# Approximation Properties andq-Statistical Convergence of Stancu-Type Generalized Baskakov-Szász Operators

Authors: Qing-Bo Cai; Adem Kilicman; Mohammad Ayman Mursaleen
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this article, we introduce Stancu-type modification of generalized Baskakov-Szász operators. We obtain recurrence relations to calculate moments for these new operators. We study several approximation properties andq-statistical approximation for these operators.


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## Body

## 1. Introduction

In 1912, Bernstein [1] proposed the famous polynomial known as the Bernstein polynomial to give a simple, short, and most elegant proof of the Weierstrass approximation theorem. Since then, several papers have appeared to study approximation properties in different settings and spaces. Many new operators were constructed, e.g., Szász [2], Mirakjan [3], Kantorovic [4], Durrmeyer [5], Stancu [6], and many more [7–9]. These operators provide the improvement of approximating functions of different classes and give better and better estimates. For example, the Baskakov operators were given in [10]:
(1)Vph;v==∑l=0∞p+l−1lvl1+vp+lhlp.For h∈C0,∞, the space of all continuous functions on 0,∞ normed with standard sup-norm ∥·∥∞.Devore and Lorentz [11] introduced a generalization of operators (1) dependent on a constant a>0 and independent of p as follows:
(2)Bph;u=∑j=0∞Wp,jauhjp,where
(3)Wp,jau=e−au/1+uGjp,aj!uj1+up+j,∑j=0∞Wp,jau=1,Gjp,a=∑i=0jjipiaj−i,and pi=pp+1⋯p+i−1,p0=1.Recently, Agrawal et al. [12] studied the following operators (2):
(4)Lp,a∗h;u=p∑j=0∞Wp,jau∫0∞sp,jthtdt,for h∈Cγ0,∞≔h∈C0,∞:ht≤Mheγt, for some γ>0,Mh>0, where sp,jt=e−ptptj/j!.Inspired by Stancu’s work [6], we have studied recently the Stancu-type generalization in [13]. Now, we propose the Stancu-type generalization of operators (4) as follows:
(5)Lp,aλ,μh;v=p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μdt,for any bounded and integrable function h defined on 0,∞, where 0≤λ≤μ. For λ=μ=0, the operators (5) reduce to operators (4).We establish recurrence relations to find moments and central moments. We study some approximation properties and the Voronovskaja-type asymptotic formula. We also study weighted approximation.


## 2. Auxiliary Results

Our first result is the recurrence formula for moments.Theorem 1.
Themth order moment for (5) is defined by
(6)Tp,a,mλ,μv≔Lp,aλ,μtm;v=p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μmdt.
m=0,1,2,⋯. Then, Tp,a,0λ,μv=1, and
(7)m+11+vTp,a,m+1λ,μv=v1+v2Tp,a,mλ,μv′+1+vλ+pv+m+1+avTp,a,mλ,μv−λmp+μ1+vTp,a,m−1λ,μv.Proof.
We use the identity(8)v1+v2Wp,kav′=k−pv1+v−avWp,kav.
Then,(9)v1+v2Tp,a,mλ,μv′=p∑k=0∞v1+v2Wp,kav′∫0∞sp,kthpt+λp+μmdt=p∑k=0∞k−pv1+v−avWp,kav∫0∞sp,ktpt+λp+μmdt=p1+v∑k=0∞k−pvWp,kav∫0∞sp,ktpt+λp+μmdt−av.p∑k=0∞Wp,kav∫0∞sp,ktpt+λp+μmdt,(10)v1+v2Tp,a,mλ,μv′=I−axTp,a,mλ,μv,where
(11)I=p1+v∑k=0∞k−pvWp,kav∫0∞sp,ktpt+λp+μmdt=p1+v∑k=0∞k−pt+λ+pt+λ−pvWp,kav∫0∞sp,ktpt+λp+μmdt=p1+v∑k=0∞k−pt+pt+λ−λ+pvWp,kav∫0∞sp,ktpt+λp+μmdt=p1+v∑k=0∞Wp,kav∫0∞k−ptsp,ktpt+λp+μmdt+p1+vp+μ∑k=0∞Wp,kav∫0∞k−ptsp,ktpt+λp+μm+1dt−p1+vλ+pv∑k=0∞Wp,kav∫0∞sp,ktpt+λp+μmdt,(12)I=∑1+∑2+∑3,say
where(13)∑2=1+vp+μTp,a,m+1λ,μv,(14)∑3=−1+vλ+pvTp,a,mλ,μv,(15)∑1=p1+v∑k=0∞Wp,kav∫0∞k−ptsn,ktpt+λp+μmdt=1+v∑k=0∞Wp,kav∫0∞ptsp,kt′pt+λp+μmdt,usingtsp,kt′=∫0∞k−ptsp,kt,=p+μ1+v∑k=0∞Wp,kav∫0∞pt+λp+μ−λp+μsp,kt′pt+λp+μmdt=p+μ1+v∑k=0∞Wp,kav∫0∞sp,kt′pt+λp+μm+1dt−λ1+v∑k=0∞Wp,kav∫0∞sp,kt′pt+λp+μmdt=J1+J2,saywhere
(16)J1=p+μ1+vWp,kav∫0∞sp,kt′pt+λp+μm+1dt=−1+vm+1p∑k=0∞Wp,kav∫0∞sp,ktpt+λp+μmdt−1+vm+1Tp,a,mλ,μv,J2=−λ1+v∑k=0∞Wp,kav∫0∞sp,kt′pt+λp+μmdt=−λ1+v−mnp+μ∑k=0∞Wp,kav∫0∞sp,ktpt+λp+μm−1dt=−mλp+μ1+vTp,a,m−1λ,μv.
Therefore,(17)∑1=−mλp+μ1+vTp,a,m−1λ,μv−1+vm+1Tp,a,mλ,μv.
Substituting (17), (14), and (13) in (12), we get
(18)I=−mλp+μ1+vTp,a,m−1λ,μv−1+vpv+λ+m+1Tp,a,mλ,μv+1+vp+μTp,a,m+1λ,μv.
Further, substituting (18) in (10), we get the result.Corollary 2.
From the above theorem, we get(i)
Lp,aλ,μ1;v=1(ii)
Lp,aλ,μt;v=1/p+μpv+av/1+v+1+λ(iii)
Lp,aλ,μt2;v=1/p+μ2p2+p+a2/1+v2+2ap/1+vv2+4p+4a/1+v+2λp+2aλ/1+vv+λ2+2λ+2Theorem 3.
Themth order central moment is defined by
(19)Mp,a,mλ,μv≔Lp,aλ,μt−vm;v=p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μ−umdt.
m=0,1,2,⋯. The following recurrence relation holds:
(20)v1+v2Mp,a,m+1λ,μv′=1+vp+μλ−pv−μv−mpv−mμv−mpv2−mμv2Mp,a,m−1λ,μv−m+mv+1+v−μv+λ−μv2+λv+avMp,a,mλ,μv+1+vp+μMp,a,m+1λ,μv.Corollary 4.
From the above theorem, we get(a)
Lp,aλ,μt−v;v=1/p+μ−μv+av/1+v+λ+1(b)
Lp,aλ,μt−v2;v=1/p+μ2p+μ2+a2/1+v2−2aμ/1+vv2+2/p+μ2p−μ−λμ+2+λa/1+vv+1/p+μ2λ2+2λ+2Corollary 5.
We further get(a)
limp⟶∞pLp,aλ,μt−v;v=−μu+av/1+v+λ+1(b)
limp⟶∞pLp,aλ,μt−v2;v=vv+2


## 3. Main Results

Peetre’sK-functional is defined as
(21)K2h,δ≔inf∥h−g∥+δ∥g″∥:g∈CB20,∞,for h∈CB0,∞,δ>0, where CB0,∞≔h∈CB0,∞:h is bounded on 0,∞ and CB20,∞≔g∈CB0,∞:g′,g″∈CB0,∞. Note that
(22)K2h;δ≤Mω2h;δ,M>0,where ω2h;δ is the second-order modulus of continuity [11].
(23)ω2h,δ=sup0<l≤δsupv∈0,∞hv+2l−2hv+l+hv,δ>0.The usual modulus of continuity ofh∈CB0,∞ is defined as
(24)ω1h,δ=sup0<l≤δsupv∈0,∞hv+l−hv.Theorem 6.
Forh∈CB0,∞,
(25)Lp,aλ,μh;v−hv≤Mω2h;ϕp,aλ,μ+ω1h;1p+μ1+λ−μv+av1+v,where M>0 and
(26)ϕp,aλ,μ=1p+μ2p2+μ2+a21+v2−2aμ1+vv2+2p−2μ−2λμ+2a2+λ1+vv+1p+μ21+λ−μv+av1+v2+λ2+2λ+2.Proof.
Put(27)L~p,aλ,μh;v=Lp,aλ,μh;v+hv−h1+λp+μ+pvp+μ+1p+μav1+v.
Note thatL~p,aλ,μ1;v=1 and L~p,aλ,μt;v=v. Let E∈CB20,∞. Then, by using Taylor’s theorem, we may write
(28)Et=Ev+t−vE′x∫vtt−yE″ydy,which gives
(29)L~p,aλ,μE;v−Ev=E′vL~p,aλ,μt−v;v+L~p,aλ,μ∫vtt−yE″ydy;v=L~p,aλ,μ∫vtt−yE″ydy;v=Lp,aλ,μ∫vtt−yE″ydy;v−∫v1/p+μ1+λ+pv+av/1+v1+λp+μ+pvp+μ+1p+μav1+v−vE″ydy.
Hence,(30)L~p,aλ,μE;v−Ev≤Lp,aλ,μ∫vtt−yE″ydy;v+∫v1/p+μ1+λ+pv+av/1+v1+λp+μ+pvp+μ+1p+μav1+v−yE″ydy.
Since∫vtt−yE′′ydy≤t−v2∥E″∥ and
(31)∫v1/p+μ1+λ+pv+av/1+v1+λp+μ+pvp+μ+1p+μav1+v−yE″ydy≤1+λp+μ−μvp+μ+1p+μav1+v2∥E″∥,we have
(32)L~p,aλ,μE;v−Ev≤Lp,aλ,μt−v2;v+1+λp+μ−μvp+μ+1p+μav1+v2∥E″∥.
Now, by Corollary4 (b), we get
(33)L~p,aλ,μE;v−Ev≤ϕp,aλ,μv∥E′′∥.
By (27), we get
(34)∣Lp,aλ,μh;v−hv∣≤L~p,aλ,μh−E;v+∣h−Ev∣+L~p,aλ,μE;v−Ev+h1+λp+μ+pvp+μ+1p+μav1+v−hv.
SinceL~p,aλ,μh;v≤3∥h∥, we get
(35)∣Lp,aλ,μh;v−hv∣≤4∥h−E∥+L~p,aλ,μE;v−Ev+h1+λp+μ+pvp+μ+1p+μav1+v−hv.
From (33), we get
(36)∣Lp,aλ,μh;v−hv∣≤4∥h−E∥+ϕp,aλ,μv∥E″∥+ω1h:1+λp+μ−μvp+μ+1p+μav1+v.
Now, taking the infimum over allE∈CB20,∞, we obtain
(37)∣Lp,aλ,μh;v−hv∣≤4K2h;ϕp,aλ,μv+ω1f:1+λp+μ−μvp+μ+1p+μav1+v.
Hence, by using (22), we get the result.For our next result, we consider the functions belonging to the Lipschitz class:(38)lipMγ=h∈CB0,∞:ht−hv≤Mt−vγt+vγ/2,whereM>0 and 0<γ≤1; v,t∈0,∞.Theorem 7.
Forh∈lipMγ, we have
(39)∣Lp,aλ,μh;v−hv∣≤Mφpλ,μvvγ/2,where φp,aλ,μv=Lp,aλ,μe1−v2;v.Proof.
First, we prove forγ=1. For h∈lipMγ, we get
(40)Lp,aλ,μh;v−hv≤p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μ−hvdt≤Mp∑k=0∞Wp,kav∫0∞sp,kt∣pt+λ/p+μ−v∣pt+λ/p+μ+vdt.
Sincev<pt+λ/p+μ+v, we get by the Cauchy-Schwarz inequality:
(41)Lp,aλ,μh;v−hv≤Mvp∑k=0∞Wp,kav∫0∞sn,ktpt+λp+μ−udt=MvLp,aλ,μe1−v2;v≤Mφp,aλ,μvv.
For0<γ<1, applying Hölder’s inequality, we get
(42)Lp,aλ,μh;v−hv≤p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μ−hvdt≤p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μ−hvdt1/γγ≤p∑k=0∞Wp,kav∫0∞sp,kthpt+λp+μ−hv1/γdtγ.
Sinceh∈lipMγ, we have
(43)∣Lp,aλ,μh;v−hv≤Mvγ/2p∑k=0∞Wp,kav∫0∞sp,kt∣pt+λp+μ−v∣dtγ=Mvγ/2Lp,aλ,μ∣e1−v∣;vγ=Mvγ/2Lp,aλ,μe1−v2;vγ≤Mφp,aλ,μvvγ.
Therefore, we get (39).Next, we obtain a Voronovskaja-type asymptotic formula.Theorem 8.
Ifh′′ exists at a point v∈0,∞ for h∈Cγ0,∞, then
(44)limp⟶∞pLp,aλ,μh≃v−hv=1+λ−μv+av1+vh′v+v22+vh″v.Proof.
From Taylor’s expansion ofv, we may write
(45)ht=hv+t−vh′v+12t−v2h″v+Rt,vt−v2,where Rt,v⟶0t⟶v. By operating Lp,aλ,μ, we obtain
(46)Lp,aλ,μh≃v−hv=Lp,aλ,μt−v;vh′v+Lp,aλ,μt−v2;vh″v2+Lp,aλ,μRt,vt−v2;v.
By the Cauchy-Schwarz inequality, we get(47)Lp,aλ,μRt,vt−v2;v≤Lp,aλ,μR2t,v;v1/2Lp,aλ,μt−v4;v1/2.
SinceLp,aλ,μh≃v⟶hv, we get
(48)limp⟶∞Lp,aλ,μR2t,v;v=R2v,v=0,limp⟶∞pLp,aλ,μRt,vt−v2;v=0.
Now, combining the above equations and using Corollary5, we get
(49)limp⟶∞pLp,aλ,μh≃v−hv=limp⟶∞pLp,aλ,μt−v;vh′v+limp⟶∞pLp,aλ,μt−v2;vh″v2+limp⟶∞pLp,aλ,μRt,vt−v2;v=1+λ−μv+av1+vh′v+v22+vh″v.LetBσ0,∞=h:0,∞⟶ℝhv≤Khσv,v≥0, where Kh is a constant which depends only on h, and
(50)∥h∥σ=supv∈0,∞∣hv∣σv.Also, letCσ0,∞=h∈Bσ0,∞:h be continuous on 0,∞, and
(51)Cσ00,∞=h∈Cσ0,∞:limv⟶∞hvσvexists,where σv=1+v2.The weighted modulus of continuity [14] is defined by
(52)Ωl,δ≔sup0<l≤δsupv∈0,∞∣hv+l−hv∣1+v+l2.Lemma 9 (see [14]).
Leth∈Cσ00,∞. Then,
(i)
Ωl,δ is a monotone increasing function of δ(ii)
Ωl,δ⟶0 as δ⟶0(iii)
Ωl,kδ≤kΩl,δ for each k∈ℕ(iv)
Ωl,αδ≤1+αΩl,δ for each α∈ℝ+Theorem 10.
Forh∈Cσ00,∞, we have
(53)supv∈0,∞∣Lp,aλ,μh;v−hv∣1+v25/2≤MΩl,1p,M>0.Proof.
By Lemma9, we have
(54)ht−hv≤1+v+∣t−v∣2Ωl,t−v≤21+v21+t−v21+t−vδΩl,δ.
OperatingLp,aλ,μ, we get
(55)Lp,aλ,μh;v−hv≤21+v2Ωl,δ1+Lp,aλ,μt−v2;v+Lp,aλ,μ1+t−v2v−tδ;v.
Using a second-order central moment, we get(56)Lp,aλ,μt−v2;v≤M11+v2p+μ≤M11+v2p,M1>0.
Applying the Cauchy-Schwarz inequality, we obtain(57)Lp,aλ,μ1+t−v2∣v−t∣δ;u≤1δLp,aλ,μt−v2;v1/2+1δLp,aλ,μt−v4;v1/2Lp,aλ,μt−v2;v1/2.
Again, using the central moment of order 4, we get(58)Lp,aλ,μt−v4;v1/2≤M21+v2p+μ≤M21+v2p,M2>0.
Combining the estimates (55)–(58) and choosing M=21+M1+M1+M2M1, δ=1/p, we get the required result.


## 4.q-Statistical Convergence

Defining aq-analog of the Cesàro matrix C1 is not unique (see [15, 16]). Here, we consider the q-Cesàro matrix, C1q=cnk1qkn,k=0∞, defined by
(59)cnk1qk=qkn+1q,ifk≤n,0,otherwise,which is regular for q≥1.LetK⊆ℕ (the set of natural numbers). Then, δK=limr1/r#k≤r:k∈K is called the asymptotic density of K, where # denotes the cardinality of the enclosed set. A sequence η=ηk is called statistically convergent to the number s if δKε=0 for each ε>0, where Kε=k≤r:∣ηk−s∣>ε (see [17]).Recently, Aktuğlu and Bekar [16] defined q-density and q-statistical convergence. The q-density is defined by
(60)δqK=δC1qK=liminfn⟶∞C1qχKn=liminfn⟶∞∑k∈Kqk−1n,q≥1.A sequenceη=ηk is said to be q-statistically convergent to the number L if δqKε=0, where Kε=k≤n:∣ηk−L∣≥ε for every ε>0. That is, for each ε>0,
(61)limn1n#k≤n:qk−1∣ηk−L∣≥ε=0,and we write Stq−limηk=L.IfδK=0 for an infinite set K, then δqK=0; hence, statistical convergence implies q-statistical convergence but not conversely (c.f. [16, Example 15]). Recently in [18], authors proved Korovkin’s type theorem via q-statistical convergence. Using the same technique we prove the following theorem.Theorem 11.
For allh∈Cρ0, we have
(62)Stq−limpLp,aλ,μh;v−hvσ=0,v∈0,∞.Proof.
It is sufficient to show thatStq−limp∥Lp,aλ,μei;v−ei∥σ=0, for i=0,1,2, where eiv=vi. It is clear that
(63)Stq−limpLp,aλ,μe0;v−e0σ=0.
By Corollary2 (ii), we have
(64)Lp,aλ,μt;v−vσ=supu∈0,∞11+v21p+μpv+av1+v+1+λ−v≤1p+μ−μ+1+λ+a.
Forε>0, define the sets:
(65)E1≔p∈ℕ:∥Lp,aλ,μe1;v−e1∥σ≥ε,E2≔p∈ℕ:1+λ−μ+ap+μ≥ε.
Then,(66)δqE2=limp⟶∞infC1qχE2p=limp⟶∞inf∑k∈E2qk−1p=0.
SinceE1⊆E2, we have δqE1≤δqE2. Hence,
(67)Stq−limpLp,aλ,μe1;v−e1σ=0.
Again, by Corollary2 (iii), we obtain
(68)Lp,aλ,μt2;v−v2σ≤supv∈0,∞11+v21p+μ2p+μ2+a21+v2−2aμ1+vv2+supv∈0,∞11+v22p+μ2p−μ−λμ+2+λa1+vv+1p+μ2λ2+2λ+2≤1p+μ2p+μ2+a2−2aμ+2p+μ2p−μ−λμ+2+λa+1p+μ2λ2+2λ+2.
Forε>0, define the sets:
(69)D1≔p∈ℕ:∥Lp,aλ,μe2;v−e2∥σ≥ε,D2≔p∈ℕ:1p+μ2p+μ2+a2−2aμ≥ε3,D3≔p∈ℕ:2p+μ2p−μ−λμ+2+λa≥ε3,D4≔p∈ℕ:1p+μ2λ2+2λ+2≥ε3.
Then,δqD2=0=δqD3=δqD4. Since D1⊆D2∪D3∪D4 which implies that δqD1≤δqD2+δqD3+δqD4,
(70)Stq−limpLp,aλ,μe2;v−e2ρ=0.
Hence, the proof is completed.Example 12.
Letη=ηp be defined by
(71)ηp=122ntimes022n−1timesn=0,1,2,⋯.
That is,1 occurs 22n times and 0 occurs 22n−1 times n=0,1,2,⋯, respectively. Let K=k∈ℕ:ηk=1. Then, limn⟶∞C1qχK22n−1=0, i.e., Stq−limηk=0, but δK does not exist, so η is not statistically convergent.
DefineApλ,μ=1+ηpLp,aλ,μ, where it is defined by (71). Then, obviously st−limp∥Apλ,μei;v−ei∥σ=0i=0,1,2. Applying the above theorem, we have
(72)Stq−limpApλ,μh;v−hσ=0forallh∈Cρ0.
On the other hand, sinceη=ηp is q-statistically convergent but neither convergent nor statistically convergent, the sequence Apλ,μ can not be convergent, while it is q-statistically convergent.


## 5. Graphical Analysis

In this section, we will give some numerical examples with illustrative graphics with the help of MATLAB.Example 13.
Letfx=x2+1, λ=3,μ=4,a=0.8, and p∈10,40,80. The convergence of the operator towards the function fx is shown in Figure 1.Example 14.
Letfx=x−1/2x−3/4, λ=2,μ=5,a=3, and p∈15,45,75. The convergence of the operator towards the function fx is shown in Figure 2.Figure 1
Convergence of the operator towards the functionfx=x2+1.Figure 2
Convergence of the operator towards the functionfx=x−1/2x−3/4.From these examples, we observe that the approximation of function by the operators becomes better when we take larger values ofn.Notice that forλ=μ=0, the operators (5) reduce to operators (4).Example 15.
Letfx=x2−4x+7. For a=3,p=30, comparison of convergence of the constructed operator (5) (green and pink) with the previously defined operator (4) (blue) is shown in Figure 3. From this figure, it is clear that the constructed operator gives a better approximation to fx than the previously defined operator.Figure 3
Comparison of convergence of the operator.



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