# On Shape Parameterα-Based Approximation Properties and q-Statistical Convergence of Baskakov-Gamma Operators

Authors: Ming-Yu Chen; Md Nasiruzzaman; Mohammad Ayman Mursaleen; Nadeem Rao; Adem Kilicman
Journal: Journal of Mathematics
 (2022)

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## Abstract

We construct a novel family of summation-integral-type hybrid operators in terms of shape parameterα∈0,1 in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre’s K-functional, Lipschitz class, and second-order modulus of smoothness. The approximation results are then obtained in weighted space. Finally, for these operators q-statistical convergence is also investigated.


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## Body

## 1. Introduction

The theory of linear positive operators deals with a question that arises in the approximate representation of an arbitrary function by the simplest one. In the recent past, the operator theory is a growing and fascinating field of research in approximation theory with the advent of computer. Several researchers constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in different function spaces in terms of several generating functions. In a variety of big data applications, problems with approximating multivariate functions are becoming increasingly significant. They are necessary for machine learning algorithms’ generalisation analysis. Machine learning [1, 2] and deep learning [3] are two areas where such operators/polynomials are useful. Recently, Chen et al. [4] constructed a sequence of new linear positive operators known as the α-Berstein operators of order m:(1)Tm,αg;y=∑i=0mgimpm,iαy,y∈0,1,for g∈C0,1,m∈ℕ, and α∈−1,1, where p1,0α=1−y, p1,1α=y, and pm,iαy(2)=1−αym−2i+1−α1−ym−2i−2+αy1−ymiyi−11−ym−i−1,m≥2.Remark 1.
One can note that, forα=1, equation (1) is reduced to classical Bernstein operators [5].
These operators are restricted for the space of continuous functions only. To approximate the wider class than the class of continuous function,i.e., space of Lebesgue integrable functions, Mohiuddine et al. constructed Kantorovich-type of α-Bernstein operators [6] and Stancu-type α-Bernstein–Kantorovich operators [7]. Cai et al. [8] introduced a generalisation of classical Bernstein operators based on shape parameter α∈0,1. These operators are termed as α−Bernstein operators of degree m and defined by (1).Remark 2.
Note thatpm,iα in the relation (2) is called α-Berstein polynomials of order m and the binomial coefficients(3)pq=p!q!(p−q)!,0≤q≤p,0,otherwise.
Later on, Aral and Erbay [9] introduced the parametric form of Baskakov–Durrmeyer operators as(4)Lm,αg;w=∑s=0∞Pm,sαwgsm,where f∈CB[0,∞)≔g∈C[0,∞):‖g‖=sup0≤x<∞|g(x)|<∞, m≥1, and w∈[0,∞) and for α∈0,1,(5)Pm,sαgw=ws−11+wm+s−1αw1+wm+s−1s−1−α1+wm+s−3s−2+1−αwm+s−1s.with m−3−2=m−2−1=0. The sequences (5) are restricted for the space of continuous functions only. Motivated by the above development, we construct a sequence of hybrid operators to approximate in a wider class, i.e., the space of Lebesgue integrable functions, as follows:(6)Am,α∗g;w=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λe−mtgtdt,where Pm,sαw is given by (5) and the gamma function as(7)Γw=∫0∞wm−1e−wdw,Γw=w−1Γw−1=w−1!.
In the subsequent sections, we establish basic lemmas, rate of convergence, order of approximation locally and globally in terms of modulus of smoothness, Peetre’s K-functional, second modulus of smoothness, Lipschitz space maximal function, and weighted modulus of smoothness. Lastly, we study theq-statistical convergence. For more basic concepts and related articles, we refer to see [10–16].Remark 3.
One can note that, forα=λ=0, the operators constructed by us in (6) are reduced to the classical Baskakov–Durrmeyer operators [17].


## 2. Basic Estimates and Approximation

Lemma 1 (see [9]).
Form∈ℕ, the α-Baskakov operator has the following identities:(8)Lm,α1;w=1,Lm,αt;w=w+2mα−1w,Lm,αt2;w=w2+4α−3mw2+wm2m+4α−4.Lemma 2.
For test functioneiw=wi,i∈0,1,2, we have the following identities:(9)Am,α∗e0;w=1,Am,α∗e1;w=w+2mα−1w+λ+1m,Am,α∗e2;w=w21+4α−3m+z2λ+3m+4α−4+2λ+3α−1m2+λ2+3λ+2m2,where m∈M and α∈−1,1.Proof.
In view of Lemma2, for e0=1, we have(10)Am,α∗e0;ω=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λe−mtdt=∑s=0∞Pm,sαwms+λ+1Γs+λ+1Γs+λ+1ms+λ+1=1.
Fore1=t, we have(11)Am,α∗e1;w=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λe−mttdt=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λ+1e−mtdt=∑s=0∞Pm,sαwms+λ+1Γs+λ+1Γs+λ+2ms+λ+2=∑s=0∞Pm,sαw1ms+λ+1Γs+λ+1Γs+λ+1=∑s=0∞Pm,sαwsm+λ+1m=z+2mα−1w+λ+1m.
Fore2=t2,(12)Am,α∗e2;w=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λe−mtt2dt=∑s=0∞Pm,sαwms+λ+1Γs+λ+1∫0∞ts+λ+2e−mtdt=∑s=0∞Pm,sαwms+λ+1Γs+λ+1Γs+λ+3ms+λ+3=∑s=0∞Pm,sαwms+λ+1Γs+λ+1s+λ+2s+λ+1Γs+λ+1ms+λ+3=∑s=0∞Pm,sαws+λ+2s+λ+3m2=∑s=0∞Pm,sαws2+2λ+3s+λ2+3λ+2m2=∑s=0∞Pm,sαws2m2+2λ+3sm2+λ2+3λ+2m2,Am,α∗e2;w=w2+4α−3w2m+zm+4α−4m2+2λ+3mw+2α−1wm+λ2+3λ+2m2=w21+4α−3m+w2λ+3m+4α−4+2λ+3α−1m2+λ2+3λ+2m2.Lemma 3.
Letηjw=t−wj,j∈0,1,2. Then,(13)Am,α∗η0;w=1,Am,α∗η1;w=2λ−1wm+λ+1m,Am,α∗η2;w=01mw2+w+1.Proof.
In the light of linearity property ofAm,α∗.;. and Lemma 2, we get the desired Lemma 3.Definition 1.
The modulus of continuity for a uniformly continuous functionf on 0,∞ is defined as(14)ω(f;δ)=sup|t1−t2|≤δ|f(t1)−f(t2)|,δ>0,t1,t2∈[0,∞).
Also, we get(15)|ft1−ft2|≤1+t1−t22δ2ωf;δ.Theorem 1.
Suppose thatQφ=φ:w≥0,φw/1+w2converges whenw⟶∞. Then, for anyφ∈C[0,∞)∩Qφ, the operatorsAm,α∗.;.given by (6) converges to functionφ uniformly.Proof.
Taking into account property (vi) of Theorem 4.1.4 in [18], it is enough to show that(16)Am,α∗ej;w⟶ejw,ifj=0,1,2.
From Lemma2, we get Am,α∗ej;w⟶ejw for j=0,1,2 when m⟶∞, which gives the proof of Theorem 1.Theorem 2.
(see [19]).Letℒ:Ca,b⟶Ba,bbe a linear and positive operator andf∈CBa,b. Then, for δ>0,(17)ℒfx−fx≤fx‖ℒe0x−1+ℒe0x+δ−1ℒe0xℒφx2xωfδ,where(18)φxt=|t−x|,x,t∈a,b×a,b.Theorem 3.
For anyg∈CB[0,∞), the sequence of operatorsAm,α∗.;.defined by (6) verifies the inequality(19)|Am,α∗g;w−gw|≤2ωg;δ,where δ=Am,α∗η2;w and CB[0,∞) stands for space of all continuous and bounded functions on the interval 0,∞.Proof.
In the light of Lemma2, Lemma 3, and Theorem 2, it is easy to obtain(20)|Am,α∗g;w−gw|≤1+δ−1Am,α∗η2;wω∗g;δ.
On takingδ=Am,α∗η2;w, we arrive at the required result.


## 3. Local and Global Approximation Results

The Peetre’s K-functional is defined by(21)K2(g,δ)=inf‖f−h‖+δ‖h′′‖:h∈CB2[0,∞),for any f∈CB[0,∞) and δ>0, where CB2[0,∞)=h∈CB[0,∞):h′,h′′∈CB[0,∞). From DeVore and Lorentz ([20], p.177, Theorem 2.4), there exists an absolute constant C>0 such that(22)K2f;δ≤Cω2f;δ.We define the auxiliary operatorA^m,αg;w as(23)A^m,αg;w=Am,α∗g;w+gw−gw+2mα−1w+λ+1m.Theorem 4.
Forg∈CB2[0,∞),w≥0andi,λ≥0,α∈0,1, one gets(24)|A^m,αg;w−gw|≤ξmw‖g′′‖,where(25)ξmw=A^m,αt−w2;w+2mα−1w+λ+1m.Proof.
From the auxiliary operator (23), we get that(26)A^m,αe0;w=1,A^m,αη1w;w=0and|A^m,αg;w|≤3‖g‖.
From Taylor series expansion, for anyg∈CB2[0,∞), we have(27)gt=gw+t−wg′w+∫wtt−ρg′′ρdρ.
On operatingA^m,αf;w in (27), we obtain(28)A^m,αg;w−gw=g′wA^m,αt−w;w+A^m,α∫wtt−ρg′′ρdρ;w.
Therefore, in the view of (26), we get(29)A^m,αg;w−gw=A^m,α∫wtt−ρg′′ρdρ;w=Am,α∗∫wtt−ρg′′ρdρ;z−∫ww+2mα−1w+λ+1mz+2mα−1w+λ+1m−ρg′′ρdρ,A^m,αg;w−gw≤Am,α∗∫wtt−ρg′′ρdρ;w+∫ww+2mα−1w+λ+1mw+2mα−1w+λ+1m−ρg′′ρdρ.
Since(30)∫wtt−ρg′′ρdρ≤t−w2∥g′′∥,then(31)∫ww+2/mα−1+λ+1/mw+2mα−1w+λ+1m−ρg′′ρdρ≤2mα−1w+λ+1m2∥g″∥.
Combining the equalities (29)–(31), we see that(32)A^m,αg;w−gw≤A^m,αt−w2;w+2mα−1w+λ+1m2‖g′′‖=ξmw‖g′′‖.which completes the proof.Theorem 5.
For anyg∈CB2[0,∞), there exists a positive numberCsatisfying the inequality(33)∣Am,α∗g;w−gw∣≤Cω2∗g;ξmw+ωg;A^m,αη1;w,whereξmwis given by (25).Proof.
Leth∈CB2[0,∞) and g∈CB[0,∞). Then, using the definition of A^m,α.;., we get(34)Am,α∗g;w−gw≤A^m,αg−h;w|+|g−hw|+|D^mhw−hw+gw+2mα−1w+λ+1m−gw.
In view of Theorem5 and relation (26), one has(35)Am,α∗g;w−gw≤4‖g−h‖+|A^m,α∗h;w−hw|+gz+2mα−1w+λ+1m−gw≤4‖g−h‖+ξmw‖h′′‖+ωg;Am,α∗η1;w.
Using the definition ofK-functional, we obtain(36)|Am,α∗g;w−gw|≤Cω2∗g;ξmw+ω∗g;Am,α∗η1;z.
This gives the proof of Theorem5.For any fixed two real positive numberss1 and s2, the Lipschitz-class of functions [21] is defined by(37)LipMs1,s2(β)≔g∈CB[0,∞):|g(t)−g(w)|≤C|t−w|βt+s1w+s2w2β/2:w,t∈(0,∞),with the positive constant C and 0<β≤1.Theorem 6.
Letg∈LipMs1,s2β. Then, it follows that(38)|Am,α∗(g;w)−g(w)|≤CAm,α∗(η2;w)s1z+s2z2β/2,where z>0.Proof.
Forβ=1, we have(39)Am,α∗g;w−gw≤Am,α∗|gt−gw|;w≤CAm,α∗t−wt+s1w+s2w21/2;w.
Since1/t+s1w+s2w2<1/s1w+s2w2, for every w∈0,∞, we get that(40)Am,α∗(g;w)−g(w)≤Ms1w+s2w21/2Am,α∗(t−w2;w)1/2≤CAm,α∗(η2;w)w1z+s2w21/2.
Thus, Theorem6 holds good when β=1. Next, on choosing 0<β<1 and applying Hölder’s inequality for p1=2/β and p2=2/2−β, we have(41)Am,α∗g;w−gw≤Am,α∗|gt−gw|;w2/ββ/2≤MAm,α∗|t−w|2t+s1w+s2w2;wβ/2.
Since1/t+s1w+s2w2<1/s1w+s2w2 for every w∈0,∞, we get(42)|Am,α∗(g;w)−g(w)|≤CAm,α∗((|t−w|2;w)s1w+s2w2β/2.
Thus, we get the proof of Theorem6.To obtain the local type approximation results in therth order, the Lipschitz-maximal function (see Lenze [22]) is given by(43)ω˜r(g;w)=supt≠w,t∈(0,∞)g(t)−g(w)|t−w|r,w∈[0,∞) and0<r≤1.Theorem 7.
Letw∈[0,∞)andg∈CB[0,∞). Then, for anyr∈(0,1], we have(44)|Am,α∗(g;w)−g(w)|≤ω˜r(g;w)Am,α∗((|t−w|2;w)r/2.Proof.
Since we know that(45)|Am,α∗g;w−gw|≤Am,α∗|gt−gw|;w.
Therefore, from equality (43), we get(46)|Am,α∗g;w−gw|≤ω˜rg;wAm,α∗|t−w|r;w.
Using the Ḧolder’s inequality withp1=2/r and p2=2/2−r, we have(47)|Am,α∗g;w−gw|≤ω˜rg;w(Am,α∗|t−w|2;w)r/2,which gives the desired result.From [23], we recall some notations. For the weight function 1+x2 and 0≤x<∞, we denote(48)B1+x2[0,∞)=f:|f(x)|≤Mf(1+x2),Mfis  a constant depending onf,C1+x2[0,∞)=f∈C[0,∞):‖f‖1+x2=supx∈[0,∞)f1+x2<∞,C1+x2k[0,∞)=f∈C1+x2[0,∞):limx⟶∞f(x)1+x2=k,wherekis a constant.Note thatC1+x2[0,∞)⊂B1+x2[0,∞). Moreover, the modulus of smoothness for any function ϕ on closed interval 0,μ, μ>0 is defined by(49)ωμ∗ϕ,δ=sup|t−w|≤δsupw,t∈0,μ|ϕt−ϕw|.Theorem 8.
Letωμ+1∗ϕ;δbe the modulus of smoothness defined on0,μ+1]⊂[0,∞. Then, for allϕ∈C1+w2[0,∞),(50)‖Am,α∗ϕ;z−ϕw‖C0,μ≤6Mϕ1+μ2δmμ+2ωμ+1∗ϕ;δmμ,where δmμ=Am,α∗η2;μ.Proof.
For allw∈0,μ and t∈[0,∞), one has(51)|ϕt−ϕw|≤6Mϕ1+μ2t−w2+1+t−zδωμ+1∗ϕ;δ.
On applying the operatorsAm,α∗, we see that(52)|Am,α∗ϕ;w−ϕw|≤6Mϕ1+μ2Am,α∗t−w2;w+1+Am,α∗|t−w|;wδωμ+1∗ϕ;δ.
Thus, forw∈0,μ, applying Lemma 2, we get(53)|Am,α∗ϕ;w−ϕw|≤6Mϕ1+μ2δmμ+1+δmμδωμ+1∗ϕ;δ.
Takingδ=δmμ, we get the desired result.Theorem 9.
Suppose the operatorsAm,α∗.;.acting fromC1+w2k[0,∞)toB1+w2[0,∞). Then, forϕ∈C1+w2k[0,∞), we have(54)limm⟶∞‖Am,α∗ϕ−ϕ‖1+w2=0.Proof.
For the results of Theorem9, we have to show that(55)limm⟶∞‖Am,α∗ei−wi‖1+w2=0,i=0,1,2.
By using Lemma2, it is enough to show(56)‖Am,α∗(e0)−w0‖1+w2=supw∈[0,∞)Am,α∗(e0;w)−11+w2=0fori=0.
Fori=1,(57)‖Am,α∗(e1)−w1‖1+w2=supw∈[0,∞)2/m(α−1)w+λ+1/m1+w2=2m(α−1)supw∈[0,∞)w1+w2+λ+1msupw∈[0,∞)11+w2,which gives us ‖Am,α∗e1−w1‖1+w2⟶0 as m⟶∞. For i=2,(58)‖Am,α∗(e2)−w2‖1+w2=supw∈[0,∞)w2(4α−3/m)+w(2λ+3/m+4α−4+(2λ+3)(α−1)/m2)1+w2+supw∈[0,∞)λ2+3λ+2/m21+w2=4α−3msupw∈[0,∞)w21+z2+(2λ+3m+4α−4+(2λ+3)(α−1)m2)supw∈[0,∞)w1+w2+λ+1)(λ+2m2supw∈[0,∞)11+w2,and we get ‖Am,α∗e2−w2‖1+w2⟶0 as m⟶∞. Hence, we get the result. □Here, we study the approximation of locally integrable functions belonging toC1+w2k[0,∞). Such type of result is investigated by Gadjiev [23].Theorem 10.
Supposeϕ∈C1+w2k[0,∞). Then, for anyθ>0, we have(59)limm⟶∞supw∈[0,∞)Am,α∗(ϕ;w)|−ϕ(w1+w21+θ=0.Proof.
Letw0 be the fixed positive real number. Then,(60)supw∈[0,∞)Am,α∗(ϕ;w)|−ϕ(w1+w21+θ≤supw≤w0Am,α∗(ϕ;w)|−ϕ(w1+w21+θ+supw≥w0Am,α∗(ϕ;w)|−ϕ(w1+z21+θ≤‖Am,α∗ϕ;w|−ϕw‖C0,z0+‖ϕ‖1+w2supw≥w0Am,α∗1+t2;w1+w21+θ+supw≥w0ϕw1+w21+θ=I1+I2+I3say.
Sincesupw∈[0,∞)Am,α∗(ϕ;w)|−ϕ(w/1+w21+θ≤supw≥w0Am,α∗(ϕ;w)|−ϕ(w1+w21+θ+ssupw≥w0Am,α∗(ϕ;w)|−ϕ(w/1+z21+θ≤‖Am,α∗ϕ;w|−ϕw‖C0,z0+‖ϕ‖1+w2supw≥w0Am,α∗1+t2;w1+w21+θ+supw≥w0ϕw1+w21+θ=I1+I2+I3say., we get(61)I3=supw≥w0ϕw1+w21+θ≤supw≥w0‖ϕ‖1+w21+w21+w21+θ≤‖ϕ‖1+w21+w2θ.
For an arbitrary real numberϵ>0, in view of Theorem 1, there exists m1∈ℕ satisfying(62)I2<11+w2θ‖ϕ‖1+w21+w2+ϵ3‖ϕ‖1+w2for allm1≥m,<‖ϕ‖1+w21+w2θ+ϵ3,for allm1≥m.
This implies that(63)I2+I3<2‖ϕ‖1+w21+w2θ+ϵ3.
For any sufficiently large numberw0, we have ‖ϕ‖1+w2/1+w2θ<ϵ/6.(64)I2+I3<2ϵ3for allm1≥m.
In the light of Theorem9 and for any m2>m, one has(65)I1=‖Am,α∗ϕ;w−ϕ‖C0,z0<ϵ3for allm2≥m.
Takem3=maxm1,m2 and taking (61) by values of I1, I2, and I3, we easily get(66)supw∈[0,∞)Am,α∗(ϕ;w)|−ϕ(w1+w21+θ<ϵ.
Thus, the proof Theorem10 is completed.


## 4.q-Density and q-Statistical Convergence

Recently, theq-analog of density and statistical convergence is studied in [24]. Let E⊆ℕ (the set of natural numbers). Then, the q-density is defined by(67)δqE=δC1qE=liminfn⟶∞C1qχEn=liminfn⟶∞∑k∈Eqk−1n,q≥1,where C1q=cnk1qkn,k=0∞ is the q-Cesàro matrix (see [24, 25]) defined by(68)cnk1qk=qkn+1q,if k≤n,0,otherwise.which is regular for q≥1, where the q-integer q>0 of any positive integer n is defined by(69)n=nq=1−qn1−q,q≠1,n,q=1.A sequenceη=ηk is said to be q-statistically convergent to the number ℒ if δqℒε=0, where ℒε=k≤n:|ηk−l|≥ε for every ε>0. That is,(70)limn1n#k≤n:qk−1|ηk−l|≥ε=0,and we write Stq−limηk=l.IfδE=0 for an infinite set E, then δqE=0; hence, statistical convergence [26] implies q-statistical convergence but not conversely (c.f. [2], Example 15).Theorem 11.
Forϕ∈C1+z2+λk[0,∞), we have(71)Stq−limm‖Am,α∗ϕ;w−ϕ‖1+w2+λ=0,for allλ>0.Proof.
Forλ=0, it is easy to get that(72)Stq−limm‖Am,α∗ei;w−ei‖1+w2=0,fori∈0,1,2.
From Lemma2, we see that(73)‖Am,α∗(e1;w)−w‖1+w2=supw∈[0,∞)w1+w22m(α−1)+supw∈[0,∞)11+w2λ+1m=2m(α−1)supw∈[0,∞)w1+w2+λ+1msupw∈[0,∞)11+w2.
Now, for a given positiveϵ>0, let(74)M1≔m:‖Am,α∗e1;w−w‖≥ϵ,M2≔m:2mα−1≥ϵ2,M3≔m:λ+1m≥ϵ2.
This implies thatM1⊆M2∪M3 which shows that δqM1≤δqM2+δqM3. Hence, we have(75)Stq−limm‖Am,α∗e1;w−w‖1+w2=0.
Fori=2, in the view of Lemma 2, we get(76)‖Am,α∗(e2;w)−w2‖1+w2=supw∈[0,∞)11+w2w2(4α−3m)+w2λ+3m+4α−4+2λ+3α−1m2+λ+1λ+2m2=4α−3msupw∈[0,∞)w1+w2+2λ+3msupw∈[0,∞)w21+w2+4α−4+(2λ+3)(α−1)m2supw∈[0,∞)w1+w2+λ+1)(λ+2m2supw∈[0,∞)11+w2.
For a given positiveϵ>0, let(77)T1≔m:Am,α∗e2;w−w2≥ϵ,T2≔m:4α−3m≥ϵ4,T3≔m:2λ+3m≥ϵ4,T4≔m:4α−4+2λ+3α−1m2≥ϵ4,T5≔m:λ+1λ+2m2≥ϵ4.
Then,T1⊆T2∪T3∪T4∪T5, and we get(78)δqT1≤δqT2+δqT3+δqT4+δqT5.
Lettingm⟶∞, we have(79)Stq−limm‖Am,α∗e2;w−w2‖1+w2=0.
This gives the desired proof of Theorem11.



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