# Theq-Chlodowsky and q-Szasz-Durrmeyer Hybrid Operators on Weighted Spaces

Authors: Harun Çiçek; Aydın İzgi
Journal: Journal of Mathematics
 (2020)

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## Abstract

The main aim of this article is to introduce a new type ofq-Chlodowsky and q-Szasz-Durrmeyer hybrid operators on weighted spaces. To this end, we give approximation properties of the modified new q-Hybrid operators. Moreover, in the weighted spaces, we examine the rate of convergence of the modified new q-Hybrid operators by means of moduli of continuity. In addition, we derive Voronovskaja’s type asymptotic formula for the related operators.


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## Body

## 1. Introduction

Polynomial approach and the classical approximation theory constitute a basic research area in applied mathematics. The development of the approximation theory plays an important role in the numerical solution of partial differential equations, data processing sciences, and many other disciplines. For example, it is widely used in geometric modelling in the aerospace and automotive industries to calculate approximate values with basic functions. Works in this field go back to the 18th century and still continue as a powerful tool in scientific calculations. Furthermore, it is used in civil engineering projects to analyze the energy efficiency and earthquake resistance data of different types of buildings in thermography calculations and earthquake engineering.Many generalized versions of these polynomials have been studied by many authors. Some of these studies are [1–13]. During the studies of approximation theory, many new operators and their generalizations were introduced with different spaces and variables. For example, on the interval 0,∞, 0,1 etc., q−analog type, K−Petree type, King’s type, and Weighted space type operators are introduced.In this study, theq− analog type operator is defined. The studies regarding the q−analog type operator are as follows. First, the q−Bernstein polynomials were produced by Phillips [14]. When q=1 is used, the results are the same as for classical operators. However, new operators with different properties are obtained for q≠1. Gupta [15] introduced and analyzed the approach characteristics of q-Durrmeyer operators. Gupta and Heping [16] identified the q-Durrmeyer operators and estimated the rate of convergence for continuous functions with the help of the moduli of continuity. In [2, 17], Mahmudov defined the King-type q-Szász operators. He obtained the rate of convergence on weighted spaces and a Voronovskaya-type theorem for these operators. Some other studies based on classical q−theory are [17–24].In light of the above studies, the following result has been obtained. For a real-valued function,fu defined on the interval 0,∞, the operators(1)∑k=0nPn,k,qu∫0∞Pn,k,qthtdqt,∑k=0nSn,k,qu∫0∞Sn,k,qthtdqt,∑k=0nSn,k,qu∫0∞Pn,k,qthtdqt,for the general operator kernels(2)Pn,k,qx=nkqxk1−xqn−k=nkqxk∏s=0n−k−11−qsx,(3)Sn,k,qy=1Eqnqyqkk−1/2nqykkq!,have been studied by many authors. On the contrary, the operator(4)∑k=0nPn,k,qu∫0∞Pn,k,qthtdqthas not been studied yet.Therefore, we introduced the following operator:(5)Hn,qh;u=nqbn∑k=0nPn,k,qubnqk∫0∞/1−qSn,k,qtbnhqktdqt,which is q-Chlodowsky and q-Szasz-Durrmeyer hybrid operators on weighted spaces. Here, Pn,k,qx and Sn,k,qy are defined as in (2)-(3), u∈0,bn, 0<q<1, h∈C0,bn, and bn is an increasing and positive sequence with properties limn⟶∞bn=∞ and limn⟶∞bn/nq=0. In this article, we intend to study the approximation properties of the operator Hn,qh;u. We produced our study by making use of [25].The important terms of q-analysis which are used in this paper are given below, see [26, 27],Given value ofq>0 and n∈ℕ, we define the q-integer nq by(6)nq=1−qn1−q,ifq≠1,n,ifq=1.Theq-factorial nq! is defined by(7)nq!=nqn−1q,…,1q,ifn=1,2,3,…,1,ifn=0,for 0≤k≤n; we define the q-binomial coefficients nkq by(8)nkq=nq!kq!n−kq!.Theq-binomial can be written in the following forms:(9)nkq=n−1k−1q+qkn−1kq,nkq=qn−kn−1k−1q+n−1kq.Exponential functionez has two q-analogs, see [26, 27]:(10)eqz=∑k=0∞zkkq!=11−1−qzq∞,z<11−q,q<1,Eqz=∏s=0∞1+1−qqsz=∑k=0∞qkk−1/2zkkq!=1+1−qzq∞,q<1,where 1−uq∞=∏s=0∞1−qsu. It is easily observed that(11)eqzEq−z=eq−zEqz=1.The definiteq−Jackson integrals and the q−improper integrals of the function h are defined by(12)∫0bhtdqt≔b1−q∑k=0∞hbqnqn,b>0,(13)∫0∞htdqt≔1−q∑k=0∞hqnBqnB,B>0.The series on the right-hand side in (12) is guaranteed to be convergent if the function h has the property hu<Cuα in a right neighborhood of u=0 for C>0, α>−1.Theq−Gamma functions are defined in two ways. These are(14)Γqu=∫01/1−qtu−1Eq−qtdqt,(15)γqBu=∫0∞/B1−qtu−1eq−tdqt.For everyB and u>0, one has(16)Γqu=KB;uγqBu,where KB;u=1/1+BBu1+1/Bqu1+Bq1−u. Especially, for any positive integer m,(17)KB;m=qmm−1/2andΓqm=qmm−1/2γqBm,see [28].The present note deals with the study of theq-Chlodowsky and q-Szasz-Durrmeyer hybrid operators on weighted spaces. Firstly, we estimate the moments for the Hn,qh;u operators. We also study the rate of convergence for these operators Hn,qh;u. Furthermore, definitions and some properties for weighted spaces are given. We guess the order of approximation by weighted Voronovskaya-type theorem.


## 2. Estimation Moments

Here, we will proveHn,qtm;u for m=0,1,2. By the definition of the q−Gamma function γq in (15), we have(18)∫0∞/1−qqktmSn,k,qtbndqt=∫0∞/1−qqktm1Eqnqt/bnqkk−1/2nqt/bnkkq!dqt=qkm∫0∞/1−qtmeq−nqtbnqkk−1/2nqktkkq!bnkdqt=qkmbnmnqm+1kq!qkk−1/2∫0∞/1−qnqtbnk+meq−nqtbnnqdqt=qkmbnm+1nqm+1kq!qkk−1/2∫0∞/1−qnq−1vk+meq−vdqv=qkmbnm+1nqm+1kq!qkk−1/2γqnq−1k+m+1=qkmbnm+1nqm+1kq!qkk−1/2Γqk+m+1qk+m+1k+m/2=qkmbnm+1nqm+1kq!qkk−1/2k+mq!qk+m+1k+m/2,where, for m=0,(19)∫0∞/1−qqkt0Sn,k,qtbndqt=bnnqq−k,for m=1,(20)∫0∞/1−qqkt1Sn,k,qtbndqt=k+1qbn2nq2q−k−1,and for m=2,(21)∫0∞/1−qqkt2Sn,k,qtbndqt=k+1qk+2qbn3nq3q−k−3.Lemma 1.
Foru∈0,∞, 0<q<1, and h∈C0,∞, we have(22)Hn,q1;u=1,(23)Hn,qt;u=u+bnqnq,(24)Hn,qt2;u=u2+uq3+2q2+1bn−q3uq3nq+q+1bn2q3nq2.Proof.
Using (19), we obtain(25)Hn,q1;u=nqbn∑k=0nPn,k,qubnqk∫0∞/1−qSn,k,qtbndqt=nqbn∑k=0nPn,k,qubnqkbnnqq−k=∑k=0nPn,k,qubn=1.
Using (20) and k+1q=qkq+1, we obtain(26)Hn,qt,u=nqbn∑k=0nPn,k,qubnqk∫0∞/1−qSn,k,qtbnqktdqt=nqbn∑k=0nPn,k,qubnqkk+1qbn2nq2q−k−1=∑k=0nPn,k,qubnk+1qbnnqq=bnq∑k=0nPn,k,qubnqkqnq+∑k=0nPn,k,qubn1nq=bnq∑k=0nPn,k,qubnqkqnq+1nq=bnqqubn∑k=0nPn,k,qubn+1nq=u+bnqnq.
Using (20) and k+1qk+2q=q3kq2+2q2+1kq+q+1, we obtain(27)Hn,qt2;u=nqbn∑k=0nPn,k,qubnqk∫0∞/1−qSn,k,qtbnqkt2dqt=nqbn∑k=0nPn,k,qubnqkk+1qk+2qbn3nq3q−k−3=∑k=0nPn,k,qubnk+1qk+2qbn2nq2q3=bn2q3∑k=0nPn,k,qubnq3kq2nq2+∑k=0nPn,k,qubn2q2+1kqnq2+∑k=0nPn,k,qubnq+1nq2=bn2q3q3bn2nq−1nqu2+q3+2q2+1nqbnu+q+1nq2=u2−1nqu2+q3+2q2+1bnq3nqu+q+1bn2q3nq2=u2+uq3+2q2+1bn−q3uq3nq+q+1bn2q3nq2,which completes the proof.Lemma 2.
Letu∈0,∞, 0<q<1, and h∈C0,∞, and we have(28)Hn,qt−u;u=bnqnq,Hn,qt−u2;u=uq3+1bn−q3unqq3+q+1bn2q3nq2.Proof.
By Lemma1, we have(29)Hn,qt−u;u=Hn,qt;u−uHn,q1;u=u+bnqnq−u=bnqnq,Hn,qt−u2;u=Hn,qt2;u−2uHn,qt;u+u2Hn,q1;u=u2+uq3+2q2+1bn−q3uq3nq+q+1bn2q3nq2−2uu+bnqnq+u2=uq3+1bn−q3unqq3+q+1bn2q3nq2,which completes the proof.By simple calculations, we obtain(30)Hn,qt−u2;u≤bn2q6nq+2bn2q3nq2here; since n⟶∞ will be qn⟶1 and nq=n, the following equation is written:(31)Hn,qt−u2;u≤3bn2n,where Hn,qt−u2;u appears to be constrained in the above inequality, and this result is available in [25]. Likewise, A. İzgi obtained the following result in [25]:(32)Hn,qt−u4;u≤144bn4n2.


## 3. Approximation ofHn,qh;u in Weighted Spaces

In this section, we use Gadjiev’s Korovkin-type theorems on the weighted spaces [4, 29]. Let Bp be the set of all functions h over the real line, pu=1+u2, where u∈−∞,∞=ℝ such that(33)hu≤Mhpu,where Mh is a positive constant depending on the function h. Now, let(34)Cpℝ=h∈Bpℝ,his continuous,Cpkℝ=h∈Cpℝ,limx⟶∞hupu=Kh<∞.It is clear thatCpkℝ⊂Cpℝ⊂Bpℝ, where Bpℝ is the linear normed space with the norm(35)hp=supu∈−∞,∞hupu,h∈Bp.Letbn be an increasing and positive sequence with features(36)limn⟶∞qn=1,limn⟶∞bn=∞andlimn⟶∞bnnq=0.Lemma 3.
Hn,qh;u defined in (5) is a sequence of positive and linear operators that move Cρ0,bn to βρ0,bn. So, there is(37)limn⟶∞Hn,qρt;u=ρu,for ρu∈Cρ0,bn on 0,bn.Proof.
Taking advantage of equations (22) and (24), we have(38)Hn,qρt;u=ρu+uq3+2q2+1bn−q3uq3nq+q+1bn2q3nq2.
Therefore,Hn,qh;uρ,0,bn is uniformly bounded on 0,bn, since(39)limn⟶∞supu∈0,bnuq3+2q2+1bn−q3uq3nq+q+1bn2q3nq2=0,under the condition in (36) that completes the proof.Theorem 1.
Leth∈Cρk0,bn. Then,(40)limn⟶∞Hn,qh−hρ,0,bn=0.Proof.
From Lemma1, we have(41)limn⟶∞Hn,q1;u−1ρ,0,bn=0,limn⟶∞Hn,qt;u−uρ,0,bn=limn⟶∞supu∈0,bnbnqnq11+u2=limn⟶∞bnqnq=0,limn⟶∞Hn,qt2;u−u2ρ,0,bn=limn⟶∞supu∈0,bnuq3+2q2+1bn−q3uq3nq+q+1bn2q3nq211+u2=limn⟶∞q3+2q2+12bn24q6nq+q+1bn2q3nq2=0,according to (36), which completes the proof.


## 4. Main Results

Here, we estimate the rate of approximation of theHn,qh;u hybrid operators. The following theorem gives the rate of approximation of the sequence of Hn,qh;u operators in terms of moduli of continuity of a function h∈Cρk0,bn. For h∈Cρk0,bn, the moduli of continuity is defined as follows:(42)ωh,δ=supht−hu:t,u∈a,b,t−u≤δ,where δ⟶0.The weighted moduli of continuity is defined as follows:(43)Λnh;δ=suph≤δu∈0,bnhu+a−hu1+u21+a2.It is seen that they provide the characteristics of the continuity module. In what follows,(44)limδ⟶0Λnh,δ=0,ht−hu≤1+t−u21+u21+t−uδΛnh,δ,for every h∈Cρk0,bn. A similar definition can be found in [5, 25].Theorem 2.
Leth∈Cρk0,bn, and we have(45)Hn,qh;u−huρ,0,bn≤μbn,n,qΛnh,bn2q3nq,where μbn,n,q is the positive constant for limn⟶∞qn=1 and limn⟶∞bn/nq=0.Proof.
From the well-known features of modulus of continuity, we have(46)ht−hu≤1+t−u21+u21+t−uδΛnh,δ.
Then,Hn is a linear-positive operator, and we write(47)Hn,qht;u−hu≤Hn,qht−hu;u≤Λnh,δ1+u2Hn,q1+t−u21+t−uδ;u≤Λnh,δ1+u2Hn,q1+t−u2;u+Hn,q1+t−u2t−uδ;u,from Lemmas 1-2, (31), and (32), we obtain(48)Hn,q1+t−u2;u=Hn,q1;u+Hn,qt−u2;u≤1+bn2q6nq+2bn2q3nq2≤μ1bn,n,q,and using Cauchy–Schwartz inequality(49)Hn,q1+t−u21+t−uδ;u≤Hn,q1+t−u22;u1/2Hn,qt−uδ2;u1/2.
We can write by (32)-(33) that(50)Hn,q1+t−u22;u=Hn,q1;u+2Hn,qt−u2;u+Hn,qt−u4;u≤1+2μ1+144bn4n2≤μ2bn,n,q,Hn,qt−uδ2;u1/2=1δHn,qt−u2;u1/21δbn2q3nq1q3+2nq≤μ3bn,n,q,where δ=bn2/q3nq.
As a result(51)Hn,qh;u−huρ,0,bn≤μbn,n,qΛnh,bn2q3nq,where(52)μbn,n,q=μ1bn,n,q+μ2bn,n,qμ3bn,n,q,which completes the proof.
Now, we prove a Voronovskaja-type result for theHn,qht;u operators.Theorem 3.
Leth, Dqhu,Dq2hu∈Cρk0,bn, and u∈0,bn be fixed, and we have(53)limn⟶∞nqbnHn,qh;u−hu=Dqhu+uDq2hu.Proof.
Byq−Taylor formula for h∈Cρk0,bn and 0<q<1,(54)ht=hu+t−uqDqhu+12qt−uq2Dq2hu+℧t;ut−uq2,where t−uq2=t−ut−qu and(55)℧t;u=ht−hu−t−uqDqhu−1/2qt−uq2Dq2hut−uq2,t≠u,0,t=u.
By implementingq−L’Hospital’s rule twice,(56)limt⟶u℧t;u=limt⟶uDq2ht−Dq2hu2=0.
Then,℧u;u=0 and ℧…;u∈Cρk0,bn. As a result, we can write(57)Hn,qh;u−hu=Hn,qt−uq;uDqhu+12qHn,qt−uq2;uDq2hu+Hn,q℧t;ut−uq2;u.
Now, let us use the Cauchy–Schwarz inequality in the last term of the last equation:(58)Hn,q℧t;ut−uq2;u≤Hn,qt−uq4;u1/2Hn,q℧2t;u;u1/2.
Since℧.,.;u∈Cρk0,bn and ℧t;u⟶℧u;u⟶0 as t⟶u, applying Lemma 2 and (31) and (32),(59)limn⟶∞Hn,q℧2t;u;u=℧2u;u=0,limn⟶∞Hn,qt−uq4;u<∞.
As a result(60)Hn,qh;u−hu=Hn,qt−uq;uDqhu+12qHn,qt−uq2;uDq2hu=Dqhubnqnq+12qDq2huuq3+1bn−q3unqq3+q+1bn2q3nq2,from (33), and we have(61)limn⟶∞nqbnHn,qh;u−hu=Dqhu+uDq2hu,which completes the proof.


## 5. Conclusion

In this paper, the approximation properties and rate of convergence ofq-Chlodowsky and q-Szasz-Durrmeyer hybrid operators in weighted spaces are investigated.For further research in this topic, it would be interesting to study whether the quality of the approximation pythagorean fuzzy set operators,q-statistical convergence operators, and q-complex operators directly influence the quality of the approximation of the characteristics. Some studies can be used for future research, such as [30–34].



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