Convergence of $\lambda$ -Bernstein-Kantorovich operators
in the $L_{p}$ -norm

Purshottam N. Agrawal¹ and Behar Baxhaku²

(Date: November 15, 2023; accepted: May 30, 2024; published online: July 11, 2024.)

Abstract.

We show the convergence of $\lambda$ -Bernstein-Kantorovich operators defined by Acu et al. [J. ineq. Appl. 2018], for functions in $L_{p}[0,1]$ , $p\geq 1$ . We also determine the convergence rate via integral modulus of smoothness.

Key words and phrases:

Bernstein-Kantorovich type operators, Peetre’s

K

-functional, integral modulus of smoothness.

2005 Mathematics Subject Classification:

41A10, 41A25, 41A36.

¹Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India, e-mails: pnappfma@gmail.com.

²Department of Mathematics, University of Prishtina ”Hasan Prishtina”, Prishtina, Kosovo, e-mail: behar.baxhaku@uni-pr.edu, corresponding author.

1. Introduction

Bernstein [4] gave a marvellous proof of the Weierstrass approximation theorem by defining a sequence of polynomials as follows:

(1)

\displaystyle\mathcal{B}_{m}(\varphi;z)=\sum_{\nu=0}^{m}q_{m,\nu}(z)\varphi%\Big{(}\tfrac{\nu}{m}\Big{)},\;\;\forall\;z\in I,\;m\in\mathbb{N},

where $q_{m,\nu}(z)={m\choose\nu}z^{\nu}(1-z)^{m-\nu},\;\;0\leq\nu\leq m$ , $I=[0,1]$ and $\varphi\in\mathcal{C}(I)$ , $\mathcal{C}(I):=\left\{\phi:\;\phi\;\mbox{is continuous on}\;I\right\}$ with the sup-norm $\|.\|_{C(I)}$ . Later, many researchers [8, 9, 17, 12, 11], etc. introduced new sequences of operators based on (1) and studied their approximation behaviour for functions in several function spaces. Ye et al. [16] proposed the following Bezier basis through a parameter $\lambda\in[-1,1]$ :

	$\displaystyle\tilde{q}_{m,0}(z)$	$\displaystyle=q_{m,0}(z)-\tfrac{\lambda}{m+1}q_{m+1,1}(z)$
	$\displaystyle\tilde{q}_{m,\nu}(z)$	$\displaystyle=q_{m,\nu}(z)+\lambda\Big{(}\tfrac{m-2\nu+1}{m^{2}-1}q_{m+1,\nu}(%z)-\tfrac{m-2\nu-1}{m^{2}-1}q_{m+1,\nu+1}(z)\Big{)},$
	$\displaystyle 1$	$\displaystyle\leq\nu\leq m-1$
(2)		$\displaystyle\tilde{q}_{m,m}(z)$	$\displaystyle=q_{m,m}(z)-\tfrac{\lambda}{m+1}q_{m+1,m}(z).$

In the particular case $\lambda=0$ , it is evident that (1) leads us to the Bernstein basis $q_{m,\nu}(z)$ , $\forall\;0\leq\nu\leq m$ . We remark here that addition of the parameter $\lambda$ provides better modeling flexibility to the basis (1). Cai et al. [6] generalized the operators (1) by involving the Bezier basis (1) in the following manner:

(3)

\displaystyle\mathfrak{B}_{m}(\varphi;z)=\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)%\varphi\Big{(}\tfrac{\nu}{m}\Big{)},\;\;\forall\;z\in I,\;m\in\mathbb{N}

and studied some direct approximation results.

For $\varphi\in L_{p}(I)$ , $p\geq 1$ , Acu et al. [1] presented a Kantorovich variant of the operators (3) as

(4)

\displaystyle\mathcal{L}_{m,\lambda}(\varphi;z)=(m+1)\sum_{\nu=0}^{m}\tilde{q}%_{m,\nu}(z)\int_{\frac{\nu}{m+1}}^{\frac{\nu+1}{m+1}}\varphi(u)du,

and studied a quantitative Voronovskaja type theorem by means of the Ditzian-Totik modulus of smoothness and a Grüss-Voronovskaja type theorem. Rahman et al. [14] investigated the convergence properties of a generalized case of (3) by shifting the nodes, for functions in $C(I)$ . Agrawal et al. [2] considered a two dimensional version of the operators defined in [14] and obtained the degree of approximation. Further, the authors [2] also examined the approximation behavior of the associated generalized boolean sum operators. Kumar [10] considered another generalization of (3) along the lines of [13] and discussed some direct theorems in the continuous functions space $C(I)$ . Aslan [3] derived the approximation properties for a new class of (3) in the univariate and the bivariate cases. Bodur et al. [5] discussed a Stancu type variant of (3) and established some results in local approximation.

Sucu and Ibikli [15] investigated the convergence of the Bernstein-Stancu- Kantorovich type operators in the spaces $C(I)$ and $L_{p}(I)$ , $p\geq 1$ . Encouraged by their work, our objective in this paper is to prove that $\mathcal{L}_{m,\lambda}(\varphi;z)$ converge to $\varphi(z)$ in the $L_{p}$ -norm, as $m\rightarrow\infty$ , $\forall\;\varphi\in L_{p}(I)$ , $p\geq 1$ and also estimate the approximation error via integral modulus of smoothness.

2. Preliminaries

In our study, the following results are needed:

Lemma 1 ([1]).

For all $m>2$ , there hold the inequalities:

\displaystyle\left|\mathcal{L}_{m,\lambda}(u-z;z)\right|\leq\left(\tfrac{1}{2(%m+1)}+\tfrac{|\lambda|}{m^{2}-1}\right);

and

(5)

\displaystyle\left|\mathcal{L}_{m,\lambda}((u-z)^{2};z)\right|\leq\left(\tfrac%{3m+4}{12(m+1)^{2}}+\tfrac{|\lambda|}{2(m^{2}-1)}\right)=\zeta(m,\lambda).

Theorem 2 ([1]).

For $\varphi\in C(I)$ , the operators (4) verify

\lim_{m\to\infty}\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\|_{C(I)}=0.

For $\varphi\in L_{p}(I)$ , $1\leq p<\infty$ , the integral modulus of smoothness is given by

\omega_{p}^{1}(\varphi;\delta)=\sup_{0<\eta\leq\delta}\|\varphi(.+\eta)-%\varphi(.)\|_{L_{p}(\Omega_{\eta})},

where $\|.\|_{L_{p}(\Omega_{\eta})}$ is the $L_{p}$ -norm over the interval $\Omega_{\eta}=[0,1-\eta]$ .
For $\varphi\in L_{p}(I)$ , $1\leq p<\infty$ , the Peetre’s $K$ - functional is defined as

\mathfrak{K}_{p}(\varphi;\delta)=\inf_{g\in W_{p}^{1}(I)}\Big{(}\|\varphi-g\|_%{L_{p}(I)}+\delta\|g^{\prime}\|_{L_{p}(I)}\Big{)},

where $W_{p}^{1}(I)=\left\{\psi\in L_{p}(I):\psi~{}\mbox{is absolutely continuous and%}\;{\psi}^{\prime}\in L_{p}(I)\right\}.$
It is well known [7, Thm. 2.4, p. 177] that for some positive constants $c_{1}$ and $c_{2}$ , there holds the following relation:

(6)

\displaystyle c_{1}\omega_{p}^{1}(\varphi;\delta)\leq\mathfrak{K}_{p}(\varphi;%\delta)\leq c_{2}\omega_{p}^{1}(\varphi;\delta).

Lemma 3.

For $\psi\in W_{p}^{1}(I),~{}p>1$ , we have

\|\mathcal{L}_{m,\lambda}(\psi)-\psi\|_{L_{p}(I)}\leq 2^{\frac{1}{p}}\left(%\tfrac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}\|{\psi}^{\prime}\|_{L_{p}(I)}.

where $\zeta(m,\lambda)$ is given by (5).

Proof.

For any $z\in I$ , we have

	$\displaystyle\left\|\mathcal{L}_{m,\lambda}(\psi;z)-\psi(z)\right\|$	$\displaystyle=(m+1)\left\|\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac{\nu}{%m+1}}^{\frac{\nu+1}{m+1}}\left(\psi(u)-\psi(z)\right)dz\right\|$
		$\displaystyle\leq(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac{\nu}{m+1%}}^{\frac{\nu+1}{m+1}}\left\|\int_{z}^{u}{\psi}^{\prime}(t)dt\right\|du$
(7)			$\displaystyle\leq\theta_{{\psi}^{\prime}}(z)(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,%\nu}(z)\int_{\frac{\nu}{m+1}}^{\frac{\nu+1}{m+1}}\|u-z\|dz,$

where

\theta_{{\psi}^{\prime}}(z)=\sup_{u\in I,u\neq z}\tfrac{1}{|u-z|}\left|\int_{z%}^{u}{\psi}^{\prime}(t)dt\right|,

is the Hardy-Littlewood majorant of ${\psi}^{\prime}$ . Now, applying the Cauchy-Schwarz inequality to (2) and Lemma 2.1 of [6], one gets

	$\displaystyle\left\|\mathcal{L}_{m,\lambda}(\psi;z)-\psi(z)\right\|\leq$
	$\displaystyle\leq\theta_{{\psi}^{\prime}}(z)\sqrt{(m+1)}\left(\sqrt{\sum_{\nu=%0}^{m}\tilde{q}_{m,\nu}(z)}\right)\times\left(\sqrt{\sum_{\nu=0}^{m}\tilde{q}_%{m,\nu}(z)\int_{\frac{\nu}{m+1}}^{\frac{\nu+1}{m+1}}(u-z)^{2}du}\right)$
(8)		$\displaystyle\leq\theta_{{\psi}^{\prime}}(z)\max_{z\in I}\left(\sqrt{\mathcal{%L}_{m,\lambda}((u-z)^{2};z)}\right)\leq\theta_{{\psi}^{\prime}}(z)\sqrt{\zeta(%m,\lambda)}.$

Using Hardy-Littlewood theorem (see [18]), one has

(9)

\displaystyle\int_{0}^{1}\left(\theta_{{\psi}^{\prime}}(z)\right)^{p}dz\leq 2%\left(\tfrac{p}{p-1}\right)^{p}\int_{0}^{1}|{\psi}^{\prime}(z)|^{p}dz,~{}p>1.

From (2) and (9), we get

\displaystyle\int_{0}^{1}\left|\mathcal{L}_{m,\lambda}(\psi;z)-\psi(z)\right|^%{p}dz\leq\left(\sqrt{\zeta(m,\lambda)}\right)^{p}\left\{2\left(\tfrac{p}{p-1}%\right)^{p}\int_{0}^{1}|{\psi}^{\prime}(z)|^{p}dz\right\}.

Hence,

\|\mathcal{L}_{m,\lambda}(\psi)-\psi\|_{L_{p}(I)}\leq 2^{\frac{1}{p}}\left(%\tfrac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}\|{\psi}^{\prime}\|_{L_{p}(I)}.

∎

3. Main Results

The following result shows that the operator (4) is an approximation method for functions in $L_{p}(I)$ .

Theorem 4.

For $\varphi\in L_{p}(I),~{}1\leq p<\infty$ , the operators (4), verify

\lim_{m\to\infty}\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\|_{L_{p}(I)}=0.

Proof.

By Luzin theorem, we know that for a given $\epsilon>0$ , $\exists$ a function $g\in C(I)$ satisfying

(10)

\displaystyle\|\varphi-g\|_{L_{p}(I)}<\epsilon.

From Theorem 2, we have

\lim_{m\to\infty}\|\mathcal{L}_{m,\lambda}(g)-g\|_{C(I)}=0,

hence for $\epsilon>0$ , $\exists$ an integer $m_{0}\in\mathbb{N}$ in such a way that

(11)

\displaystyle\|\mathcal{L}_{m,\lambda}(g)-g\|_{C(I)}<\epsilon,~{}\forall~{}m%\geq m_{0}.

Next, we show that $\exists$ a constant $M>0$ satisfying $\|\mathcal{L}_{m,\lambda}\|\leq M$ , for all $m\geq 2$ .
By Jensen’s inequality

	$\displaystyle\|\mathcal{L}_{m,\lambda}(\varphi;z)\|^{p}$	$\displaystyle\leq\bigg{\{}(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac%{\nu}{m+1}}^{\frac{\nu+1}{m+1}}\varphi(u)du\bigg{\}}^{p}$
		$\displaystyle\leq(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac{\nu}{m+1%}}^{\frac{\nu+1}{m+1}}\|\varphi(u)\|^{p}du.$

Hence,

	$\displaystyle\int_{0}^{1}\|\mathcal{L}_{m,\lambda}(\varphi;z)\|^{p}dz$	$\displaystyle\leq(m+1)\sum_{\nu=0}^{m}\left(\int_{0}^{1}\tilde{q}_{m,\nu}(z)dz%\right)\int_{\frac{\nu}{m+1}}^{\frac{\nu+1}{m+1}}\|\varphi(u)\|^{p}du$
		$\displaystyle\leq 2\\|\varphi\\|_{L_{p}(I)}^{p},\forall m\geq 2.$

Hence,

\|\mathcal{L}_{m,\lambda}(\varphi)\|_{L_{p}(I)}\leq 2^{1/p}\|\varphi\|_{L_{p}(%I)}.

Consequently, $\exists$ a constant $M>0$ satisfying

(12)

\displaystyle\|\mathcal{L}_{m,\lambda}\|\leq M,\;\forall m\geq 2.

Let us define $m^{\prime}=\max(m_{0},2)$ . Then in view of (10)–(12), we have

	$\displaystyle\\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\\|_{L_{p}(I)}$	$\displaystyle\leq\\|\mathcal{L}_{m,\lambda}(\varphi)\!\!-\!\!\mathcal{L}_{m,%\lambda}(g)\\|_{L_{p}(I)}\!\!+\!\!\|\mathcal{L}_{m,\lambda}(g)\!-\!g\\|_{C(I)}\!+%\!\\|\varphi-g\\|_{L_{p}(I)}$
		$\displaystyle\leq\left(\\|\mathcal{L}_{m,\lambda}\\|+1\right)\\|\varphi\!\!-\!\!g%\\|_{L_{p}(I)}+\|\mathcal{L}_{m,\lambda}(g)-g\\|_{C(I)}$
		$\displaystyle\leq(M+2)\epsilon,~{}\forall~{}m\geq m^{\prime}.$

Due to the arbitrariness of $\epsilon>0$ , the result follows. ∎

The next result yields the approximation degree for the operators (4) via integral modulus of smoothness.

Theorem 5.

For $\varphi\in L_{p}(I)$ , $p>1$ , the operators $\mathcal{L}_{m,\lambda}$ verify the following inequality

\displaystyle\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\|_{L_{p}(I)}\leq C%\omega_{p}^{1}(\varphi;\sqrt{\zeta(m,\lambda)}),

where $\zeta(m,\lambda)$ is given by (5) and the constant $C$ is independent of $\varphi$ and $m$ .

Proof.

From the proof of Theorem 4 and 3, we have

\|\mathcal{L}_{m,\lambda}(g)-g\|_{L_{p}(I)}\leq\begin{cases}3\|g\|_{L_{p}(I)},%&g\in L_{p}(I)\\[5.69054pt]2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}\|{g}^{\prime}%\|_{L_{p}(I)},&g\in W_{p}^{1}(I)\end{cases}.

Then for $\varphi\in L_{p}(I)$ and any $\psi\in W_{p}^{1}(I)$ , we may write

	$\displaystyle\\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\\|_{L_{p}(I)}$	$\displaystyle\leq\\|\mathcal{L}_{m,\lambda}(\varphi-\psi)-(\varphi-\psi)\\|_{L_{%p}(I)}+\\|\mathcal{L}_{m,\lambda}(\psi)-\psi\\|_{L_{p}(I)}$
		$\displaystyle\leq 3\left(\\|\varphi-\psi\\|_{L_{p}(I)}+2^{\frac{1}{p}}\left(%\tfrac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}\\|{\psi}^{\prime}\\|_{L_{p}(I)}%\right).$

Hence due to the arbitrariness of $\psi\in W_{p}^{1}(I)$ , we get

\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\|_{L_{p}(I)}\leq 3\mathfrak{K}\left%(\varphi;2^{\frac{1}{p}}\left(\tfrac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}%\right).

Finally using (6), we have

	$\displaystyle\\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\\|_{L_{p}(I)}$	$\displaystyle\leq 3c_{2}\omega_{p}^{1}\left(\varphi;2^{\frac{1}{p}}\left(%\tfrac{p}{p-1}\right)\sqrt{\zeta(m,\lambda)}\right)$
		$\displaystyle\leq 3c_{2}\left(1+2^{\frac{1}{p}}\left(\tfrac{p}{p-1}\right)%\right)\omega_{p}^{1}(\varphi;\sqrt{\zeta(m,\lambda)})$
		$\displaystyle\leq C\omega_{p}^{1}(\varphi;\sqrt{\zeta(m,\lambda)}),$

whence the result follows. ∎

Acknowledgements.

The authors are extremely grateful to the learned reviewer for the invaluable suggestions and comments leading to a considerable improvement in the presentation of the paper.

References

[1] A.M. Acu, N. Manav, D.F. Sofonea, Approximation properties of $\lambda$ -Kantorovich operators, J. Inequal. Appl., 202 2018, 2018.
[2] P.N. Agrawal, B. Baxhaku and R. Shukla, A new kind of bi-variate $\lambda$ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form, Math. Found. Comput., 5 (2022) no. 3, pp. 157–172.
[3] R. Aslan, Approximation properties of univariate and bivariate new class $\lambda$ -Bernstein–Kantorovich operators and its associated GBS operators, Comput. Appl. Math., 42, art. no. 34, 2023. https://doi.org/10.1007/s40314-022-02182-w
[4] S.N. Bernstein, Démonstration du théoréme de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkow, 2 (1912) no. 13, pp. 1–2.
[5] M. Bodur, N. Manav, F. Taşdelen, Approximation properties of $\lambda$ -Bernstein-Kantorovich-Stancu operators, Math. Slovaca, 72 (2022) no. 1, pp. 141–152. https://doi.org/10.1515/ms-2022-0010
[6] Q.B. Cai, B.Y. Lian, G. Zhou, Approximation properties of $\lambda$ -Bernstein operators, J. Inequal. Appl., 61 (2018), 2018.
[7] R.A. DeVore, G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg New York, 1993.
[8] Z. Ditzian, K. Ivanov, Bernstein-type operators and their derivatives, J. Approx. Theory, 56 (1989) no. 1, pp. 72–90. https://doi.org/10.1016/0021-9045(89)90134-2
[9] S. Guo, C. Li, X. Liu, Z. Song, Pointwise approximation for linear combinations of Bernstein operators, J. Approx. Theory, 107 (2000) no. 1, pp. 109–120. https://doi.org/10.1006/jath.2000.3504
[10] A. Kumar, Approximation properties of generalized $\lambda$ -Bernstein-Kantorovich type operators, Rend. Circ. Mat. Palermo, II. Ser, 70 (2021), pp. 505–520, https://doi.org/10.1007/s12215-020-00509-2
[11] M. Mursaleen, K.J. Ansari, A. Khan, On $(p,q)$ -analogue of Bernstein operators, Appl. Math. Comput., 266 (2015), pp. 874–882.
[12] G. Nowak, Approximation properties for generalized $q$ -Bernstein polynomials, J. Math. Anal. Appl., 350 (2009), pp. 50–55. https://10.1016/j.jmaa.2008.09.003
[13] M.A. Özarslan, O. Duman, Smoothness properties of modified Bernstein-Kantorovich operators, Numer. Funct. Anal. Optimiz., bf 37 (2016) no. 1, pp. 92–105. https://doi.org/10.1080/01630563.2015.1079219
[14] S. Rahman, M. Mursaleen, A.M. Acu, Approximation properties of $\lambda$ -Bernstein-Kantorovich operators with shifted knots, Math. Methods. Appl. Sci., 42 (2019), pp. 4042–4053. https://doi.org/10.1002/mma.5632
[15] S. Sucu, E. Ibikli, Approximation by means of Kantorovich-Stancu type operators, Numer. Funct. Anal. Optimiz., 34 (2013) no. 5, pp. 557–-575. https://doi.org/10.1080/01630563.2012.716806
[16] Z. Ye, X. Long, X.M. Zeng, Adjustment algorithms for Bézier curve and surface In: International Conference on Computer Science and Education, 2010, pp. 1712–1716.
[17] X.M. Zeng, F.F. Cheng, On the rates of approximation of Bernstein type operators, J. Approx. Theory, 109 (2001) no. 2, pp. 242–256. https://doi.org/10.1006/jath.2000.3538
[18] A. Zygmund, Trigonometric Series $I$ , $I I$ , Cambridge University Press, Cambridge, U.K., 1959.

	$\displaystyle\left\|\mathcal{L}_{m,\lambda}(\psi;z)-\psi(z)\right\|$	$\displaystyle=(m+1)\left\|\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac{\nu}{%m+1}}^{\frac{\nu+1}{m+1}}\left(\psi(u)-\psi(z)\right)dz\right\|$
		$\displaystyle\leq(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,\nu}(z)\int_{\frac{\nu}{m+1%}}^{\frac{\nu+1}{m+1}}\left\|\int_{z}^{u}{\psi}^{\prime}(t)dt\right\|du$
(7)			$\displaystyle\leq\theta_{{\psi}^{\prime}}(z)(m+1)\sum_{\nu=0}^{m}\tilde{q}_{m,%\nu}(z)\int_{\frac{\nu}{m+1}}^{\frac{\nu+1}{m+1}}\|u-z\|dz,$

	$\displaystyle\\|\mathcal{L}_{m,\lambda}(\varphi)-\varphi\\|_{L_{p}(I)}$	$\displaystyle\leq\\|\mathcal{L}_{m,\lambda}(\varphi)\!\!-\!\!\mathcal{L}_{m,%\lambda}(g)\\|_{L_{p}(I)}\!\!+\!\!\|\mathcal{L}_{m,\lambda}(g)\!-\!g\\|_{C(I)}\!+%\!\\|\varphi-g\\|_{L_{p}(I)}$
		$\displaystyle\leq\left(\\|\mathcal{L}_{m,\lambda}\\|+1\right)\\|\varphi\!\!-\!\!g%\\|_{L_{p}(I)}+\|\mathcal{L}_{m,\lambda}(g)-g\\|_{C(I)}$
		$\displaystyle\leq(M+2)\epsilon,~{}\forall~{}m\geq m^{\prime}.$

Convergence of $\lambda$ -Bernstein-Kantorovich operators
in the $L_{p}$ -norm

Abstract.

Key words and phrases:

2005 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

Lemma 1 ([1]).

Theorem 2 ([1]).

Lemma 3.

Proof.

3. Main Results

Theorem 4.

Proof.

Theorem 5.

Proof.

Acknowledgements.

References

Information

Indexing and abstracting

Publisher

Convergence of λ-Bernstein-Kantorovich operators in the Lp-norm

Abstract.

Key words and phrases:

2005 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

Lemma 1 ([1]).

Theorem 2 ([1]).

Lemma 3.

Proof.

3. Main Results

Theorem 4.

Proof.

Theorem 5.

Proof.

Acknowledgements.

References

Information

Indexing and abstracting

Publisher

Convergence of $\lambda$ -Bernstein-Kantorovich operators
in the $L_{p}$ -norm