Abstract
In this paper, we construct a Durrmeyer variant of the modified \(\alpha \)-Bernstein-type operators introduced by Kajla and Acar (Ann Funct Anal 10(4):570–582, 2019), for \(\alpha \in [0,1]\). We investigate the degree of approximation via the approach of Peetre’s K-functional and the Lipschitz-type maximal function. The quantitative Voronovskaja- and Grüss Voronovskaja-type theorems are discussed. Further, we determine the rate of convergence by the above operators for the functions with derivatives of bounded variation.
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1 Introduction
Let \(\psi :J\rightarrow {\mathbb {R}}\) be bounded on \(J=[0,1]\). In 1912, to prove Weierstrass approximation theorem (Weierstrass 1885), Bernstein (1912) defined a sequence of linear positive operators, known as Bernstein operators, as follows:
where
It is easily verified that the Bernstein basis functions \(P_{n,m}(\varkappa )\) satisfy the recurrence relation
There are many applications of Bernstein operators in fields such as mathematics, physics, computer science and engineering. Due to their useful structure, many researchers discovered their various approximation properties and made significant contributions, the interested reader may refer to Pǎltǎnea (2004), Gonska (2007), Gonska and Raşa (2009), Gavrea and Ivan (2012), Tachev (2012), etc. Chen et al. (2017) defined a generalization of Bernstein operators (1.1) based on a real parameter ‘\(\alpha \)’ satisfying \(0\le \alpha \le 1\), as
where the \(\alpha \)-Bernstein basis function \(P_{n,m,\alpha }(\varkappa ),\) for \(n\ge 2,\) is given by
with \({n-2\atopwithdelims ()-2}={n-2\atopwithdelims ()-1}=0\). It is easily seen that \(P_{n,m,\alpha }(\varkappa ),\) verifies the recurrence relation
Chen et al. (2017) studied the uniform convergence theorem and the Voronovskaja-type asymptotic formula, etc. For the special case \(\alpha =1\), the operators (1.2) reduce to (1.1). Acar and Kajla (2018) introduced a bivariate extension of (1.2) and studied some approximation properties of these operators and the associated GBS (Generalized Boolean Sum) operators. Kajla and Acar (2018) defined the Durrmeyer-type modification of (1.2) and studied the local and global approximation properties and Voronovskaja-type asymptotic theorem. For more details of the research work in this direction, we refer to cf. (Lupas and Lupas 1987; Nowak 2009; Aral et al. 2013; Acar et al. 2016; Gupta and Tachev 2017; Mohiuddine et al. 2017; Acu et al. 2018; Mohiuddine et al. 2020; Kajla et al. 2020; Rao et al. 2021; Mishra and Gandhi 2019; Mohiuddine 2020; Mohiuddine and Özger 2020; Özger et al. 2020, etc.). Khosravian-Arab et al. (2018) introduced a new family of Bernstein operators as follows:
where
and
\(a_0(n)\) and \(a_1(n)\) being two unknown sequences, which may be defined in an appropriate manner. If \(a_0(n)=1,\) and \(a_1(n)=-1,\) then (1.4) reduces to (1.1). Gupta et al. (2019) defined a Kantorovich version of the operators (1.4) and studied some better approximation properties. For \(\psi \in C(J),\) the space of continuous functions on J with the uniform norm denoted by \(\Vert .\Vert \), Kajla and Acar (2019) introduced a modification of the generalized Bernstein operators (1.4) by means of a parameter \(\alpha \), \(0\le \alpha \le 1\) as follows:
where
and \(P_{n,m,\alpha }(\varkappa )\) is the same as defined in (1.3) and examined the uniform convergence and the asymptotic approximation. It is clear that for \(\alpha =1\), the operators (1.6) include the operators (1.4).
Inspired by the above research work, for \(\psi \in C(J),\) we define a Durrmeyer-type modification of the operators (1.6) as follows:
where \(P_{n,m,\alpha }(\varkappa )\) is the \(\alpha \)-Bernstein basis function as defined in (1.3). It is evident that for \(a_0(n)=1\), \(a_1(n)=-1\), and \(\alpha =1\), the operator (1.8) includes the Bernstein–Durrmeyer operator introduced by Durrmeyer (1967) and subsequently studied by many other researchers (Derriennic 1981; Berens et al. 1992; Zhou 1992; Karsli 2019, etc.). Throughout this paper, we assume that the sequences \(a_0(n)\) and \(a_1(n)\) satisfy the relation \(2a_0(n)+a_1(n)=1\).
The purpose of this paper is to investigate the approximation degree of the operators (1.8) with the aid of the Peetre’s K-functional and the Lipschitz-type maximal function. We also discuss the quantitative Voronovskaya- and Grüss–Voronovskaya-type theorem. The rate of convergence for functions with the derivative of bounded variation is also derived.
2 Preliminaries
Throughout this paper, we assume \({\mathbb {N}}_{0}:={\mathbb {N}}\cup \{0\}\). Let \(e_j({\mathfrak {t}})={\mathfrak {t}}^j\), \(j\in {\mathbb {N}}_{0}\). By a simple computation, we get
Lemma 1
(Kajla and Acar 2019) For \(\sum _{m=0}^{n}m^jP_{n,m,\alpha }^{M,1}(\varkappa )\), \(j\in {\mathbb {N}}_{0}\), we have the following identities:
As a consequence of Lemma 1, we obtain:
Lemma 2
The operators \(K_{n,\alpha }^{M,1}(.;\varkappa ),\) verify the following equalities:
Let \(\mu _{n,\alpha ,r}^{M,1}(\varkappa )=K_{n,\alpha }^{M,1}(( {\mathfrak {t}}-\varkappa )^r;\varkappa )\), \(r\in {\mathbb {N}}_{0}\). Further, let us assume that \({\lim }_{n\rightarrow \infty } a_i(n)=p_i,\) for \(i=0,1\). Then by a simple computation, using Lemma 2, we reach the following crucial result:
Lemma 3
For the operators \(K_{n,\alpha }^{M,1}(.;\varkappa )\), we have
Remark
From Lemma 3, for all sufficiently large n and \(\varkappa \in J\), we have \(\mu _{n,\alpha ,2}^{M,1}(\varkappa )\le \frac{C}{n}\phi ^2(\varkappa )\) and \(\mu _{n,\alpha ,4}^{M,1}(\varkappa )\le \frac{C}{n^2}\phi ^4(\varkappa )\), where \(\phi ^2(\varkappa )=\varkappa (1-\varkappa )\) and C is a positive constant. Our following result shows that the operators \(K_{n,\alpha }^{M,1}\) are bounded operators.
Lemma 4
For \(\psi \in C(J),\) and for each \(\varkappa \in J\), we have
Proof
Applying the definition of the operators (1.8) and using Lemma 2, we have
which completes the proof. \(\square \)
3 Main Results
First, we show that the operator \(K_{n,\alpha }^{M,1}(\psi )\) is an approximation process for function \(\psi \in C(J).\)
3.1 Basic Uniform Convergence Theorem
Theorem 1
If \(\psi \in C(J)\) and \(\alpha \in J \), then
uniformly in \(\varkappa \in J\).
Proof
Applying Lemma 2, it follows that \(\lim _{n\rightarrow \infty }K_{n,\alpha }^{M,1}( {\mathfrak {t}}^j;\varkappa )={\varkappa }^j,\; j=0,1,2\) uniformly in \(\varkappa \in J\). Hence, by the well-known Bohman–Korovkin theorem (Korovkin 1960), we obtain the desired result. \(\square \)
3.2 Local Approximation
For \(\psi \in C(J)\) and any \(\delta >0,\) the Peetre’s K-functional \(K(\psi ;\delta )\) is defined by
where \(C^2(J)=\{g:g''\in C(J)\}\). By DeVore and Lorentz (1993), \(\exists \) a constant \(M>0\) such that
where \(\omega _2(\psi ;\sqrt{\delta })\) is the second-order modulus of continuity of \(\psi \in C(J)\), defined as
Also, for \(\psi \in C(J),\) the first-order modulus of continuity is given by
Theorem 2
For \(\psi \in C(J),\) \(\exists \) a constant \(M>0\) such that
where \(\xi _{n,\alpha }^{M,1}(\varkappa )=\mu _{n,\alpha ,2}^{M,1}(\varkappa )+(\mu _{n,\alpha ,1}^{M,1}(\varkappa ))^2.\)
Proof
Let us define an auxiliary operator as follows:
where \(\alpha _n(\varkappa )\) is defined as in Lemma 2. From (3.3), it is clear that \({\bar{K}}_{n,\alpha }^{M,1}(\psi ;\varkappa )\) is a linear operator and applying Lemma 2
Let \(g\in C^2(J)\) and \(\varkappa \in J\) be arbitrary. Then by Taylor’s formula,
Now, applying the linear operator \({\bar{K}}_{n,\alpha }^{M,1}(.;\varkappa )\) on both sides of the above equation and using Eq. (3.4), we get
Hence,
Now from Eq. (3.3) and using Lemma 2, we have
For \(\varkappa \in J,\; \psi \in C(J)\) and any \(g\in C^2(J)\), from Eqs. (3.5) and (3.6), we obtain
Taking infimum on the right-hand side over all \(g\in C^2(J)\) and using the definition of Peetre’s K-functional given by (3.1), we obtain
Hence considering the relation (3.2), we get
which completes the proof. \(\square \)
Lipschitz-type space: In view of Szász (1950), for \(\varkappa \in J\)
where \(\rho \in (0,2],\;and\;M_{\psi }>0\) is a constant dependent only on \(\psi \).
Theorem 3
Let \(\psi \in Lip_M^{(\rho )}\). Then for all \(\varkappa \in (0,1]\), we have
Proof
First of all, we show the result for the case \(\rho =2\). By our hypothesis, we have
Now using the inequality \(\frac{1}{ {\mathfrak {t}}+\varkappa }\le \frac{1}{\varkappa }, \;\forall \; \;t\in J\;and\;\varkappa \in (0,1]\), we get
This proves the result for \(\rho =2.\) Now, we show the above theorem for \(0<\rho <2\). By using Hölder inequality with \(q_1=\frac{2}{\rho }\) and \(q_2= \frac{2}{2-\rho }\), we get
This completes the proof. \(\square \)
Next, we study a local direct estimate for the operators defined in (1.8). First of all, we define the Lipschitz-type maximal function of order \(\rho ,\) given by Lenze (1988) as
For similar studies, one can refer to Kajla and Agrawal (2015), Kajla et al. (2017), Kajla (2017) and Neer et al. (2017).
Theorem 4
Let \(\psi \in C(J)\) and \(0<\rho \le 1\). Then, \(\forall \; \varkappa \in J,\) we have
Proof
In view of (3.7),
Applying the operator \(K_{n,\alpha }^{M,1}(.;\varkappa )\) on the above inequality, then using Lemma 3 and the Hölder inequality with \(q_1=\frac{2}{\rho }\), \(q_2= \frac{2}{2-\rho }\), we have
\(\square \)
3.3 Asymptotic approximation by \(K_{n,\alpha }^{M,1}\)
We present a Voronovskaja-type asymptotic theorem for the operators \(K_{n,\alpha }^{M,1}(\psi ;\varkappa )\), defined in (1.8).
Theorem 5
Let \(\psi \in C(J)\) and \(\alpha \in J\). If \(\psi ''(\varkappa )\) exists at a given point \(\varkappa \in J\) then we have
Furthermore, if \(\psi '' \in C(J)\) then Eq. (3.8) holds uniformly in \(\varkappa \in J\).
Proof
By using Taylor’s formula, we have
where \(\phi ( {\mathfrak {t}}, \varkappa )\in C(J)\) and \(\phi ( {\mathfrak {t}}, \varkappa )\rightarrow 0, \) as \( {\mathfrak {t}}\rightarrow \varkappa \). Now operating by \(K_{n,\alpha }^{M,1}(.;\varkappa )\) on both sides of (3.9) and using Lemma 3, we get
Hence,
Since \(\phi ( {\mathfrak {t}}, \varkappa )\rightarrow 0\) as \( {\mathfrak {t}}\rightarrow \varkappa \), for a given \(\epsilon >0, \;\exists \) a \(\delta >0\) such that \(|\phi ( {\mathfrak {t}}, \varkappa )|<\epsilon \) whenever \(| {\mathfrak {t}}-\varkappa |<\delta ,\) and for \(| {\mathfrak {t}}-\varkappa |\ge \delta \), we have \(|\phi ( {\mathfrak {t}}, \varkappa )|\le M\frac{( {\mathfrak {t}}-\varkappa )^2}{\delta ^2}\), where \(M>0\). Let \(\chi _{\delta }({\mathfrak {t}})\) be the characteristic function of the interval \((\varkappa -\delta , \varkappa +\delta )\). Now using Lemma (3), we obtain
Hence, due to the arbitrariness of \(\epsilon >0\), we have \(\lim _{n\rightarrow \infty } nK_{n,\alpha }^{M,1}(\phi ( {\mathfrak {t}}, \varkappa )( {\mathfrak {t}}-\varkappa )^2;\varkappa )=0.\) Thus by (3.10), we get
This completes the proof of the first assertion of the theorem.
The second assertion follows due to the uniform continuity of \(\psi ''\) on J enabling \(\delta \) to become independent of \(\varkappa ,\) and all the other estimates hold uniformly in \(\varkappa \in J\). \(\square \)
3.4 Quantitative Voronovskaja-Type Theorem
Next, we calculate the order of approximation for the operators (1.8) by means of the Ditzian–Totik modulus of smoothness. Let \(\phi (\varkappa )=\sqrt{\varkappa (1-\varkappa )}\). For \(\psi \in C(J),\) the Ditzian–Totik modulus of smoothness of first order (Ditzian and Totik 2012) is defined as
and the corresponding Peetre’s K-functional is given by
where \(W_\phi (J)=\{g:g\in AC_{loc}(J),\;\Vert \phi g'\Vert < \infty \}\) and \(AC_{loc}(J)\) is the space of locally absolutely continuous functions on every closed and bounded interval \([a,b]\subseteq J\). It is well-known from Ditzian and Totik (2012) that \(\exists \) a constant \(M>0,\) such that
Theorem 6
For any \(\psi \in C^2(J)\) and sufficiently large n, there holds the following two inequalities:
where \(M'\) is some positive constant.
Proof
For \(\psi \in C^2(J)\) and \( {\mathfrak {t}}, \varkappa \in J,\) using Taylor’s formula, we have
Hence,
Now applying the linear positive operator \(K_{n,\alpha }^{M,1}(.;\varkappa )\) on both sides of the above equation, we have
For any \(g\in C^2(J),\) Finta (2011, p. 337) estimated the right-hand quantity of Eq. (3.12) as follows:
Now, combining (3.12) and (3.13)
Now, using Cauchy–Schwarz’s inequality and Remark 2, we obtain
Since \(\phi ^2(\varkappa )\le \phi (\varkappa )\le 1, \;\forall \; \varkappa \in J\), we get
and
Taking the infimum on the right-hand side of the above relations over all \(g\in W_\phi (J),\) we obtain
where \(M'=2C\). Now, applying relation (3.11), we reach the required result. \(\square \)
3.5 Grüss–Voronovskaya-Type Theorem for the Operators \(K_{n,\alpha }^{M,1}\)
Theorem 7
For \(\psi , g \in C^2(J)\), there holds the following equality:
uniformly \(\varkappa \in J.\)
Proof
We have the double derivative of the product of two functions \(\psi \) and g as
By making an appropriate arrangement, we get
Using Theorem 1, for any \(\psi \in C(J),\) \(K_{n,\alpha }^{M,1}(\psi ;\varkappa )\rightarrow \psi (\varkappa )\), as \(n\rightarrow \infty \), uniformly in \(\varkappa \in J\), and for \(\psi ''(\varkappa )\in C(J),\) from the proof of Theorem 5, it is clear that
uniformly in \(\varkappa \in J\). Hence, applying Lemma 3, we obtain the desired result. \(\square \)
3.6 Approximation of functions with derivatives of bounded variation (DBV)
Cheng (1983) obtained the rate of convergence of Bernstein polynomials for functions of bounded variation (BV). Bojanic and Cheng (1989) obtained the rate of convergence of Bernstein polynomials for functions with DBV. Bojanic and Khan (1991) estimated the rate of convergence of some operators for functions with DBV. Subsequently, many mathematicians studied in this direction (Bojanic and Cheng 1989, 1992; Zeng and Chen 2000; Gupta et al. 2003; Ibikli and Karsli 2005; Gupta et al. 2005). We shall obtain the rate of convergence of the operators \(K_{n,\alpha }^{M,1}(\psi ;\varkappa )\) defined by (1.8) for functions \(\psi ({\mathfrak {t}};\varkappa )\) having DBV. We show that the operators \(K_{n,\alpha }^{M,1}(\psi ;\varkappa )\) converge to the function \(\psi (\varkappa )\), where \(\psi '(\varkappa +)\) and \(\psi '(\varkappa -)\) exist. Let DBV(J) be the class of all absolutely continuous functions f on J, having a derivative \(\psi '\) equivalent with a function of BV on every finite subinterval of J. Note that the function \(\psi \in DBV(J)\) can be represented as
where g is a function of BV on every finite subinterval of J. We observe that the operator (1.8) may be rewritten as
where \(N_{n,\alpha }^{M,1}( {\mathfrak {t}}, \varkappa )\) is the kernel defined as
Lemma 5
For \(\forall \; \varkappa \in (0,1)\) and sufficiently large n, we have
Proof
(i) By definition, we have
Similarly, we can prove the other inequality (ii). \(\square \)
Let
Theorem 8
Let \(\psi \in DBV (J),\; \varkappa \in (0,1)\) and n be sufficiently large. Then, we have
Proof
By the hypothesis (3.16), we have
where
Now using Lemma 2, Eqs. (3.14) and (3.17), we get
Since \(\displaystyle {\int _0^1\bigg (\int _{\varkappa }^{{\mathfrak {t}}} \bigg (\psi '(v)-\frac{1}{2}(\psi '(\varkappa +)+\psi '(\varkappa -))\bigg ) \delta _{\varkappa }(v)\mathrm{d}v\bigg ) N_{n,\alpha }^{M,1}( {\mathfrak {t}}, \varkappa )\mathrm{d}{\mathfrak {t}}=0},\) we have
Now, we break the second term on the right-hand side of (3.18) as follows:
Then from (3.18), we get
Now applying Cauchy–Schwarz inequality, we have
Using Lemma 5 and integration by parts,
Consequently,
Since \(\psi '_{\varkappa }(\varkappa )=0\) by the hypothesis 3.16, it follows that
Now, using Lemma 5, we get
By the definition of total variation of \(\psi \) and putting \({\mathfrak {t}}=(\varkappa -\frac{\varkappa }{v}),\)
Hence,
Since by Lemma 5, \(\lambda _{n,\alpha }^{M,1}( {\mathfrak {t}}, \varkappa )\le 1\) and \(\psi '_{\varkappa }(\varkappa )=0,\) we obtain
Now through the definition of total variation of \(\psi \)
Hence combining the estimates of \(K_1(\varkappa )\) and \(K_2(\varkappa ),\) we get
Using Lemma 5, we can write
Now applying integration by parts and hypothesis (3.16)
Now, using hypothesis (3.16)
and
Now in view of Lemma 5, \(1-\lambda _{n,\alpha }^{M,1}( {\mathfrak {t}}, \varkappa )\le 1\) and from hypothesis (3.16), \(\psi '_{\varkappa }(\varkappa )=0\), therefore
Now again using Lemma 5 and definition (3.16), we get
From the definition of total variation of \(\psi \) and put \({\mathfrak {t}}=\varkappa +\frac{\varkappa }{v}\),
Hence,
Now, we calculate \(E_4\). Using Cauchy–Schwarz inequality,
In order to estimate \(E_3\), we note that \(t\ge 2\varkappa \), hence \(( {\mathfrak {t}}-\varkappa )\ge \varkappa ,\) therefore
Hence collecting the estimates of \(E_1-E_4,\) we have
Now combining Eqs. (3.19)–(3.21), we reach the desired result. \(\square \)
References
Acar T, Aral A, Raşa I (2016) The new forms of Voronovskaya\(^{\prime }\)s theorem in weighted spaces. Positivity 20(1):25–40
Acar T, Kajla A (2018) Degree of approximation for bivariate generalized Bernstein type operators. Results Math. 73:79
Acu AM, Hodiş S, Raşa I (2018) A survey on estimates for the differences of positive linear operators. Constr Math Anal 1(2):113–127
Aral A, Gupta V, Agarwal RP (2013) Applications of q-calculus in operator theory. Springer, New York
Berens H, Schmid HJ, Xu Y (1992) Bernstein–Durrmeyer polynomials on a simplex. J Approx Theory 68:247–261
Bernstein SN (1912) Démonstration du théorém de Weierstrass fondée sur la calcul desprobabilitiés. Commun Soc Math Charkow Sér 2(13):1–2
Bojanic R, Cheng F (1989) Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation. J Math Anal Appl 141:136–151
Bojanic R, Cheng F (1992) Rate of convergence of Hermite–Fejer polynomials for functions with derivatives of bounded variation. Acta Math Hungar 59:91–102
Bojanic R, Khan MK (1991) Rate of convergence of some operators of functions with derivatives of bounded variation. Atti Sem Mat Fis Univ Modena 39(2):495–512
Chen X, Tan J, Liu Z, Xie J (2017) Approximation of functions by a new family of generalized Bernstein operators. J Math Anal Appl 450:244–261
Cheng F (1983) On the rate of convergence of Bernstein polynomials of functions of bounded variation. J Approx Theory 39(3):259–274
Derriennic MM (1981) Sur l\(^{\prime }\) approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies. J Approx Theory 31:325–343
DeVore RA, Lorentz GG (1993) Constructive approximation, ser. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 303. Springer, Berlin
Ditzian Z, Totik V (2012) Moduli of smoothness, vol 9. Springer, Berlin
Finta Z (2011) Remark on Voronovskaja theorem for \(q\) Bernstein operators. Stud Univ Babeş Bolyai Math 56:335–339
Durrmeyer JL (1967) Une formule d’inversion de la transform\(\acute{e}\)e de Laplace: Applications a la theorie des moments. Ph.D. Thesis, These de 3 cycle, Faculte des Sciences, I Universite de Paris
Gavrea I, Ivan M (2012) An answer to a conjecture on Bernstein operators. J Math Anal Appl 390(1):86–92. https://doi.org/10.1016/j.jmaa.2012.01.021.570
Gonska H (2007) On the degree of approximation in Voronovskaja\(^{\prime }\)s theorem. Stud Univ Babeş Bolyai Math 52(3):103–115
Gonska H, Raşa I (2009) Asymptotic behaviour of differentiated Bernstein polynomials. Mat Vesnik 61(1):53–60
Gupta V, Abel U, Ivan M (2005) Rate of convergence of Beta operators of second kind for functions with derivatives of bounded variation. Int J Math Math Sci 23:3827–3833
Gupta V, Tachev G, Acu AM (2019) Modified Kantorovich operators with better approximation properties. Numer Algorithms 81(1):125–149
Gupta V, Tachev G (2017)Approximation with positive linear operators and linear combinations, Dev. Math. 50. Springer, Cham
Gupta V, Vasishtha V, Gupta MK (2003) Rate of convergence of summation-integral type operators with derivatives of bounded variation, JIPAM, 4(2). Article 34
Ibikli E, Karsli H (2005) Rate of convergence of Chlodowsky type Durrmeyer operators. JIPAM 6(4). Article 106
Kajla A (2017) Direct estimates of certain Mihesan–Durrmeyer type operators. Adv Oper Theory 2(2):162–178
Kajla A, Acar T (2018) Blending type approximation by generalized Bernstein–Durrmeyer type operators. Miskolc Math Notes 19(1):319–336
Kajla A, Acar T (2019) Modified \(\alpha \)-Bernstein operators with better approximation properties. Ann Funct Anal 10(4):570–582
Kajla A, Acu AM, Agrawal PN (2017) Baskakov–Sz\(\acute{a}\)sz-type operators based on inverse Polya–Eggenberger distribution. Ann Funct Anal 8(1):106–123
Kajla A, Agrawal PN (2015) Approximation properties of Sz\(\acute{a}\)sz type operators based on Charlier polynomials. Turk J Math 39(6):990–1003
Kajla A, Mohiuddine SA, Alotaibi A, Goyal M, Singh KK (2020) Approximation by \(\vartheta \)-Baskakov–Durrmeyer-type hybrid operators. Iran J Sci Technol Trans Sci 44:1111–1118
Karsli H (2019) Some properties of \(q\)-Bernstein–Durrmeyer operators. Tbilisi Math J 12(4):189–204
Khosravian-Arab H, Dehghan M, Eslahchi MR (2018) A new approach to improve the order of approximation of the Bernstein operators: theory and application. Numer Algorithms 77(1):111–150
Korovkin PP (1960) Linear operators and approximation theory. Hindustan Publishing Corporation, Delhi
Lenze B (1988) On Lipschitz-type maximal functions and their smoothness spaces. Nederl Akad Wetensch Indag Math 50(1):53–63
Lupas L, Lupas A (1987) Polynomials of binomial type and approximation operators. Stud Univ Babeş-Bolyai Math 32(4):61–69
Mishra VN, Gandhi RB (2019) Study of sensitivity of parameters of Bernstein–Stancu operators. Iran J Sci Technol Trans A Sci 43:2891–2897
Mohiuddine SA (2020) Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators. Adv Differ Equ 676:17
Mohiuddine SA, Ahmad N, Özger F, Alotaibi A, Hazarika B (2020) Approximation by the parametric generalization of Baskakov–Kantorovich operators linking with Stancu operators. Iran J Sci Technol Trans A Sci 45:593–605
Mohiuddine SA, Özger F (2020) Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter \(\alpha \). Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114:70
Mohiuddine SA, Acar T, Alotaibi A (2017) Construction of a new family of Bernstein–Kantorovich operators. Math Methods Appl Sci 40(18):7749–7759
Neer T, Agrawal PN, Araci S (2017) Stancu–Durrmeyer type operators based on q-integers. Appl Math Inf Sci 11(3):767–775
Nowak G (2009) Approximation properties for generalized q-Bernstein polynomials. J Math Anal Appl 350(1):50–55
Özger F, Srivastava HM, Mohiuddine SA (2020) Approximation of functions by a new class of generalized Bernstein–Schurer operators. Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 114:173
Pǎltǎnea R (2004) Approximation theory using positive linear operators. Birkhäuser, Boston
Rao N, Nasiruzzaman M, Heshamuddin M, Shadab M (2021) Approximation properties by modified Baskakov–Durrmeyer operators based on shape parameter-\(\alpha \). Iran J Sci Technol Trans Sci. https://doi.org/10.1007/s40995-021-01125-0
Szász O (1950) Generalization of S. Bernstein\(^{\prime }\)s polynomials to the infinite interval. J Res Nat Bur Standards 45:239–245
Tachev G (2012) The complete asymptotic expansion for Bernstein operators. J Math Anal Appl 385:1179–1183. https://doi.org/10.1016/j.jmaa.2011.07.042
Weierstrass KG (1885) Über Die Analytische Darstellbarkeit Sogenannter Willkürlicher Functionen Einer Reellen Ver\(\ddot{a}\)nderlichen, Sitzungsber. Acad. Berlin, 633–639. [Also In: Mathematische Werke, vol. 3, 1–37, Berlin: Mayer and Müller 1903]
Zeng XM, Chen W (2000) On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation. J Approx Theory 102:1–12
Zhou DX (1992) Inverse theorems for multidimensional Bernstein–Durrmeyer operators in \(L_p^*\). J Approx Theory 70:68–93
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Agrawal, P.N., Kajla, A. & Singh, S. Modified \(\alpha \)-Bernstein–Durrmeyer-Type Operators. Iran J Sci Technol Trans Sci 45, 2049–2061 (2021). https://doi.org/10.1007/s40995-021-01197-y
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DOI: https://doi.org/10.1007/s40995-021-01197-y
Keywords
- Peetre’s K-functional
- Ditzian–Totik modulus of smoothness
- Voronovskaja-type theorem
- Grüss Voronovskaja-type theorem
- Functions of bounded variation