Citation for this paper:
Latif Braha, N., Mansour, T., & Srivastava, H. M. (2021). A Parametric
Generalization of the Baskakov-Schurer-Szász-Stancu Approximation Operators.
Symmetry, 13(6), 1-24. https://doi.org/10.3390/sym13060980.
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A Parametric Generalization of the Baskakov-Schurer-Szász-Stancu Approximation
Operators
Naim Latif Braha, Toufik Mansour, & Hari Mohan Srivastava
May 2021
© 2021 Naim Latif Braha et al. This is an open access article distributed under the terms of the Creative Commons Attribution License. https://creativecommons.org/licenses/by/4.0/
This article was originally published at:
https://doi.org/10.3390/sym13060980
Article
A Parametric Generalization of the
Baskakov-Schurer-Szász-Stancu Approximation Operators
Naim Latif Braha1 , Toufik Mansour2and Hari Mohan Srivastava3,4,5,6,*
Citation: Braha, N.L.; Mansour, T.; Srivastava, H.M. A Parametric Generalization of the Baskakov-Schurer-Szász-Stancu Approximation Operators. Symmetry 2021, 13, 980. https://doi.org/10.3390/sym13060980
Academic Editor: Carmen Violeta Muraru
Received: 28 April 2021 Accepted: 25 May 2021 Published: 31 May 2021
Publisher’s Note:MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil-iations.
Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
1 Department of Mathematics and Computer Sciences, University of Prishtina, Avenue “Mother Tereza” Nr. 5,
10000 Prishtinë, Kosova; nbraha@yahoo.com
2 Department of Mathematics, University of Haifa, Haifa 3498838, Israel; tmansour@univ.haifa.ac.il 3 Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada 4 Department of Medical Research, China Medical University Hospital, China Medical University,
Taichung 40402, Taiwan
5 Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street,
Baku AZ1007, Azerbaijan
6 Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy * Correspondence: harimsri@math.uvic.ca
Abstract:In this paper, we introduce and investigate a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators, which considerably extends the well-known class of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For this new class of approxi-mation operators, we present a Korovkin type theorem and a Grüss-Voronovskaya type theorem, and also study the rate of its convergence. Moreover, we derive several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we prove some shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and, as a special case, we deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.
Keywords:approximation operators; parametric generalization; Baskakov-Schurer-Szász-Stancu op-erators; Korovkin type theorem; Voronovskaya type theorem; rate of convergence; Grüss-Voronovskaya type theorem; shape-preserving properties
MSC:Primary 40C15, 40G10, 41A36; Secondary 40A35
1. Introduction
One of the most powerful theorems in the approximation theory is known as the Weierstrass Approximation Theorem, which states that any continuous function f(x)defined on the closed interval[a, b]can be approximated by an algebraic polynomial P(x)with real coefficients for each x∈ [a, b].
The idea of finding concrete algebraic functions for better approximation has been studied extensively, and a number of polynomial operators have been used directly. The first results are given for the Bernstein operators, which were generalized by Szász [1] as follows: Sn(f , x) =e−nx ∞
∑
j=0 (nx)j j! f j nfor x∈ [0,∞). Baskakov [2] defined the following sequence of linear operators: Ls(f , x) = 1 (1+x)s ∞
∑
r=0 s+r−1 r xr (1+x)rf x sfor s∈ Nand x ∈ [0,∞),Nbeing the set of positive integers. Subsequently, Schurer [3]
generalized the Bernstein operators in the following form: Lm,p(f , x) = m+p
∑
k=0 m+p k xk(1−x)m+p+k f k m . Stancu [4] defined the following sequence of operators:Sαs,β(f , x) = s
∑
r=0 s r xr(1−x)s−rf r+α s+βfor 05α5β. More recently, the following form of the Baskakov-Schurer-Szász-Stancu
operators was introduced by Sofyalioglu and Kanat [5]: Ms,pα,β(f ; x):= (s+p) ∞
∑
r=0 s+p+r−1 r xr (1+x)s+p+r · Z ∞ 0 s −(s+p)t [(s+p)t]r r! f (s+p)t+α s+p+β dt, where s is a positive integer, p is a non-negative integer, and 05α5β.2. The New Generalized Baskakov-Schurer-Szász-Stancu Operators
In this paper, we are interested in investigating a more generalized new class of operators, namely, the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. We define these operators as follows:
Mγn,p,α,β(f ; x) = (n+p) ∞
∑
k=0 bk,γn,p(x) Z ∞ 0 f (n+p)t+ α n+p+β skn,p(t)dt (1) with bk,γn,p(x) = " γx 1+x n+p+k−1 k − (1−γ)(1+x)n+p+k−3 k−2 + (1−γ)xn+p+k−1 k # xk−1 (1+x)n+p+k−1, where n, p, k∈ N, 05α5β, γ∈ Rand x∈ [0,∞).Remark 1. It is clearly seen that M1,α,βn,p (f ; x) =Mαn,p,β(f ; x).
The aims of this paper are to first study the Korovkin type theorem, the Grüss-Voronovskaya type theorem, and the rate of the convergence for the parametric general-ization of the Baskakov-Schurer-Szász-Stancu operators. We then present some results related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, in the last section, we give some preserving properties of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators such as convexity.
3. Preliminary Results
By simple applications of the principle of mathematical induction, one can obtain Lemmas1and2below:
Lemma 1. For all` =0, Z ∞ 0 t `sk n,p(t)dt= `!(k+`` ) (n+p)`+1.
Proof. We proceed to the proof by the principle of mathematical induction on N given by
N := ` +k=0. First of all, for N=0 (` =k=0), we have
Z ∞ 0 s 0 n,p(t)dt= Z ∞ 0 e −(n+p)tdt= 1 n+p, as claimed in Lemma1.
We now assume that the claimed result holds true for some N given by N := ` +k=0.
We then prove the claimed result for
N+1 := (` +1) +k. Indeed, by partial integration, we have
Z ∞ 0 t `+1sk n,p(t)dt= Z ∞ 0 t `+k+1e−(n+p)t (n+p)k k! dt = − Z ∞ 0 t `+k+1 d dt ( e−(n+p)t (n+p) k−1 k! ) dt = ` +k+1 n+p Z ∞ 0 t `+ke−(n+p)t (n+p)k k! dt. Thus, by the induction hypothesis, we have
Z ∞ 0 t `+1sk n,p(t)dt= (` +1)!(k+`+1`+1 ) (n+p)`+2 ,
which shows that the claimed result also holds true for N+1= ` +k+1. This evidently completes our proof of Lemma1by the principle of mathematical induction.
Lemma 2. For all m=0,
∞
∑
k=0 bk,γn,p(x) m ∏ j=1 (k+j) (n+p)m+1 = fm−gm+hm (n+p)m+1 , where fm, gmand hmare defined as follows:fm=
∑
k=0 γn+p+k−1 k m∏
j=1 (k+j) x k (1+x)n+p+k, gm =∑
k=2 (1−γ)n+p+k−3 k−2 m∏
j=1 (k+j) x k−1 (1+x)n+p+k−2and hm=
∑
k=0 (1−γ)n+p+k−1 k m∏
j=1 (k+j) x k (1+x)n+p+k−1, and they satisfy the following recurrence relations:fm+1= (m+1+x(n+p))fm+x(1+x) d dx{fm}, gm+1= (m+2+x(n+p−1))gm+x(1+x) d dx{gm} and hm+1 = (m+1+x(n+p−1))hm+x(1+x) d dx{hm}, with f0=γ, g0=x(1−γ)and h0= (1+x)(1−γ).
Proof. By using similar arguments as in the proof of Lemma1, we can establish the result
asserted by Lemma2. We choose to skip the details involved.
By means of Lemmas1and2, and, by using the principle of mathematical induction on m, we are led to the following result.
Proposition 1. For all m=0,
fm=
∑
k=0 γn+p+k−1 k m∏
j=1 (k+j) x k (1+x)n+p+k =m! γ m∑
j=0 m j n+p−1+j j xj, gm =∑
k=2 (1−γ)n+p+k−3 k−2 m∏
j=1 (k+j) x k−1 (1+x)n+p+k−2 = (1−γ)m! m∑
j=0 m+2 j+2 n+p−1+j j xj+1 and hm =∑
k=0 (1−γ)n+p+k−1 k m∏
j=1 (k+j) x k (1+x)n+p+k−1 = (1−γ)(1+x)m! m∑
j=0 m j n+p−1+j j xj.Furthermore, for all m=0,
∞
∑
k=0 bk,γn,p(x) m ∏ j=1 (k+j) (n+p)m+1 = m! ∑m j=0 " (mj) + (1−γ) (mj) − (m+2j+2) ! x # (n+p−1+jj )xj (n+p)m+1 .By (1), we have Mn,pγ,α,β(t`; x) = (n+p) ∞
∑
k=0 bk,γn,p(x) Z ∞ 0 ((n+p)t+α)` (n+p+β)` s k n,p(t)dt = `∑
j=0 (n+p)j+1α`−j(`j) (n+p+β)` ∞∑
k=0 bk,γn,p(x) Z ∞ 0 t jsk n,p(t)dt.Thus, by Lemma1, we obtain Mγn,p,α,β(t`; x) = `
∑
j=0 α`−j (n+p+β)` ` j ∞∑
k=0 bk,γn,p(x) k+j j j!,which, by Lemma2, implies that Mn,pγ,α,β(t`; x) = `
∑
j=0 α`−j (n+p+β)` ` j (fj−gj+hj).Hence, by applying the above Proposition, we can prove the following result.
Theorem 1. For all` =0,
Mγn,p,α,β(e`; x) = ` ∑ j=0 α`−jj!(`j) j ∑ i=0 (ji) + (1−γ) h (ji) − (j+2i+2)ix(n+p−1+ii )xi (n+p+β)` .
For instance, Theorem1for` =0, 1, 2 gives the following moments: 1. Mγn,p,α,β(e0; x) =1, 2. Mγn,p,α,β(e1; x) = α+1 n+p+β+ n+p+2γ−2 n+p+β x, 3. Mγn,p,α,β(e2; x) = α2+2α+2 (n+p+β)2 + 2[(n+p+2γ)α+2n+2p−2α+5γ−5] (n+p+β)2 x +(n+p)(n+4γ+p−3) (n+p+β)2 x 2.
In what follows, we will prove the Korovkin type theorem for the parametric gener-alization of the Baskakov-Schurer-Szász-Stancu operators. In the last several years, this subject is widely studied, and it is treated, among others, in the following references (see, for example, Refs. [6–20]). Some other related recent developments on this subject can be found in [21–24].
Theorem 2. Let(Mn,pγ,α,β)be a sequence of positive linear operators defined on C[0, R]for any finite R such that, for every i∈ {0, 1, 2},
lim n→∞ Mγ ,α,β n,p (ei; x) −ei =0, (2)
where ei =xi. Then, for every f ∈C[0, R], lim n→∞ Mγ ,α,β n,p (f ; x) − f =0. (3)
Proof. From Theorem1, we have
Mγn,p,α,β(e1; x) −e1 = α+1 n+p+β+ n+p+2γ−2 n+p+β x−x =0 and Mγ ,α,β n,p (e2; x) −e2 = α2+2α+2 (n+p+β)2+ 2((n+p+2γ)α+2n+2p−2α+5γ−5) (n+p+β)2 x +(n+p)(n+4γ+p−3) (n+p+β)2 x 2−x2 =0.
By means of the basic form of the Korovkin type theorem (see, for example, Ref. [25]), we complete the proof of Theorem2.
Lemma 3. For all` =0,
Mγn,p,α,β (y−x)`; x= `
∑
j=0 ` j (−x)`−jMn,pγ,α,β(ej; x).Proof. Lemma3follows immediately from (1).
Example 1. By Theorem1and Lemma3for` =0, 1, 2, we obtain
Mγn,p,α,β (y−x)0; x=1, Mγn,p,α,β (y−x)1; x= α +1 n+p+β+ 2(γ−1) −β n+p+β x and Mγn,p,α,β (y−x)2; x= α 2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x + β 2−4βγ+n+p+4β (n+p+β) x 2.
Moreover, if we consider Lemma3for` 56 and n→∞, we obtain lim n7→∞nM γ,α,β n,p (y−x)1; x=α+1+ (2γ−2−β)x, lim n7→∞nM γ,α,β n,p (y−x)2; x=2x+x2, lim n7→∞n 2Mγ,α,β n,p (y−x)3; x=6(α+2)x+3(α−2β+4γ−1)x2+ (6γ−3β−4)x3, lim n7→∞n 2 Mγ,α,β n,p (y−x)4; x=12x2+12x3+3x4, lim n7→∞n 3Mγ,α,β n,p (y−x)5; x =60(α+3)x2+60(α−β+2γ+2)x3 +5(3α−12β+24γ−1)x4+5(6γ−3β−2)x5 and lim n7→∞n 3 Mγ,α,β n,p (y−x)6; x=120x3+180x4+90x5+15x6.
4. Direct Estimates
With B[0,∞), C[0,∞), and CB([0,∞)), we will denote the space of all bounded func-tions, the space of all continuous funcfunc-tions, and the space of all continuous and bounded functions defined in the interval[0,∞), respectively, endowed with the norm given by
kfk = sup x∈[0,∞)
|f(x)|.
The modulus of continuity of the function f ∈C[0,∞)is defined by
ω(f ; δ):=sup{|f(x) −f(y)|: x, y∈ [0,∞) and |x−y| 5δ}.
It is known that, for any value of the|x−y|, we have
|f(x) − f(y)| 5ω(f ; δ) | x−y| δ +1 .
Theorem 3. Let f ∈CB[0,∞). Then, the following inequality for the operators(1)holds true: Mγ ,α,β n,p f −f 5ω(f ; √ n) 1+n√+p n " 1−(1−α) (n+p+1)x+1 (1+x)n+p #12 · " 2α+1+ (8αγ−8α−2β+14γ−14)x−8β(γ−1)x2 4(n+p)(n+p+β)2 +4α 2+ (−8αβ+4α+3)x+ (4β2−4β+2)x2 4(n+p+β)2 +2x(4xβ−x−4α)(n+p)2+ x 2(n+p)2 (n+p+β)2 #12! .
Proof. We know that operators Mγn,p,α,βare linear and positive. Let f ∈CB[0,∞). In view of the modulus of continuity, we have
Mγ ,α,β n,p (f ; x) −f(x) 5 (n+p) ∞
∑
k=0 bk,γn,p(x) Z∞ 0 f (n+p)t+α n+p+β −f(x) skn,p(t)dt 5ω(f ; δ) 1+1 δ(n+p) ∞∑
k=0 bk,γn,p(x) Z∞ 0 (n+p)t+α n+p+β −x skn,p(t)dt ! . (4) Let us set B := ∞∑
k=0 bn,pk,γ(x) Z ∞ 0 (n+p)t+α n+p+β −x skn,p(t)dt. Then, by the Cauchy-Schwarz inequality, we getB5 "∞
∑
k=0 bk,γn,p(x) #12 · "∞∑
k=0 bk,γn,p(x) Z ∞ 0 (n+p)t+α n+p+β −x 2 ·sk n,p(t) 2 dt #12 . (5) By direct calculations, we see that∑
k=0bk,γn,p(x) =1−
(1−γ)[(n+p+1)x+1]
and also that A0= Z ∞ 0 1·s k n,p(t) 2 dt= ( 2k k) 22k+1(n+p), A1= Z ∞ 0 t·s k n,p(t) 2 dt= (2k+1)( 2k k) 22k+2(n+p)2 and A2= Z ∞ 0 t 2·sk n,p(t) 2 dt= (k+1)(2k+1)( 2k k) 22k+2(n+p)3 . These last three equalities lead us to the following consequence: Z ∞ 0 (n+p)t+α n+p+β −x 2 ·sk n,p(t) 2 dt = (n+p) 2 (n+p+β)2A2+2 α n+p+β −x n+p n+p+βA1+ α n+p+β−x 2 A0. Hence, in view of the positivity of bn,pk,α(x), if we use the following expression:
Z ∞ 0 (n+p)t+α n+p+β −x 2 ·sk n,p(t) 2 dt together with the fact the
2k k 522k, we obtain ∞
∑
k=0 bk,αn,p(x) Z ∞ 0 (n+p)t+α n+p+β −x 2 ·sk n,p(t) 2 dt5U, where U= 2α+1+ (8αγ−8α−2β+14γ−14)x−8β(γ−1)x 2 4(n+p)(n+p+β)2 +4α 2+ (−8αβ+4α+3)x+ (4β2−4β+2)x2 4(n+p+β)2 +2x(4xβ−x−4α)(n+p)2+ x 2(n+p)2 (n+p+β)2From (6) and (5), we find that B5 s 1−(1−α)((n+p+1)x+1 (1+x)n+p U. Putting δ=√n, we get the result asserted by Theorem3.
In what follows, we will give an upper bound for the sequence of the parametric generalization of the Baskakov-Schurer-Szász operators.
Theorem 4. For any f ∈CB[0,∞),
Mn,pγ,α,β(f ; x) 5 kfkC.
Proof. From the definition of the parametric generalization of the
Baskakov-Schurer-Szász-Stancu operators in (1), we have Mγn,p,α,β(f ; x) 5 sup t∈R+ |f(t)| · (n+p) ∞
∑
k=0 bn,pk,γ(x) Z ∞ 0 s k n,p(t)dt = sup t∈R+ |f(t)| ·Mγn,p,α,β(e0; x) = kfkC, as asserted by Theorem4.For f ∈C[0,∞)and δ>0, the second-order modulus of smoothness of f is defined as follows: w2(f , √ δ):= sup 0<h5√δ sup x,x+h∈[0,∞) {|f(x+2h) −2 f(x) +f(x−h)|}.
The Peetre’s K-functional is defined by
K2(f , δ) =infkf −gkC[0,∞)+δkg00kC[0,∞): g∈W2 ,
where δ>0 and
W2= {g∈C[0,∞): g0, g00∈C[0,∞)}.
It is known that there exists a positive constantC >0 such that (see [26] (Theorem 3.1.2)),
K2(f , δ) 5 Cw2(f ,
√
δ) (δ>0).
Theorem 5. Let f ∈C[0, A]for any finite real number A. Then, Mγ ,α,β n,p (f ; x) − f(x) 5 2 Akfkb 2+3 4(A+b 2+2) ω2(f ; b), where b= 4 q Mγn,p,α,β (s−x)2; x.
Proof. Let fS be the Steklov function of the second order for the function f(x).
Know-ing that
Mγn,p,α,β(e0; x) =1, which follows from Theorem1, we have
Mγn,p,α,β(f ; x) −f(x) 5 Mγn,p,α,β(f−fS; x) + |M γ,α,β n,p (fS; x) −fS(x)| + |fS(x) −f(x)| 52kfS−fk + M γ,α,β n,p (fS; x) −fS(x) . (7)
Now, from the Lemmas in [27], we find that Mγ ,α,β n,p (f ; x) −f(x) 5 3 2ω2(f ; b) + Mγ ,α,β n,p (fS; x) − fS(x) . (8)
Knowing that fS∈C2[0, A], and from the Lemmas in [17], we obtain Mγ ,α,β n,p (fS; x) −fS(x) 5 fS0 q Mn,pγ,α,β (s−x)2; x+1 2 fS00 Mγ ,α,β n,p (s−x)2; x.
The following inequality is valid (see [27]):
kfS00k 5 3
2b2ω2(f ; b). (9)
In light of (8) and (9), (7) takes the following form: Mγ ,α,β n,p (fS; x) −fS(x) 5 fS0 q Mγn,p,α,β (s−x)2; x + 3 4b2ω2(f ; b)M γ,α,β n,p (s−x)2; x. From the relation (9) and the Landau inequality (see [28]), we get
kfS0k 5 2
Akfk + 3A
4b2ω2(f ; b). (10)
Using relations (9) and (10), and upon setting b= 4 q Mγn,p,α,β (s−x)2; x, we obtain Mγ ,α,β n,p (fS; x) − fS(x) 5 2 Akfkb 2+3 4(A+b 2) ω2(f ; b). Now, from relation (8), we complete the proof of Theorem5. Let
CB2[0,∞) = {f ∈CB[0,∞): f0, f00∈CB[0,∞)} with the norm given by
kfkC2
B = kfk∞+ kf
0k
∞+ kf00k∞
and the Peetre’s K-functional given by (see [29])
K(f ; δ) = {kf−gk∞+δkgkC2 B}.
Theorem 6. Let f ∈CB[0,∞). Then, the following inequality holds true: Mγ ,α,β n,p (f ; x) −f(x) 5A(n, p, α, β, γ, x)kfkC2 B,
for every x=0, where A(n, p, α, β, γ, x) = α+1 n+p+β+ 2(γ−1) −β n+p+β x + " α2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x +β 2−4βγ+n+p+4β (n+p+β) x 2 # .
Proof. By using the Taylor formula and the linearity of the operators Mγn,p,α,β(f ; x), we obtain Mn,pγ,α,β(f ; x) − f(x) =Mn,pγ,α,β (t−x); x f0(x) + 1
2M
γ,α,β
where ϕ∈ (x, t). In addition, from the above Example, we have Mγn,p,α,β(f ; x) −f(x) = kf0k · α+1 n+p+β +2(γ−1) −β n+p+β x +kf 00k 2 " α2+2α+2 (n+p+β)2 +2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x +β 2−4βγ+n+p+4β (n+p+β) x 2 # 5A(n, p, α, β, γ, x)kfkC2 B, where A(n, p, α, β, γ, x) = α+1 n+p+β+ 2(γ−1) −β n+p+β x + " α2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x +β 2−4βγ+n+p+4β (n+p+β) x 2 # , which proves Theorem6.
Theorem 7. Let f ∈C[0,∞). Then, Mγ ,α,β n,p (f ; x) − f(x) 52M " ω2 f ; r 1 2 A(n, p, α, β, γ, x) ! +min 1,1 2A(n, p, α, β, γ, x) kfk∞ # , whereMis a positive constant and A(n, p, α, β, γ, x)is defined as in Theorem6.
Proof. From the linearity of the operator Mγn,p,α,β(f ; x)and the following relation: f(t) − f(x) = f(t) −g(t) +g(t) −g(x) +g(x) −g(t), we obtain Mγn,p,α,β(f ; x) − f(x) 5 Mn,pγ,α,β(f−g; x) − f(x) + Mγ ,α,β n,p (g; x) −g(x) + |f(x) −g(x)|.
Now, from Theorems4and6, and, by considering that g∈C2B, we get Mγn,p,α,β(f ; x) − f(x) 52kf−gk +A(n, p, α, β, γ, x)kgkC2 B =2K f ;1 2A(n, p, α, β, γ, x) . It is known that K(f ; δ) 5 Chω2(f ; √ δ) +min{1, δ}kfk∞ i ,
whereC is a positive constant, holds true for every δ > 0 (see [26]). From the last two relations, we get the result asserted by Theorem7.
We will give the Voronovskaya type theorem for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators.
Theorem 8. For f ∈CB[0,∞), the following limit relation: lim n→∞n M γ,α,β n,p f(t); x −f(x) = f0(x)[1+α+ (2γ−2−β)x] + f 00(x) 2 2x+x2, holds true for every x∈ [0, M]and any finite M.
Proof. By Taylor’s expansion theorem of the function f in CB[0,∞), we obtain: f(t) = f(x) + (t−x)f0(x) +1 2(t−x) 2 f00 (x) + (t−x)2ψx(t), where ψx(t) = f(t) − f(x) − (t−x)f0(x) −12(t−x)2 f00(x) (t−x)2 (x6=t) 0 (x=t)
and the function ψx(·)is the Peano form of the remainder, ψx(·) ∈CB[0,∞)and ψx(t) →0 as t→x. Applying the operator Mγn,p,α,βon both sides of the above relation, we find that
nMn,pγ,α,β f(t); x −f(x) = f0(x)nMγn,p,α,β (t−x); x + f 00(x) 2 nM γ,α,β n,p (t−x)2; x +nMn,pγ,α,β (t−x)2ψx(t); x. In addition, from the above Example, we get
lim n→∞n M γ,α,β n,p f(t); x −f(x) = f0(x)[1+α+ (2γ−2−β)x] + f 00(x) 2 2x+x2+ lim n→∞nM γ,α,β n,p (t−x)2ψx(s); x, which, after applying the Cauchy-Schwarz inequality, yields
nMγn,p,α,β (t−x)2ψx(s); x5n2Mn,pγ,α,β (t−x)4; x 1 2 Mγ,α,β n,p ψ2x(t); x 1 2.
We now observe that ψ2x(t) →0 as t→x and ψx2(·) ∈CB[0,∞). Thus, from Theorem2, it follows that
Mγ,α,β
n,p ψ2x(t); x 1 2 →0
as n→∞. Then, by using the last relations for every x∈ [0, M], we get lim n→∞n M γ,α,β n,p f(t); x−f(x)= f0(x)[1+α+ (2γ−2−β)x] + f 00(x) 2 2x+x2. This completes the proof of Theorem8.
In what follows, we will give the Grüss-Voronovskaya type theorem (see [30]) for the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators.
Theorem 9. Let f0, f00, g0, g00∈CB[0,∞). Then, lim n→∞n Mγn,p,α,β(f g; x) −Mn,pγ,α,β(f , x)Mn,pγ,α,β(g; x) = (2x+x2)f0(x)g0(x),
for each x∈ [0, M], where M is finite.
Proof. After some calculations, we obtain
nhMn,pγ,α,β(f g; x) −Mn,pγ,α,β(f ; x)Mγn,p,α,β(g, x) i = nMn,pγ,α,β(f g; x) − f g) − [1+α+ (2γ−2−β)x](f g)0(x) − (2x+x2)(f g) 00(x) 2 −g(x) nMγn,p,α,β(f , x) − f(x) − [1+α+ (2γ−2−β)x]f0(x) − (2x+x2)f 00(x) 2 −Mn,pγ,α,β(f ; x) " nMγn,p,α,β(g; x) −g(x) − [1+α+ (2γ−2−β)x]g0(x) − (2x+x2) g 00(x) 2 # +2x+x2f0(x)g0(x) +2x+x2 g 00(x) 2 [f(x) −M γ,α,β n,p (f ; x)] + [1+α+ (2γ−2−β)x]g0(x)[f(x) −Mγn,p,α,β(f ; x)].
The proof of Theorem9now follows from Theorem8and the above Example. The following results give light to the speed of the change between the difference of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators and their derivatives, measured in terms of the modulus of continuity.
Theorem 10. Let h, h0, h00∈C[0,∞). Then, (n+p+β) Mn,pγ,α,β(h; x) −h(x)−h0(x)[α+1+ (2(γ−1) −β)x] −h 00(x) 2 α2+2α+2 n+p+β + 2(n+p−αβ+2αγ−2α−β+5γ−5) n+p+β x + (β2−4βγ+n+p+4β)x2 ! =O(1) ·ω h00;√1 n (n→∞)
for every x∈ [0, M]for any finite M.
Proof. From the Taylor’s theorem, we have
h(u) =h(x) +h0(x)(u−x) +h 00(x) 2 (u−x) 2+R(u, x), where R(u, x) = h 00( θ) −h00(x) 2 (u−x) 2 for θ∈ (u, x). We thus find that
M γ,α,β n,p (h; x) −h(x) −h0(x)Mγn,p,α,β(u−x; x) − h00(x) 2 M γ,α,β n,p (u−x)2; x 5Mγn,p,α,β(|R(u, x)|; x).
From this last relation, we get (n+p+β) Mn,pγ,α,β(h; x) −h(x)−h0(x)α+1+ 2(γ−1) −βx −h 00(x) 2 " α2+2α+2 n+p+β + 2(n+p−αβ+2αγ−2α−β+5γ−5) n+p+β x + (β2−4βγ+n+p+4β)x2 # 5 (n+p+β)Mn,pγ,α,β |R(u, x)|; x.
By the properties of the modulus of continuity, we have h00(θ) −h00(x) 2! 5 1 2! 1+|θ−x| δ ω(h00; δ).
On the other hand, it is easily seen that
h00(θ) −h00(x) 2! 5 ω(h00; δ) (|u−x| 5δ) (t−x)4 δ4 ω(h 00; δ) (|u−x| = δ).
For 0<δ<1, we obtain that
h00(θ) −h00(x) 2! 5ω (h00; δ) 1+ (u−x) 4 δ4 , which yields |R(u, x)| 5ω(h00; δ) 1+ (u−x) 4 δ4 (u−x)2=ω(h00; δ) (u−x)2+(u−x) 6 δ4 . By the linearity of Mγn,p,α,βand the above relation, we obtain
Mγ,α,β n,p (|R(u, x)|; x) 5ω(h00; δ) Mγ,α,β n,p (u−x)2; x+ 1 δ4M γ,α,β n,p (u−x)6; x . Now, in view of the above Example, for every x∈ [0, M], we have
Mγn,p,α,β(|R(u, x)|; x) 5ω(h00; δ) " O 1 n + 1 δ4 O 1 n3 # =O 1 n ω(h00; δ). Thus, for δ= √1 n, we complete the proof of Theorem10.
The next result gives an estimation of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the special Lipschitz-type space Lip∗Mα([1]), defined
as follows: Lip∗M(α):= f ∈CB[0,∞):|f(s) − f(x)| 5 M |s−x|α (x+s)α2, x ∈ (0,∞) and s∈ (0,∞) , whereMis a positive constant and α∈ (0, 1].
Theorem 11. Let f ∈Lip∗M(α). Then, for all x, t∈ (0,∞), n∈ Nand α∈ (0, 1], Mγ ,α,β n,p (f ; x) − f(x) 5 M (x+t)α2 α2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x + β 2−4βγ+n+p+4β (n+p+β) x 2 !α2 , whereMis a positive constant.
Proof. Let f ∈ Lip∗M(α) and α ∈ (0, 1]. We will distinguish between the following
two cases. I. For α=1, we have Mn,pγ,α,β f(t); x−f(x) 5 Mγn,p,α,β |f(t) −f(x)|; x 5 M ·Mγn,p,α,β |t−x| (x+t)12 ; x ! 5 M (x+t)12 Mn,pγ,α,β(|t−x|; x)
for a positive constantM.
If we apply the Cauchy-Schwarz inequality in the last expression, we get Mγ ,α,β n,p f(t); x− f(x) 5 M (x+t)12 Mγn,p,α,β(|t−x|; x) 5 M (x+t)12 q Mn,pγ,α,β (t−x)2; x = M (x+t)12 α2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x + β 2−4βγ+n+p+4β (n+p+β) x 2 !12 .
II. For α∈ (0, 1), we have Mγ ,α,β n,p f(t); x−f(x) 5 Mγn,p,α,β |f(t) − f(x)|; x 5 M ·Mn,pγ,α,β |t−x|α (x+t)α2 ; x ! 5 M (x+t)α2 Mn,pγ,α,β(|t−x|α; x).
If we apply the Hölder inequality on the last relation under the conditions that p= 1 α, q= 1 1−α, we obtain Mγn,p,α,β f(t); x−f(x) 5 M (x+t)α2 h Mn,pγ,α,β(|t−x|; x) iα
for a positive constantM. Thus, after applying the Cauchy-Schwarz inequality, obtain the following estimate: Mγn,p,α,β f(t); x−f(x) 5 M (x+t)α2 q Mγn,p,α,β (t−x)2; x α = M (x+t)α2 ( α2+2α+2 (n+p+β)2+ 2(n+p−αβ+2αγ−2α−β+5γ−5) (n+p+β)2 x + β 2−4βγ+n+p+4β (n+p+β) x 2 )α2 , which completes the proof of Theorem11.
The Ditzian-Totik uniform modulus of smoothness of the first and the second orders are defined as follows (see [26]):
ωγ(f ; δ):= sup 0<|h|5δ sup x,x+hγ(x)∈[0,∞) f x+hγ(x) − f(x) and ω2φ(f ; δ):= sup 0<|h|5δ sup x,x±hφ(x)∈[0,∞) f x+hφ(x) −2 f(x) +f x−hφ(x) ,
respectively, where φ is an admissible step-weight function on[a, b], that is,
φ(x) = [(x−a)(b−x)]1/2
if x∈ [a, b]. The corresponding K-functional is defined as follows: K2,φ(x)(f , δ) = inf g∈W2(φ) kf−gkC[0,∞)+δkφ2g00kC[0,∞) , where δ>0, W2(φ) = {g∈CB[0,∞): g0 ∈AC[0,∞), φ2g00∈CB[0,∞)} and g0∈ AC[0,∞) means that g0is absolutely continuous on[0,∞). It is known that there exists an absolute constant C>0 such that (see [26])
C−1ω2φ(f ;
√
δ) 5K2,φ(x)(f , δ) 5C ω2φ(f ;
√
Theorem 12. LetΦ= px(1−x) (x ∈ [0, 1])be a step-weight function of the Ditzian-Totik modulus of smoothness. Then, for any f ∈CB[0, 1]and x∈ [0, 1], n∈ Nand 2γ<β+2,
Mγ ,α,β n,p (f ; x) − f(x) 54K2,Φ(x) f , Mn,pγ,α,β (s−x)2; x+α1(n, p, α, β) 4Φ2(x) ! +ωγ f ;α1(n, p, α, β) γ(x) , where α1(n, p, α, β) = α+1 n+p+β+ 2(γ−1) −β n+p+β . Proof. Let Mγn,p,α,β,∗(f ; x) =Mn,pγ,α,β(f ; x) +f(x) − f x+β1(n, p, α, β, x), where β1(n, p, α, β, x) = α+1 n+p+β+ 2(γ−1) −β n+p+β x.
We then observe that
Mn,pγ,α,β,∗(1; x) =1 and Mn,pγ,α,β,∗ (s−x); x=0. Let g∈W2(φ). Then, by using Taylor’s expansion, we write
g(s) =g(x) +g0(x)(s−x) + Z s
x(s−u)g
00(u)du s∈ [0,∞),
which implies that
Mγn,p,α,β,∗(g; x) −g(x) =Mγn,p,α,β Z s x(s−u)g 00(u)du; x − Z x+β1(n,p,α,β,x) x [x+β1(n, p, α, β, x) −u]g00(u)du. Therefore, we have Mγ ,α,β,∗ n,p (g; x) −g(x) 5 Mγn,p,α,β Z s x (s−u)g00(u)du ; x + Z x+β1(n,p,α,β,x) x |x+β1(n, p, α, β, x) −u| · |g 00 (u)|du 5 φ2g00(x)Mγn,p,α,β Z s x |s−u| φ2(u) du ; x + φ2g00(x) · Z x+β1(n,p,α,β,x) x |x+β1(n, p, α, β, x) −u| φ2(u) du .
Let u = ρx+ (1−ρ)s (ρ ∈ [0, 1]). Since φ2 is concave on [0,∞), it follows that φ2(u) ≥ρφ2(x) + (1−ρ)φ2(s)and hence |s−u| φ2(u) = ρ|x−s| φ2(u) 5 ρ|x−s| ρφ2(x) + (1−ρ)φ2(s) 5 |x−s| φ2(x).
We thus obtain Mγ ,α,β,∗ n,p (g) −g 5 kφ2g00kC[0,∞) φ2(x) Mγ,α,β n,p (s−x)2; x+xβ1(n, p, α, β, x) . From the above relations, we obtain
Mγn,p,α,β,∗(f , x) −f(x) 5 Mγn,p,α,β,∗(f−g) + Mγn,p,α,β,∗(g) −g + kf−gk + f x+β1(n, p, α, β, x)−f(x) 54kf−gk + φ2g00 φ2(x) M γ,α,β n,p (s−x)2; x +xβ1(n, p, α, β, x) + f(x+β1(n, p, α, β, x)) −f(x) . We know that kf(x+β1(n, p, α, β, x)) −f(x)k 5 f x+γ(x) Mγn,p,α,β (s−x); x γ(x) ! − f(x) 5ωγ f ;β1(n, p, α, β, x) γ(x) . Therefore, we have Mn,pγ,α,β(f , x) −f(x) 54K2,Φ(x) f , Mγ,α,β n,p (s−x)2; x+xβ1(n, p, α, β, x) 4Φ2(x) ! +ωγ f ;β1(n, p, α, β, x) γ(x) . (12)
From the conditions given in Theorem12, the properties of the K-functional, and the modulus of continuity, we get
1. α+1 n+p+β + 2(γ−1) −β n+p+β x5 α+1 n+p+β+ 2(γ−1) −β n+p+β , 2. K2,Φ(x) f ,M γ,α,β n,p (s−x)2; x+xβ1(n, p, α, β, x) 4Φ2(x) ! 5K2,Φ(x) f ,M γ,α,β n,p (s−x)2; x+α1(n, p, α, β) 4Φ2(x) ! , 3. ωγ f ;β1(n, p, α, β, x) γ(x) 5ωγ f ;α1(n, p, α, β) γ(x)
for every x∈ [0, 1]. Combining the relation (12) and the other preceding relations, we obtain Mγ,α,β n,p (f ; x) − f(x) 54K2,Φ(x) f , Mn,pγ,α,β (s−x)2; x+α1(n, p, α, β) 4Φ2(x) ! +ωγ f ;α1(n, p, α, β) γ(x) , as asserted by Theorem12.
5. Weighted Approximation
Let ρ(x) =x2+1 be the weight function and letMf be a positive constant. We define the weighted space of functions as follows:
(i) Bρ[0,∞)is the space of functions f defined on[0,∞)and satisfying
|f(x)| 5 Mfρ(x).
(ii) Cρ[0,∞)is the subspace of all continuous functions in Bρ[0,∞).
(iii) Cρ∗[0,∞)is the subspace of functions f ∈ Cρ[0,∞)for which
f (x)
ρ(x) is convergent as
x→∞.
We note that the space Bρ[0,∞)is a normed linear space with the norm given by
kfkρ=sup
x=0
|f(x)|
ρ(x) .
In order to calculate the rate of convergence, we consider the weighted modulus of continuityΩ(f ; δ)defined on infinite interval[0,∞)as
Ω(f ; δ) = sup x=0; 0<|h|5δ |f(x+h) − f(x)| (1+h2)ρ(x) ∀ f ∈C ∗ ρ[0,∞).
For any µ∈ [0,∞), the weighted modulus of continuityΩ(f ; δ)verifies the follow-ing inequality:
Ω(f ; µδ) 52(1+µ)(1+δ2)Ω(f ; δ),
and, for every f ∈C∗ρ[0,∞), we get
|f(t) − f(x)| 52 | t−x| δ +1 (1+δ2)Ω(f ; δ)(1+x2)(1+ (t−x)2).
Theorem 13. Let ρ(x)be a weight function on[0,∞). Then, for each function f ∈C∗ρ[0,∞), lim n→∞ Mγ ,α,β n,p (f ; x) −f(x) ρ=0.
Proof. It suffices to check that Mγn,p,α,β(ei; x)converges uniformly to ei, for i∈ {0, 1, 2}, as n tends to∞ and applies the well-known weighted Korovkin type theorem, where ei(x) =xi. The uniform convergence arises from the fact that
lim n Mγ ,α,β n,p ei−ei ρ=0 (i=0, 1, 2).
Using Theorem1, the result for i=0 is trivial.
We now prove that the results are true for i=1 and i=2, respectively. Indeed, for f ∈Cρ∗[0,∞), we obtain Mγ ,α,β n,p e1−e1 ρ=sup x=0 |Mγ,α,β n,p e1−e1| ρ(x) 5sup x=0 |γ1(n, p, α, β, x)| ρ(x) .
By a similar consideration, we have Mγ ,α,β n,p e2−e2 ρ=sup x=0 ( Mγ ,α,β n,p e2−e2 ρ(x) ) 5sup x=0 | γ2(n, p, α, β, x)| ρ(x) ,
where γ1(n, p, α, β, x) = α+1 n+p+β+ n+p+2γ−2 n+p+β x−x and γ2(n, p, α, β, x) = α2+2α+2 (n+p+β)2+ 2((n+p+2γ)α+2n+2p−2α+5γ−5) (n+p+β)2 x +(n+p)(n+4γ+p−3) (n+p+β)2 x 2−x2.
We thus conclude that lim n→∞ Mγ ,α,β n,p ei−ei ρ=0 (i=0, 1, 2),
which completes the proof of Theorem13.
Theorem 14. Let f ∈Cρ∗[0,∞). Then, the following inequality holds true:
sup x∈[0,∞) Mγ ,α,β n,p (f ; x) −f(x) (1+x2)(1+ Cx+ Dx2+ Ex3+ Fx4) 5 KΩ f ; n−14
for a sufficiently large n, whereC,D,E andF are positive constants dependent only on n, p, α, β and γ, andKis a positive constant.
Proof. For x∈ [0,∞), we have Mγn,p,α,β(f ; x) − f(x) = (n+p) ∞
∑
k=0 bn,pk,γ(x) Z ∞ 0 f (n+p)t+α n+p+β − f(x) s k n,p(t)dt.Using the properties of the weighted modulus, we obtain Mγ ,α,β n,p (f ; x) − f(x) 5 (n+p) ∞
∑
k=0 bk,γn,p(x)2(1+δn2)Ω(f ; δn)(1+x2) · Z ∞ 0 (n+p)t+α n+p+β −x δn +1 " 1+ (n+p)t+ α n+p+β −x 2# skn,p(t)dt. Let us define S(t, x) = (n+p)t+α n+p+β −x δn +1 " 1+ (n+p)t+α n+p+β −x 2# skn,p(t).Since skn,p(t) >0 for every t∈ (0,∞), we have
S(t, x) 5 2(1+δ2n)skn,p(t) (n+p)t+α n+p+β −x 5 δn 2(1+δ2n) (n+p)t+α n+p+β −x 4 δ4n s k n,p(t) (n+p)t+α n+p+β −x = δn ,
which implies that S(t, x) 52(1+δ2n) 1+ (n+p)t+α n+p+β −x 4 δ4n skn,p(t).
Thus, clearly, we get Mγn,p,α,β(f ; x) −f(x) 54(n+p) ∞
∑
k=0 bk,γn,p(x)(1+δ2n)Ω(f ; δn)(1+x2) · Z ∞ 0 1+δn2 1+ (n+p)t+α n+p+β −x 4 δn4 skn,p(t)dt. By Lemma1, we have Z ∞ 0 1+ (n+p)t+α n+p+β −x 4 δn4 s k n,p(t)dt = Z ∞ 0 s k n,p(t)dt+ 1 δn4(n+p+β)4 4∑
j=0 4 j (n+p)j · (α− (n+p+β)x)4−j Z ∞ 0 t jsk n,p(t)dt = 1 n+p+ 1 δ4n(n+p+β)4(n+p) 4∑
j=0 (α− (n+p+β)x)4−j j!4 j k+j j . Thus, by using the above Proposition, we have∞
∑
k=0 bn,pk,α(x) Z ∞ 0 1+ (n+p)t+α n+p+β −x 4 δn4 skn,p(t)dt = 1 n+p + 4∑
j=0 j∑
i=0 24(α− (n+p+β)x)4−j (4−j)! δ4 n(n+p+β)4(n+p) · j i + (1−γ) j i − j+2 i+2 xn+p−1+i i xi = 1 n+p + 3x2(x+2)2(n+p) −2xA+n+pB (n+p+β)4δ4n , where A= −6α2−24α−36+ (−3α2+12αβ−24αγ +6α+24β−84γ+48)x+ (6αβ−12αγ−6β2+24βγ+8α−6β−54γ+38)x2 + (−3β2+12βγ−8β−8γ+5)x3and
B=α4+4α3+12α2+24α+24+ (−4α3β+8α3γ−8α3−12α2β+60α2γ−60α2
−24αβ+216αγ−216α−24β+336γ−336)x
+ [−6β(−α2β+4α2γ−4α2−2αβ+20αγ−20α−2β+36γ−36)]x2
+4β2(−αβ+6αγ−6α−β+15γ−15)x3+ [−β3(−β+8γ−8)]x4.
From the above relation, we obtain
|Mγn,p,α,β(f ; x) − f(x)| 54(n+p)(1+δn2)2Ω(f ; δn)(1+x2) · ∞
∑
k=0 bk,αn,p(x) Z ∞ 0 1+ (n+p)t+α n+p+β −x 4 δ4n skn,p(t)dt =4(n+p)(1+δ2n)2Ω(f ; δn)(1+x2) · 1 n+p + 3x2(x+2)2(n+p) −2xA+ B n+p (n+p+β)4δ4n ! . In addition, for δn =n− 1 4, we have |Mγn,p,α,β(f ; x) − f(x)| 5 KΩ(f ; δn)(1+x2)(1+ Cx+ Dx2+ Ex3+ Fx4),whereC,D,E, andF are positive constants depending only on n, p, α, β, and γ, andKis a positive constant. This proves Theorem14.
6. Shape-Preserving Properties
In this section, we will present some shape-preserving properties by proving that the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators preserves the convexity under certain conditions.
Theorem 15. Let f ∈C[0,∞). If f(x)is convex on[0,∞)and n+p+4γ>3 for α, β, γ∈ R, then, the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators are also convex.
Proof. Let us suppose that f(x)is convex and that x0and x1are distinct points in the
interval[x, y], where x < x0 < x1 < y and x, y ∈ [a, b] ⊂ [0,∞). Then, the Lagrangian interpolation polynomial through the points x0, f(x0) and x1, f(x1) is given by
P(x) = x−x1 x0−x1 f(x0) + x−x0 x1−x0 f(x1). Then, based upon Theorem1, we have
h Mγn,p,α,β(P; x)i00= f(x1) −f(x0) x1−x0 α+1 n+p+β +n+p+2γ−2 n+p+β x +x1f(x0) −x0f(x1) x1−x0 00 =0.
On the other hand, we have Mn,pγ,α,β(f ; x) =Mγn,p,α,β(P; x) + f 00( ξt) 2! h Mn,pγ,α,β(t2; x) − (x0+x1)Mγ ,α,β n,p (t, x) +x0x1 i =Mγ,α,β n,p (P; x) + f00(ξt) 2! " α2+2α+2 (n+p+β)2 +2[(n+p+2γ)α+2n+2p−2α+5γ−5] (n+p+β)2 x +(n+p)(n+4γ+p−3) (n+p+β)2 x 2 − (x0+x1) α+1 n+p+β +n+p+2γ−2 n+p+β x +x0x1 # .
From this last relation, we find that h Mγn,p,α,β(f ; x) i00 = f00(ξt) · (n+p)(n+4γ+p−3) (n+p+β)2 >0
under the given conditions. This completes the proof of Theorem15.
Corollary. The classical Baskakov-Schurer-Szász-Stancu operators preserve the property of convexity.
Proof. We know that, for γ=1, we are led to the Baskakov-Schurer-Szász-Stancu operators
from the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. Since n, p∈ N, in the special case when γ=1, we have n+p+4γ>3. The proof now follows from Theorem15.
7. Concluding Remarks and Observations
In our present investigation, we have introduced, and systematically studied the properties and relations associated with, a new class of the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators. Our findings have considerably and significantly extended the well-known family of the classical Baskakov-Schurer-Szász-Stancu approximation operators. For our new class of the Baskakov-Schurer-Szász-Baskakov-Schurer-Szász-Stancu approximation operators, we have established a Korovkin type theorem and a Grüss-Voronovskaya type theorem. We have also studied the rate of its convergence. Moreover, we have proved several results which are related to the parametric generalization of the Baskakov-Schurer-Szász-Stancu operators in the weighted spaces. Finally, we have derived a number of shape-preserving properties for the parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators. We have also appropriately specialized our results in order to deduce the corresponding shape-preserving properties for the classical Baskakov-Schurer-Szász-Stancu approximation operators.
The various results and their consequences, which we have presented in this article, will potentially motivate and encourage further researches on the subject dealing with the parametric generalization of the Baskakov-Schurer-Szász-Stancu approximation operators.
Author Contributions:Conceptualization, N.L.B., T.M. and H.M.S.; methodology, N.L.B. and T.M.; software, N.L.B. and H.M.S.; validation, N.L.B., T.M. and H.M.S.; formal analysis, N.L.B., T.M. and H.M.S.; investigation, N.L.B., T.M. and H.M.S.; resources, N.L.B. and H.M.S.; data curation, N.L.B. and T.M.; writing—original draft preparation, N.L.B. and T.M.; writing—review and editing, N.L.B. and H.M.S.; visualization, N.L.B. and T.M.; supervision, H.M.S.; project administration, H.M.S.; funding acquisition, H.M.S. All authors have read and agreed to the published version of the manuscript.
Funding:This research received no external funding.
Institutional Review Board Statement:Not applicable.
Informed Consent Statement:Not applicable.
Data Availability Statement:Not applicable.
References
1. Szász, O. Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stand. 1950, 45, 239–245. [CrossRef] 2. Baskakov, V.A. An instance of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR
1957, 113, 249–251.
3. Schurer, F. Linear Positive Operators in Approximation Theory; Report of the Mathematcal Institute of the Technical University of Delft: Delft, The Netherlands, 1962.
4. Stancu, D.D. Approximation of function by means of a new generalized Bernstein operator. Calcolo 1983, 20, 211–229. [CrossRef] 5. Sofyalioglu, M.; Kanat, K. Approximation properties of generalized Baskakov-Schurer-Szász–Stancu operators preserving
e−2ax, a>0. J. Inequal. Appl. 2019, 2019, 112. [CrossRef]
6. Atlihan, O.G.; Unver, M.; Duman, O. Korovkin theorems on weighted spaces: Revisited. Period. Math. Hungar 2017, 75, 201–209. [CrossRef]
7. Braha, N.L. Some weighted equi-statistical convergence and Korovkin type-theorem. Results Math. 2016, 70, 433–446. [CrossRef] 8. Braha, N.L. Some properties of new modified Szász-Mirakyan operators in polynomial weight spaces via power summability
method. Bull. Math. Anal. Appl. 2018, 10, 53–65.
9. Braha, N.L. Some properties of Baskakov-Schurer-Szász operators via power summability methods. Quaest. Math.
2019, 42, 1411–1426. [CrossRef]
10. Braha, N.L. Korovkin type theorem for Bernstein-Kantorovich operators via power summability method. Anal. Math. Phys.
2020, 10, 62. [CrossRef]
11. Braha, N.L. Some properties of modified Szász-Mirakyan operators in polynomial spaces via the power summability method. J. Appl. Anal. 2020, 26, 79–90. [CrossRef]
12. Braha, N.L.; Loku, V. Korovkin type theorems and its applications via αβ-statistically convergence. J. Math. Inequal.
2020, 14, 951–966. [CrossRef]
13. Braha, N.L.; Loku, V.; Srivastava, H.M.Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Appl. Math. Comput. 2015, 266, 675–686. [CrossRef]
14. Braha, N.L.; Mansour, T.; Mursaleen, M. Some properties of Kantorovich-Stancu-type generalization of Szász operators including Brenke-type polynomials via power series summability method. J. Funct. Spaces 2020, 2020, 3480607. [CrossRef]
15. Braha, N.L.; Mansour, T.; Mursaleen, M.; Acar, T. Some properties of λ-Bernstein operators via power summability method. J. Appl. Math. Comput. 2020, 65, 125–146. [CrossRef]
16. Braha, N.L.; Srivastava, H.M.; Mohiuddine, S.A. A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 2014, 228, 162–169. [CrossRef] 17. Gavrea, I.; Ra¸sa, I. Remarks on some quantitative Korovkin-type results. Rev. Anal. Numér. Théor. Approx. 1993, 22, 173–176. 18. Kadak, U.; Braha, N.L.; Srivastava, H.M. Statistical weightedB-summability and its applications to approximation theorems.
Appl. Math. Comput. 2017, 302, 80–96.
19. Loku, V.; Braha, N.L. Some weighted statistical convergence and Korovkin type-theorem. J. Inequal. Spec. Funct. 2017, 8, 139–150. 20. Mursaleen, M.; Al-Abied, A.A.H.; Ansari, K.J. On approximation properties of Baskakov-Schurer-Szász–Stancu operators based
on q-integers. Filomat 2018, 32, 1359–1378. [CrossRef]
21. Braha, N.L.; Srivastava, H.M.; Et, M. Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems. J. Appl. Math. Comput. 2021, 65, 429–450. [CrossRef]
22. Gupta, V.; Acu, A.M.; Srivastava, H.M. Difference of some positive linear approximation operators for higher-order derivatives, Symmetry 2020, 12, 915. [CrossRef]
23. Srivastava, H.M.; Ícoz, G.; Çekim, B. Approximation properties of an extended family of the Szász-Mirakjan Beta-type operators, Axioms 2019, 8, 111. [CrossRef]
24. Srivastava, H.M.; Mursaleen, M.; Alotaibi, A.M.; Al-Abied, A.A.H. Some approximation results involving the q-Szász-Mirakjan-Kantorovich type operators via Dunkl’s generalization. Math. Methods Appl. Sci. 2017, 40, 5437–5452. [CrossRef]
25. Altomare, F.; Campiti, M. Korovkin-Type Approximation Theory and Its Application; Walter de Gruyter Studies in Mathematics; De Gruyter & Company: Berlin, Germany, 1994; Volume 17.
26. Ditzian, Z.; Totik, V. Moduli of Smoothness; Springer-Verlag: Berlin/Heidelberg, Germany; New York, NY, USA, 1987.
27. Zhuk, V.V. Functions of the Lip 1 class and S. N. Bernstein’s polynomials (Russian, with English summary). Vestn. Leningr. Univ. Mat. Mekh. Astronom. 1989, 1989, 25–30. 122–123; English translation: Vestn. Leningr. Univ. Math. 1989, 22, 38–44.
28. Landau, E. Einige Ungleichungen für zweímal differentzierban funktionen. Proc. London Math. Soc. 1913, 13, 43–49. 29. Peetre, J. Theory of interpolation of normed spaces. Notas Mat. Rio Jan. 1963, 39, 1–86.
30. Gal, S.G.; Gonska, H. Grüss and Grüss-Voronovskaya-type estimates for some Bernstein-type polynomials of real and complex variables. Jaen J. Approx. 2015, 7, 97–122.
