Convergence of $\lambda$-Bernstein-Kantorovich operators
in the $L_p$-norm

Abstract.

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

1. Introduction

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

(1) ${ℬ}_m(\phi ;z)={\displaystyle \sum _{\nu =0}^m}{q}_{m,\nu }(z)\phi \left(\frac{\nu }{m}\right),\forall z\in I,m\in \mathbb{N},$

where $q_{m,\nu }(z)=\left(\genfrac{}{}{0pt}{}{m}{\nu }\right)z^{\nu }{(1-z)}^{m-\nu },\mathrm{\hspace{0.33em}\hspace{0.33em}0}\le \nu \le m$, $I=[0,1]$ and $\phi \in 𝒞(I)$, $𝒞(I):=\{\varphi :\varphi \text{is continuous on}I\}$ with the sup-norm $\parallel .{\parallel }_{C(I)}$. 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 $\lambda \in [-1,1]$:

${\tilde{q}}_{m,0}(z)$ $=q_{m,0}(z)-\frac{\lambda }{m+1}{q}_{m+1,1}(z)$

${\tilde{q}}_{m,\nu }(z)$ $=q_{m,\nu }(z)+\lambda \left(\frac{m-2\nu +1}{m^2-1}{q}_{m+1,\nu }(z)-\frac{m-2\nu -1}{m^2-1}{q}_{m+1,\nu +1}(z)\right),$

$1$ $\le \nu \le m-1$

(2) ${\tilde{q}}_{m,m}(z)$ $=q_{m,m}(z)-\frac{\lambda }{m+1}{q}_{m+1,m}(z).$

In the particular case $\lambda =0$, it is evident that (1) leads us to the Bernstein basis $q_{m,\nu }(z)$, $\forall \mathrm{\hspace{0.33em}0}\le \nu \le m$. 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) ${𝔅}_m(\phi ;z)={\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z)\phi \left(\frac{\nu }{m}\right),\forall z\in I,m\in \mathbb{N}$

and studied some direct approximation results.

For $\phi \in L_p(I)$, $p\ge 1$, Acu et al. [1] presented a Kantorovich variant of the operators (3) as

(4) ${ℒ}_{m,\lambda }(\phi ;z)=(m+1){\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}\phi (u)𝑑u,$

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 $L_p(I)$, $p\ge 1$. Encouraged by their work, our objective in this paper is to prove that ${ℒ}_{m,\lambda }(\phi ;z)$ converge to $\phi (z)$ in the $L_p$-norm, as $m\to \mathrm{\infty }$, $\forall \phi \in L_p(I)$, $p\ge 1$ 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:

$\left|{ℒ}_{m,\lambda }(u-z;z)\right|\le \left(\frac{1}{2(m+1)}+\frac{|\lambda |}{m^2-1}\right);$

and

(5) $\left|{ℒ}_{m,\lambda }({(u-z)}^2;z)\right|\le \left(\frac{3m+4}{12{(m+1)}^2}+\frac{|\lambda |}{2(m^2-1)}\right)=\zeta (m,\lambda ).$

Theorem 2 ([1]).

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

$$\underset{m\to \mathrm{\infty }}{lim}{\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{C(I)}=0.$$

For $\phi \in L_p(I)$, , the integral modulus of smoothness is given by

where $\parallel .{\parallel }_{L_p({\mathrm{\Omega }}_{\eta })}$ is the $L_p$-norm over the interval ${\mathrm{\Omega }}_{\eta }=[0,1-\eta ]$.
For $\phi \in L_p(I)$, , the Peetre’s $K$- functional is defined as

$${𝔎}_p(\phi ;\delta )=\underset{g\in W_p^1(I)}{inf}\left({\Vert \phi -g\Vert }_{L_p(I)}+\delta {\Vert g^{\prime }\Vert }_{L_p(I)}\right),$$

where $W_p^1(I)=\{\psi \in L_p(I):\psi \text{is absolutely continuous and}{\psi }^{\prime }\in L_p(I)\}.$
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) $c_1{\omega }_p^1(\phi ;\delta )\le {𝔎}_p(\phi ;\delta )\le c_2{\omega }_p^1(\phi ;\delta ).$

Lemma 3.

For $\psi \in W_p^1(I),p>1$, we have

$${\Vert {ℒ}_{m,\lambda }(\psi )-\psi \Vert }_{L_p(I)}\le 2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta (m,\lambda )}{\Vert {\psi }^{\prime }\Vert }_{L_p(I)}.$$

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

Proof.

For any $z\in I$, we have

$\left|{ℒ}_{m,\lambda }(\psi ;z)-\psi (z)\right|$ $=(m+1)\left|{\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}\left(\psi (u)-\psi (z)\right)𝑑z\right|$

$\le (m+1){\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}\left|{\displaystyle {\int }_z^u}{\psi }^{\prime }(t)𝑑t\right|𝑑u$

(7) $\le {\theta }_{{\psi }^{\prime }}(z)(m+1){\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}|u-z|𝑑z,$

where

$${\theta }_{{\psi }^{\prime }}(z)=\underset{u\in I,u\ne z}{sup}\frac{1}{|u-z|}\left|{\int }_z^u{\psi }^{\prime }(t)𝑑t\right|,$$

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

$\left|{ℒ}_{m,\lambda }(\psi ;z)-\psi (z)\right|\le$

$\le {\theta }_{{\psi }^{\prime }}(z)\sqrt{(m+1)}\left(\sqrt{{\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z)}\right)\times \left(\sqrt{{\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}{(u-z)}^2𝑑u}\right)$

(8) $\le {\theta }_{{\psi }^{\prime }}(z)\underset{z\in I}{\max }\left(\sqrt{{ℒ}_{m,\lambda }({(u-z)}^2;z)}\right)\le {\theta }_{{\psi }^{\prime }}(z)\sqrt{\zeta (m,\lambda )}.$

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

(9) ${\displaystyle {\int }_0^1}{\left({\theta }_{{\psi }^{\prime }}(z)\right)}^p𝑑z\le 2{\left(\frac{p}{p-1}\right)}^p{\displaystyle {\int }_0^1}{|{\psi }^{\prime }(z)|}^p𝑑z,p>1.$

From (2) and (9), we get

${\displaystyle {\int }_0^1}{\left|{ℒ}_{m,\lambda }(\psi ;z)-\psi (z)\right|}^p𝑑z\le {\left(\sqrt{\zeta (m,\lambda )}\right)}^p\left\{2{\left(\frac{p}{p-1}\right)}^p{\displaystyle {\int }_0^1}{|{\psi }^{\prime }(z)|}^p𝑑z\right\}.$

Hence,

$${\Vert {ℒ}_{m,\lambda }(\psi )-\psi \Vert }_{L_p(I)}\le 2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta (m,\lambda )}{\Vert {\psi }^{\prime }\Vert }_{L_p(I)}.$$

∎

3. Main Results

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

Theorem 4.

For , the operators (4), verify

$$\underset{m\to \mathrm{\infty }}{lim}{\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}=0.$$

Proof.

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

(10)

From Theorem 2, we have

$$\underset{m\to \mathrm{\infty }}{lim}{\Vert {ℒ}_{m,\lambda }(g)-g\Vert }_{C(I)}=0,$$

hence for $ϵ>0$, $\exists$ an integer $m_0\in \mathbb{N}$ in such a way that

(11)

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

${|{ℒ}_{m,\lambda }(\phi ;z)|}^p$ $\le {\left\{(m+1){\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}\phi (u)𝑑u\right\}}^p$

$\le (m+1){\displaystyle \sum _{\nu =0}^m}{\tilde{q}}_{m,\nu }(z){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}{|\phi (u)|}^p𝑑u.$

Hence,

${\displaystyle {\int }_0^1}{|{ℒ}_{m,\lambda }(\phi ;z)|}^p𝑑z$ $\le (m+1){\displaystyle \sum _{\nu =0}^m}\left({\displaystyle {\int }_0^1}{\tilde{q}}_{m,\nu }(z)𝑑z\right){\displaystyle {\int }_{\frac{\nu }{m+1}}^{\frac{\nu +1}{m+1}}}{|\phi (u)|}^p𝑑u$

$\le 2{\Vert \phi \Vert }_{L_p(I)}^p,\forall m\ge 2.$

Hence,

$${\Vert {ℒ}_{m,\lambda }(\phi )\Vert }_{L_p(I)}\le 2^{1/p}{\Vert \phi \Vert }_{L_p(I)}.$$

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

(12) $\Vert {ℒ}_{m,\lambda }\Vert \le M,\forall m\ge 2.$

Let us define $m^{\prime }=\max (m_0,2)$. Then in view of (10)–(12), we have

${\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}$ $\le {\Vert {ℒ}_{m,\lambda }(\phi )-{ℒ}_{m,\lambda }(g)\Vert }_{L_p(I)}+{|{ℒ}_{m,\lambda }(g)-g\Vert }_{C(I)}+{\Vert \phi -g\Vert }_{L_p(I)}$

$\le \left(\Vert {ℒ}_{m,\lambda }\Vert +1\right){\Vert \phi -g\Vert }_{L_p(I)}+{|{ℒ}_{m,\lambda }(g)-g\Vert }_{C(I)}$

$\le (M+2)ϵ,\forall m\ge m^{\prime }.$

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

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

Theorem 5.

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

${\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}\le C{\omega }_p^1(\phi ;\sqrt{\zeta (m,\lambda )}),$

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

Proof.

From the proof of Theorem 4 and 3, we have

Then for $\phi \in L_p(I)$ and any $\psi \in W_p^1(I)$, we may write

${\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}$ $\le {\Vert {ℒ}_{m,\lambda }(\phi -\psi )-(\phi -\psi )\Vert }_{L_p(I)}+{\Vert {ℒ}_{m,\lambda }(\psi )-\psi \Vert }_{L_p(I)}$

$\le 3\left({\Vert \phi -\psi \Vert }_{L_p(I)}+2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta (m,\lambda )}{\Vert {\psi }^{\prime }\Vert }_{L_p(I)}\right).$

Hence due to the arbitrariness of $\psi \in W_p^1(I)$, we get

$${\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}\le 3𝔎(\phi ;2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta (m,\lambda )}).$$

Finally using (6), we have

${\Vert {ℒ}_{m,\lambda }(\phi )-\phi \Vert }_{L_p(I)}$ $\le 3c_2{\omega }_p^1(\phi ;2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\sqrt{\zeta (m,\lambda )})$

$\le 3c_2\left(1+2^{\frac{1}{p}}\left(\frac{p}{p-1}\right)\right){\omega }_p^1(\phi ;\sqrt{\zeta (m,\lambda )})$

$\le C{\omega }_p^1(\phi ;\sqrt{\zeta (m,\lambda )}),$

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.