=1rforj=0,12r2+1rκ+2ηθκrforj=1,13r3+1r2κ+2ηθκr+1rκ+2ηθκr2forj=2,14r4+1r3κ+2ηθκr+32r2κ+2ηθκr2+1rκ+2ηθκr3forj=3,15r5+1r4κ+2ηθκr+2r3κ+2ηθκr2+2r2κ+2ηθκr3+1rκ+2ηθκr4forj=4.
Thus, from (7) and (9), clearly we can write
(10)Rr,η∗γ0;z=Pr,η∗γ0;z=1,Rr,η∗γ1;z=Pr,η∗γ1;z+12rPr,η∗γ0;z,Rr,η∗γ2;z=Pr,η∗γ2;z+1rPr,η∗γ1;z+13r2Pr,η∗γ0;z,Rr,η∗γ3;z=Pr,η∗γ3;z+32rPr,η∗γ2;z+1r2Pr,η∗γ1;z+14r3Pr,η∗γ0;z,Rr,η∗γ4;z=Pr,η∗γ4;z+2rPr,η∗γ3;z+2r2Pr,η∗γ2;z+1r3Pr,η∗γ1;z+1r4Pr,η∗γ0;z.
Therefore, by applying Lemma1, we get the required results.Lemma 3.
For the central momentsγ1−zi,i=1,2,4, we have the following identities:
(11)Rr,η∗γ1−z;z=12r2L′1L1+4η+1,Rr,η∗γ1−z2;z=zr+13r23L′′1L1+61+2ηL′1L1+12η2+6η+1,Rr,η∗γ1−z4;z=−7r2z2+1r3−8L′′′1L1−4L′′1L1+24η+20L′1L1+40η2+28η+12z+1r4L′v1L1+43+2ηL′′′1L1+19+36η+24η2L′′1L1+13+36η+48η2+32η3L′1L1+16η4+16η3+8η2+2η+1.


## 3. Approximations in Weighted Space

In the present section, we follow the well-known results by Gadziev [21] and recall the results in weighted spaces with some additional conditions precisely, under the analogous of P.P. Korovkin’s theorem holds. In order to define the uniformly approximations, we take z→φz be the kind of functions which is continuous and strictly increasing with the assumptions Φz=1+φ2z and limz→∞Φz=∞. For this reason, we let BΦ0,∞ be a set of all such functions which are defined on 0,∞ and verifying the results
(12)BΦ0,∞=f:∣fz∣≤KfΦz,where Kf is a constant and depending only on function f and BΦ0,∞ equipped the norm with
(13)∥f∥Φ=supz∈0,∞∣fz∣Φz.Furthermore, we denote the set all continuous functions on0,∞ by C0,∞ and its subsets be CΦ0,∞ defined as CΦ0,∞=BΦ0,∞∩C0,∞.It is well known for the sequence of linear positive operatorsKrr≥1 (see [21]) maps CΦ0,∞ into BΦ0,∞ if and only if
(14)∣KrΦ;z∣≤MΦz,where M is a positive constant. For m∈ℕ, let us denote
(15)CΦm0,∞=f∈CΦ0,∞:limz→∞fzΦz=c,existsandisfinite.Theorem 4.
LetHf=f:suchthatfz/Φzisconvergentwhenz→∞. Then, for every f∈Hf∩CΦ0,∞, operators (7) are uniformly convergent on each compact subset of 0,∞ such that
(16)Rr,η∗f;z⇒f,where ⇒ denotes the uniform convergence.Proof.
In view of Lemma2, we use Korovkin’s theorem by [22]; then, it is enough to see that for each j=0,1,2(17)Rr,η∗γj;z→zjuniformly. Thus obviously, we get limr→∞Rr,η∗γ0;z=1, limr→∞Rr,η∗γ1;z=z, and limr→∞Rr,η∗γ2;z=z2, which completes the proof of Theorem 4.Theorem 5 [21, 23].
Let the positive linear operatorsJrr≥1 acting from CΦ0,∞ to BΦ0,∞ and for j=0,1,2 if it verifies that limr→∞Jrγj−zjΦ=0, then for every f∈CΦm0,∞ it satisfies
(18)limr→∞Jrf−fΦ=0.Theorem 6.
For everyφ∈CΦm0,∞, operators Rr,η∗ satisfy
(19)limr→∞Rr,η∗φ−φΦ=0.Proof.
It is enough to prove Theorem6; we use the well-known Korovkin theorem and show
(20)limr→∞∥Rr,η∗γj−zj∥Φ=0,j=0,1,2.
Taking into account Lemma2, then it is easy to see that
(21)∥Rr,η∗γ0−1∥Φ=supz∈0,∞∣Rr,η∗1;z−1∣Φz=0.
Forj=1, we can write here
(22)∥Rr,η∗γ1−z∥Φ=supz∈0,∞Rr,η∗γ1;z−zΦz=supz∈0,∞1Φz12r2L′1L1+4η+1.
Ifr→∞, then easily we get ∥Rr,η∗γ1−z∥Φ→0. Similarly, for j=2, we conclude that
(23)∥Rr,η∗γ2−z2∥Φ=supz∈0,∞∣Rr,η∗γ2;z−z2∣Φz=supz∈0,∞z1+z21r2L′1L1+4η+2+supz∈0,∞1Φz13r23L′′1L1+61+2ηL′1L1+12η2+6η+1.
Thus, we easily get∥Rr,η∗γ2−z2∥Φ→0, as r→∞.Theorem 7.
Ifφ∈CΦm0,∞. Then, operators Rr,η∗ follow that
(24)limr→∞supz∈0,∞∣Rr,η∗φ;z−φz∣Φz1+ξ=0,where the number ξ∈0,∞.Proof.
By the virtue of∣φz∣≤φΦ1+z2 and for any positive real z0, we easily obtain
(25)limr→∞supz∈0,∞∣Rr,η∗φ;z−φz∣Φz1+ξ≤supz≤z0∣Rr,η∗φ;z−φz∣Φz1+ξ+supz≥z0∣Rr,η∗φ;z−φz∣Φz1+ξ≤Rr,η∗φ;z−φzC0,z0+φΦsupz≥z0∣Rr,η∗1+t2;z−φz∣Φz1+ξ+supz≥z0∣φz∣Φz1+ξ=J1+J2+J3,suppose.
Thus,(26)J3=supz≥z0∣φz∣Φz1+ξ≤supz≥z0φΦ1+z2Φz1+ξ≤φΦ1+z02ξ.
From Lemma2, it follows that
(27)limr→∞supz≥z0Rr,η∗1+t2;zΦz=1.
Now, for eachε>0, there exists r1∈ℕ for all r≥r1 such that
(28)supz≥z0Rr,η∗1+t2;zΦz≤1+z02ξφΦε3+1.
Therefore, for allr≥r1(29)J2=φΦsupz≥z0Rr,η∗1+t2;zΦz1+ξ≤φΦ1+z02ξ+ε3.
In view of (26) and (29), we get
(30)J2+J3≤2φΦ1+z02ξ+ε3.
If we choose anyz0 so large, such that φΦ/1+z02ξ≤ε/6, then we get
(31)J2+J3≤2ε3,forallr≥r1.
On the other hand, there existsr2≥r such that
(32)J1=Rr,η∗φ;z−φzC0,z0≤ε3.
Finally, taker3=maxr1,r2 and on combining (31) and (32) with the above expression, we get
(33)supz∈0,∞∣Rr,η∗φ;z−φz∣Φz1+ξ<ε.
This completes the proof of Theorem7.Definition 8.
For everyδ¯>0 and all f∈C0,∞, the modulus of continuity of the uniformly continuous function f on 0,∞ defined as
(34)ω¯f;δ¯=sup∣t1−t2∣≤δ¯ft1−ft2,t1,t2∈0,∞,∣ft1−ft2∣≤1+∣t1−t2∣δ¯2ω¯f;δ¯.Theorem 9 [24].
Let the sequence of positive linear operatorsKr≥1:x,y→Cu,v and u,v⊆x,y, then
(1)
for anyf∈Cx,y and z∈u,v, it follows that(35)Krf;z−fz≤fzKr1;z−1+Kr1;z+1δ¯Krt−z2;zKr1;zω¯f;δ¯,(2)
if anyφ′∈Cx,y, then for all z∈u,v one has(36)Krϕ;z−ϕz≤ϕzKr1;z−1+ϕ′zKrt−z;z+Krt−z2;zKr1;z+1δ¯Krt−z2;zω¯ϕ′;δ¯.Theorem 10.
Letf∈CΦ0,∞, then for all z∈0,∞ it follows the inequality
(37)∣Rr,η∗f;z−fz∣≤2ω¯f;δ¯r,η∗z,where δ¯=δ¯r,η∗z=Rr,η∗γ1−z2;z.Proof.
If we consider Lemma2 and Theorem 9, then we can obtain
(38)Rr,η∗f;z−fz≤fzRr,η∗γ0;z−1+Rr,η∗γ0;z+1δ¯Rr,η∗γ0−z2;zRr,η∗γ0;zω¯f;δ¯,where if we take δ¯=δ¯r,η∗z=Rr,η∗γ1−z2;z then we are easily denumerable to get results.Theorem 11.
For anyz∈0,∞, if ϕ∈CΦ′0,∞, then we have the inequality
(39)Rr,η∗ϕ;z−ϕz≤12r2L′1L1+4η+1ϕ′z+2δ¯r,η∗zω¯ϕ′;δ¯r,η∗z,where δ¯=δ¯r,η∗z=Rr,η∗γ1−z2;z.Proof.
If we consider Lemmas2 and 3 and (2) of Theorem 9, then it is obvious to get that
(40)Rr,η∗ϕ;z−ϕz≤Rr,η∗γ0;z−1ϕz+ϕ′zRr,η∗γ1−z;z+Rr,η∗γ1−z2;zRr,η∗γ0;z+1δ¯Rr,η∗γ1−z2;zω¯ϕ′;δ¯.
Putδ¯=δ¯r,η∗z=Rr,η∗γ1−z2;z, then we easily get our desired results of Theorem 11.
From [25] for an arbitrary f∈CΦm0,∞,m∈ℕ∪0, the weighted modulus of continuity introduced such that
(41)Ω¯f;δ¯=supz∈0,∞,∣h∣≤δ¯∣fz+h−fz∣1+h21+z2.
Two main properties of this modulus of continuity arelimδ¯→0Ω¯f;δ¯=0 and
(42)∣ft−fz∣≤21+∣t−z∣δ¯1+δ¯21+z21+t−z2Ω¯f;δ¯,where t,z∈0,∞ and Ω¯ weighted modulus of continuity of the function for f∈CΦm0,∞.Theorem 12.
Letf∈CΦm0,∞, then for all z∈0,∞ we have the inequality
(43)supz∈0,αrηRr,η∗f;z−fz1+z2≤M1+αrηΩf;αrη,where M=22+M1+M2>0, for M1,M2>0 and
(44)αrη=max1r,13r23L′′1L1+61+2ηL′1L1+12η2+6η+1.Proof.
We use expressions (41) and (42) and applying the Cauchy-Schwarz inequality to operators Rr,η∗, we get
(45)Rr,η∗f;z−fz≤21+δ¯21+z2Ω¯f;δ¯1+Rr,η∗γ1−z2;z+Rr,η∗1+γ1−z2∣γ1−z∣δ¯;z.
We know the expression(46)Rr,η∗1+γ1−z2∣γ1−z∣δ¯;z=1δ¯Rr,η∗∣γ1−z∣;z+Rr,η∗γ1−z2∣γ1−z∣δ¯;z≤1δ¯Rr,η∗γ1−z2;z1/2+Rr,η∗γ1−z4;z1/2Rr,η∗γ1−z2δ¯2;z1/2=1δ¯Rr,η∗γ1−z2;z1/21+Rr,η∗γ1−z4;z.
In view of Lemma3, we can obtain
(47)Rr,η∗γ1−z2;z≤αrηz+1≤M1z+1asr→∞,Rr,η∗γ1−z4;z≤βrηz2+z+1≤M2z2+z+1asr→∞,where M1 and M2 are positive constant and
(48)αrη=max1r,13r23L′′1L1+61+2ηL′1L1+12η2+6η+1,βrη=max−7r2,1r3−8L′′′1L1−4L′′1L1+24η+20L′1L1+40η2+28η+12,1r4L′v1L1+43+2ηL′′′1L1+19+36η+24η2L′′1L1+13+36η+48η2+32η3L′1L1+16η4+16η3+8η2+2η+1.
Thus, from inequality (45), we get
(49)Rr,η∗f;z−fz≤21+δ¯21+z2Ω¯f;δ¯1+Rr,η∗γ1−z2;z+1δ¯Rr,η∗γ1−z2;z1/21+Rr,η∗γ1−z4;z≤21+δ¯21+z2Ω¯f;δ¯1+M1z+1+1δ¯αrηz+11+M2z2+z+1.If we chooseδ¯=αrη and taking supremum z∈0,αrη, then we easily get the result.


## 4. Direct Approximation Results ofRr,η∗

The present section gives some direct approximation results in space ofK-functional and in Lipschitz spaces. We take Cb0,∞ be the set of all continuous and bounded functions defined on 0,∞.Definition 13.
For everyδ¯>0 and f∈C0,∞ the K-functional is defined such that
(50)Kψf;δ¯=inff−ψCb0,∞+δ¯ψ′′Cb0,∞:ψ,ψ′∈Cb20,∞,(51)Cbk0,∞=f:f∈Cb0,∞,k∈ℕ;suchthatlimz→∞fz1+z2=mf<∞.
For an absolute constantM>0, one has
(52)Kψf;δ¯≤Mω¯2f;δ¯+min1,δ¯fCb0,∞.
Letω¯2f;δ¯ denote the modulus of continuity of order two such that
(53)ω¯2f;δ¯=sup0<h<δ¯supz∈0,∞fz+2h−2fz+h+fz,while the classical modulus of continuity is given by
(54)ω¯f;δ¯=sup0<h<δ¯supz∈0,∞fz+h−fz.Theorem 14.
For an arbitraryφ∈Cb20,∞, let an auxiliary operator Sr,η∗ be such that
(55)Kr,η∗φ;z=Rr,η∗φ;z+φz−φz+12r2L′1L1+4η+1.
Then, for anyϕ∈Cb20,∞ operators (55), verify the inequality
(56)Kr,η∗ϕ;z−ϕy≤δ¯r,η∗z++14r22L′1L1+4η+12∥ψ′′∥,where δ¯r,η∗z is defined by Theorem 10.Proof.
For anyϕ∈Cb20,∞, it is easy to verify that Kr,η∗γ0;z=1 and
(57)Kr,η∗γ1;z=Rr,η∗γ1;z+z−z+12r2L′1L1+4η+1=z.
We have(58)Rr,η∗φ;z≤φ,∣Kr,η∗φ;z∣≤∣Rr,η∗φ;z∣+∣φz∣+∣φz+12r2L′1L1+4η+1∣≤3∥φ∥.
For anyϕ∈Cb20,∞, the Taylor series expression gives us
(59)ϕt=ϕz+t−zϕ′z+∫ztt−χϕ′′χdχ.
Therefore, after applying the operatorsKr,η∗, on both sides we get
(60)Kr,η∗ϕ;z−ϕz=ϕ′zKr,η∗γ1−z;z+Kr,η∗∫zγ1γ1−χϕ′′χdχ;z=Kr,η∗∫ztγ1−χϕ′′χdχ;z=Rr,η∗∫zγ1γ1−χϕ′′χdχ;z+∫zzz−χϕ′′χdχ;z−∫zz+1/2r2L′1/L1+4η+1z+12r2L′1L1+4η+1−χϕ′′χdχ;Kr,η∗ϕ;z−ϕz≤Rr,η∗∫zγ1γ1−χϕ′′χdχ;z+∫zz+1/2r2L′1/L1+4η+1z+12r2L′1L1+4η+1−χϕ′′χdχ.
We know the inequality(61)∫ztt−χϕ′′χdχ≤t−z2ϕ′′,∫zz+1/2r2L′1/L1+4η+1z+12r2L′1L1+4η+1−χϕ′′χdχ≤12r2L′1L1+4η+12ϕ′′.
Thus, we get(62)Kr,η∗ϕ;z−ϕy≤Rr,η∗γ1−z2;z+14r22L′1L1+4η+12∥ϕ′′∥.
This gives the complete proof.Theorem 15.
Ifϕ∈Cb20,∞, then for any f∈Cb0,∞ operators Rr,η∗ by (7) satisfying
(63)Rr,η∗f;z−fz≤Mω¯2f;12δ¯r,η∗z+14r22L′1L1+4η+12+min1;14δ¯r,η∗z+14r22L′1L1+4η+12fCb0,∞+ω¯f;12r2L′1L1+4η+1,where δ¯r,η∗z is defined by Theorem 10.Proof.
We prove Theorem15 in view of Theorem 14. Therefore, for all f∈Cb0,∞ and ϕ∈Cb20,∞, we get
(64)Rr,η∗f;z−fz=Kr,η∗f;z−fz+fz+12r2L′1L1+4η+1−fz≤∣Kr,η∗f−ϕ;z∣+∣Kr,η∗ϕ;z−ϕz∣+∣ϕz−fz∣+fz+12r2L′1L1+4η+1−fz≤4∥f−ϕ∥+ωf;12r2L′1L1+4η+1+δ¯r,η∗z+14r22L′1L1+4η+12∥ψ′′∥.
If we take infimum for allϕ∈Cb20,∞, then in view of (50) it is easy to conclude that
(65)Rr,η∗f;z−fz≤4K2f;14δ¯r,η∗z+14r22L′1L1+4η+12+ω¯f;12r2L′1L1+4η+1≤Mω¯2f;12δ¯r,η∗z+14r22L′1L1+4η+12+min1;14δ¯r,η∗z+14r22L′1L1+4η+12fCb0,∞+ω¯f;12r2L′1L1+4η+1.
The proof is completed here.Now, we give the local direct estimate for the operatorsRr,η∗ defined by (7) via the well-known Lipschitz-type maximal function involving the parameters μ,ν>0 and number λ∈0,1. Thus, from [26], we recall that
(66)LipLλ=f∈Cb0,∞:∣ft−fz∣≤Lt−zλμz2+νz+tλ/2;z,t∈0,∞,where L is a positive constant.Theorem 16.
For anyf∈LipLλ satisfied by (70), operators Rr,η∗ hold the inequality
(67)∣Rr,η∗f;z−fz∣≤Lδ¯r,η∗zμz2+νzλ/2,where δ¯r,η∗z is obtained by Theorem 10.Proof.
Letf∈LipLλ for 0<λ≤1; then, first we verify the results are true when λ=1. For any μ,ν≥0, it is easy to use the result μz2+νz+t−1/2≤μz2+νz−1/2 and then we apply the Cauchy-Schwarz inequality. Thus, we can write
(68)Rr,η∗f;z−fz≤Rr,η∗ft−fz;z+fz1;z−1≤Rr,η∗∣t−z∣μz2+νz+t1/2;z≤Lμz2+νz−1/2Rr,η∗t−z;z≤Lμz2+νz−1/2Rr,η∗γ1−z2;z1/2.
From these conclusions, we get that the statement holds forλ=1. Now, we check if the statement is valid if 0<λ<1. For this reason, we use monotonicity property to Rr,η∗ and apply the well-known Hölder inequality
(69)Rr,η∗f;z−fz≤Rr,η∗∣ft−fz∣;z≤Rr,η∗ft−fz2/λ;zλ/2Rr,η∗γ0;z2−λ/2≤LRr,η∗γ1−z2;zt+μz2+νzλ/2≤Lμz2+νz−λ/2Rr,η∗γ1−z2;zλ/2≤Lμz2+νz−λ/2Rr,η∗γ1−z2;zλ/2=Lδ¯r,η∗zμz2+νzλ/2,which completes the proof.Here, we obtain the other local approximation results ofRr,η∗ in Lipschitz spaces. For all Lipschitz maximal function f∈Cb0,∞,0<λ≤1 and t,z∈0,∞, from [27] we recall that
(70)ω¯λf;z=supt≠z,t∈0,∞∣ft−fz∣t−zλTheorem 17.
Letf∈Cb0,∞, then for all z∈0,∞,
(71)Rr,η∗f;z−fz≤δ¯r,η∗zλ/2ω¯λf;z,where ω¯λf;z is obtained in and δ¯r,η∗z is defined by Theorem 10.Proof.
From the well-known Hölder inequality, we get(72)Rr,η∗f;z−fz≤Rr,η∗ft−fz;z≤ω¯λf;z∣Rr,η∗t−zλ;z≤ω¯λf;zRr,η∗γ0;z2−λ/2Rr,η∗t−z2;zλ/2=ω¯λf;zRr,η∗γ1−z2;zλ/2.
Thus, we get the proof.


## 5. Voronovskaja-Type Approximation Theorems

In this section, we establish a quantitative Voronovskaja-type theorem for the operatorsRr,η∗f;z.Theorem 18.
Letf∈Cb0,∞, then for each z∈0,∞(73)limr→∞rRr,η∗ψ;z−ψz=2L′1L1+4η+1ψ′z2+zψ′′z2,where ψ′z,ψ′′z∈Cb0,∞.Proof.
From the expression of Taylor’s expansion of functionψz in Cb0,∞, we write
(74)ψt=ψz+t−zψ′z+12t−z2ψ′′z+t−z2Qzt,where Qzt is the remainder term and Qz∈0,∞ with Qzt→0 as t→z. On applying the operators Rr,η∗·;z to (74), then use the Cauchy-Schwarz inequality. Thus, we get
(75)Rr,η∗ψ;z−ψz=ψ′zRr,η∗γ1−z;z+ψ′′z2Rr,η∗γ1−z2;z+Rr,η∗γ1−z2Qzγ1;z≤ψ′zRr,η∗γ1−z;z+ψ′′z2Rr,η∗γ1−z2;z+Rr,η∗γ1−z4;zRr,η∗Qz2γ1;z.
Since we havelimr→∞Rr,η∗Qz2γ1;z=0, therefore
(76)limr→∞rRr,η∗γ1−z2Qzγ1;z=0.
Thus, we have(77)limr→∞rRr,η∗ψ;z−ψz=limr→∞rRr,η∗γ1−z;zψ′z+ψ′′z2Rr,η∗γ1−z2;z+Rr,η∗γ1−z2Qzγ1;z,which completes the proof.As a consequence of Theorem18, we immediately get the corollary.Corollary 19.
For anyψ∈C0,∞, we have
(78)limr→∞rRr,η∗ψ;z−ψz−12r2L′1L1+4η+1ψ′z−δ¯r,η∗z2ψ′′z=0.


## 6. Conclusion

Motivated by article [5], we have introduced a Kantorovich generalization of the Szász-Mirakjan operators by Dunkl analogue involving the Appell polynomials. These types of generalizations enable to give the generalized results rather than the earlier study demonstrations by [3, 5, 7]. Lastly, we have also discussed the Voronovskaja-type approximation theorems of these new operators.



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