Approximation of some classes of functions by Landau type operators

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(original), Approximation Theory, ISI/JCR, Linear and positive approximation operators, paper

Abstract

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in $L_p$ spaces and in weighted $L_p$ spaces (1≤p<∞). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness.

Authors

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Ali Aral
Kirikkale University, Turkey

Keywords

Landau operator; weighted space; Korovkin theorem, modulus of smoothness

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Cite this paper as:

O. Agratini, A. Aral, Approximation of some classes of functions by Landau type operators, Results in Mathematics, 76 (2021) art. no. 12, https://doi.org/10.1007/s00025-020-01319-9

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Results in Mathematics

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Springer

DOI

https://doi.org/10.1007/s00025-020-01319-9

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1422-6383

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1420-9012

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Approximation of some classes of functions by Landau type operators

Octavian Agratini Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
and
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania agratini@math.ubbcluj.ro and Ali Aral Kirikkale University, Department of Mathematics,
71450 Yahşihan, Kirrikale, Turkey aliaral73@yahoo.com

Abstract.

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in $L_{p}$ spaces and in weighted $L_{p}$ spaces $(1\leq p<\infty)$ . Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness.

Keywords and phrases: Landau operator, weighted space, Korovkin theorem, modulus of smoothness.

Mathematics Subject Classification: 41A36, 41A25.

1. Introduction

A strongly rooted part of Approximation Theory is the approximation of various signals by linear and positive operators. Over time, many mathematicians have begun to generalize and modify classic processes of this type, thus providing a deeper study of them. This note follows this path, investigating a generalization of Landau operators. In [11, Eq. (2)], Edmund Landau proved the following relation

\lim\limits_{n\to\infty}\displaystyle\frac{\displaystyle\int_{0}^{1}f(\xi)(1-(% \xi-x)^{2})^{n}d\xi}{2\displaystyle\int_{0}^{1}(1-u^{2})^{n}du}=f(x),

the convergence being uniform over an interval $[a,b]$ , where $0<a<b<1$ . However, as is stated in the paper, the function $f$ can be extended by continuity on a larger interval. Further we point out some results obtained about Landau’s operators without claiming to include all important achievements related to these operators.

In [10], Jackson obtained their order of approximation. Sikkema [17, p. 44] proved a qualitative Voronovskaja result. A discrete version of this class of operators was studied by Gao [8]. Pendina [14] investigated iterations of Landau’s operators. Generalizations of operators in different forms have been introduced. Approximation by Landau operators relative to the Hausdorff metric appeared in [18]. Mamedov [12] presented the following generalization

(M_{n}f)(x)=\displaystyle\frac{kn^{1/(2k)}}{\Gamma(1/(2k))}\int_{0}^{1}f(t)(1-% (t-x)^{2k})^{n}dt,\ x\in[0,1],

where $n\in\mathbb{N}$ is fixed and $\Gamma$ stands for Gamma function. The deeper investigation of $M_{n}$ , $n\in\mathbb{N}$ , operators was continued by the same author in [13]. Two other generalizations were constructed by Chen and Shih [4]. Gal and Iancu [7] replaced in the classical operators, the usual integral by the nonlinear Choquet integral.

Another generalization of Landau operators was proposed in [1]. These new operators reproduce the affine functions, a less common feature of integral type operators. The approximation properties were studied in two spaces, namely in $C(\mathbb{R})$ , the space of real valued continuous functions on $\mathbb{R}$ , and in the weighted space $B_{\rho_{m}}(\mathbb{R})$ , where

B_{\rho_{m}}(\mathbb{R})=\{f:\mathbb{R}\to\mathbb{R}\mid\ |f(x)|\leq M_{f}\rho% _{m}(x),\ x\in\mathbb{R}\},

with the polynomial weight $\rho_{m}(x)=1+x^{2m}$ , $m\in\mathbb{N}$ fixed.

In this paper we pursue two aims.

The first purpose is to continue the investigation of the operators introduced in [1] by their study in $L_{p}(\mathbb{R})$ , $1\leq p<\infty$ , normed spaces which play a central role in many questions in analysis. The special importance of them derive from the fact that they offer a partial but useful generalization of the fundamental $L_{2}(\mathbb{R})$ space of square integrable functions. The second purpose is to extend the operators for vector functions, this assuming new challenges for error evaluation.

The architecture of this article still includes three paragraphs, in the first we present the class of operators and in the other two we investigate the aspects depicted above.

2. Preliminaries

Trying to realize a self-contained exposure, we present the operators introduced in [1], specifing the meaning of the notations as well as some formulas.

Let $\alpha\geq 1$ be fixed and $\lambda=1/(2\alpha)\in(0,1/2]$ . For each $n\in\mathbb{N}$ we set

a_{n}=\int_{0}^{1}(1-y^{2\alpha})^{n}dy\quad\mbox{and}\quad\tau_{p,n}=% \displaystyle\frac{B(n+1,p\lambda)}{B(n+1,\lambda)},\ p\in\mathbb{N},

(2.1)

where $B$ stands for Beta function. Using elementary calculations we get

a_{n}=\lambda B(n+1,\lambda),

\int_{0}^{1}y^{i}(1-y^{2\alpha})^{n}dy=\lambda B(n+1,(1+i)\lambda).

The sequences $(a_{n})_{n\geq 1}$ and $(\tau_{p,n})_{n\geq 1}$ for $p\geq 2$ , are strictly decreasing. Moreover, for $p\geq 2$ ,

\lim\limits_{n\to\infty}\tau_{p,n}=0.

(2.2)

We consider the linear and positive operators defined as follows

	$\displaystyle(L_{n}f)(x)$	$\displaystyle=\displaystyle\frac{1}{a_{n}}\int_{0}^{1}f(x-y+\tau_{2,n})(1-y^{2% \alpha})^{n}dy$		(2.3)
		$\displaystyle=\displaystyle\frac{1}{a_{n}}\int_{x-1}^{x}f(t+\tau_{2,n})(1-(x-t% )^{2\alpha})^{n}dt,\ x\in\mathbb{R},$

where $f$ is Lebesgue measurable on the domain.

The behavior of the operators on the Korovkin type test functions $e_{j}$ , $e_{j}(x)=x^{j}$ for $j\in\{0,1,2\}$ , is described by the following relations

L_{n}e_{0}=e_{0},\quad L_{n}e_{1}=e_{1},\quad L_{n}e_{2}=e_{2}+\tau_{3,n}-\tau% _{2,n}^{2}.

(2.4)

Further, we focus on the study of approximation properties in unweighted and weighted $L_{p}$ spaces, $1\leq p<\infty$ .

3. Approximation in $L_{p}$ spaces

As usual, we denote the norm of the $L_{p}$ space, $1\leq p<\infty$ , by $\|\cdot\|_{L_{p}}$ .

Lemma 3.1.

Let $L_{n}$ , $n\in\mathbb{N}$ , be defined by (2.3). If $f\in L_{p}(\mathbb{R})$ , where $1\leq p<\infty$ , then we get

\|L_{n}f\|_{L_{p}}\leq\|f\|_{L_{p}}.

Proof. We can write successively

	$\displaystyle\\|L_{n}f\\|_{L_{p}}$	$\displaystyle=\left(\int_{\mathbb{R}}\|(L_{n}f)(x)\|^{p}dx\right)^{1/p}$
		$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int_{0% }^{1}f(x-y+\tau_{2,n})(1-y^{2\alpha})^{n}dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\|f(x-y+\tau_{2,n}\|^{p}dx\right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\\|f\\|_{L_{p}}\ \displaystyle\frac{1}{a_{n}}\int_{0}^{1}(1-y^{% 2\alpha})^{n}dy$
		$\displaystyle=\\|f\\|_{L_{p}},$

see (2.1). $\square$

The above property says that our operators are non-expansive on the spaces $L_{p}(\mathbb{R})$ .

A subtle measurement of the error of approximation is provided by modulus of smoothness. Let $T_{h}$ , $h\in\mathbb{R}$ , denote the translation operator, $T_{h}(f;x)=f(x+h)$ , and $\Delta_{h}:=T_{h}-I$ the difference operator, where $I$ is the identity operator. The first modulus of smoothness of $f\in L_{p}(\mathbb{R})$ is defined by

\omega(f;\delta)_{p}=\sup_{0\leq h\leq\delta}\|\Delta_{h}(f;\cdot)\|_{L_{p}},% \ \delta\geq 0.

Among its properties, see, e.g., the monograph [5, Chapter 2, §7], we recall: it is a continuous and increasing function of $\delta$ with $\omega(f;0)_{p}=0$ . It also takes place

\omega(f;\lambda\delta)_{p}\leq(\lambda+1)\omega(f;\delta)_{p},\ \lambda>0.

(3.1)

Actually, $\omega(\cdot;\delta)_{p}$ meets the conditions that ensure it is a seminorm on $L_{p}(\mathbb{R})$ , $p\geq 1$ .

Theorem 3.1.

Let $L_{n}$ , $n\in\mathbb{N}$ , be defined by (2.3). If $f\in L_{p}(\mathbb{R})$ for some $1\leq p<\infty$ , then the following relation

\|L_{n}f-f\|_{L_{p}}\leq 2\omega\left(f;\sqrt{\tau_{3,n}-\tau_{2,n}^{2}}\right% )_{p}

holds, where $\tau_{2,n}$ and $\tau_{3,n}$ are defined at (2.1).

Proof. Since the operators reproduce the constants, see (2.4), we have

(L_{n}f)(x)-f(x)=\displaystyle\frac{1}{a_{n}}\int_{0}^{1}(f(x-y+\tau_{2,n})-f(% x))(1-y^{2\alpha})^{n}dy.

(3.2)

We deduce

	$\displaystyle\\|L_{n}f\!-\!f\\|_{L_{p}}$	$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int_{0% }^{1}(f(x-y+\tau_{2,n})-f(x))(1-y^{2\alpha})^{n}dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|f(x-y+\tau_{2,n})-f(x)\right\|^{p}dx\right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\omega(f;\|\tau_{2,n}-% y\|)_{p}(1-y^{2\alpha})^{n}dy.$

The first increase represents Minkowski’s integral inequality for two measure spaces, see [9, Problem 202]. The inversions of the order of integration are justified by Fubini’s Theorem.

Based on (3.1), for any $\delta>0$ we can write

	$\displaystyle\\|L_{n}f-f\\|_{L_{p}}$	$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\omega(f;\delta)_{p}\int_{0}^{1}% \left(1+\displaystyle\frac{\|\tau_{2,n}-y\|}{\delta}\right)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\omega(f;\delta)_{p}\left(1+\displaystyle\frac{1}{\delta}% \left(\displaystyle\frac{1}{a_{n}}\int_{0}^{1}(\tau_{2,n}-y)^{2}(1-y^{2\alpha}% )^{n}dy\right)^{1/2}\right)$
		$\displaystyle\leq\omega(f;\delta)_{p}\left(1+\displaystyle\frac{1}{\delta}(% \tau_{3,n}-\tau_{2,n}^{2})^{1/2}\right).$

We used both Cauchy-Schwarz inequality for integrals and the identity

\displaystyle\frac{1}{a_{n}}\int_{0}^{1}(\tau_{2,n}-y)^{2}(1-y^{2\alpha})^{n}% dy=(L_{n}e_{2})(0),

see (2.4).

Taking in view the relations (2.2), the choice of $\delta=(\tau_{3,n}-\tau_{2,n}^{2})^{1/2}$ is appropriate and the conclusion of the theorem is completed. $\square$

Next, we explore the approximation properties of the operators in some weighted spaces $L_{p}$ . For a given $p\in[1,\infty)$ , we consider the weight

\nu:\mathbb{R}\to(0,1],\ \nu(x)=(1+x^{2m})^{-p},

where $m>1$ is a fixed integer. We denote by $L_{p,\nu}(\mathbb{R})$ the linear space of $p$ -absolutely integrable functions on $\mathbb{R}$ with respect to the weight $\nu$ , i.e., for $1\leq p<\infty$ ,

L_{p,\nu}(\mathbb{R})=\{f:\mathbb{R}\to\mathbb{R}:\ f\nu^{\frac{1}{p}}\in L_{p% }(\mathbb{R})\}

endowed with the norm $\|\cdot\|_{p,\nu}$ defined as follows

\|f\|_{p,\nu}=\|f\nu^{\frac{1}{p}}\|_{L_{p}}=\left(\int_{\mathbb{R}}|f(t)|^{p}% \nu(t)dt\right)^{1/p}.

Lemma 3.2.

Each operator $L_{n}$ defined by (2.3) maps $L_{p,\nu}(\mathbb{R})$ into $L_{p,\nu}(\mathbb{R})$ .

Proof. Let $n\in\mathbb{N}$ be fixed. It is enough to show that

\|L_{n}f\|_{p,\nu}\leq M\|f\|_{p,\nu},\ f\in L_{p,\nu},

(3.3)

where $M$ is a positive constant.

Using again Minkowski’s inequality, we can write

	$\displaystyle\\|L_{n}f\\|_{p,\nu}$	$\displaystyle=\left(\int_{\mathbb{R}}\left\|\displaystyle\frac{(L_{n}f)(x)}{1+x% ^{2m}}\right\|^{p}dx\right)^{1/p}$
		$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\displaystyle% \frac{1}{(1+x^{2m})^{p}}\left\|\int_{0}^{1}f(x-y+\tau_{2,n})(1-y^{2\alpha})^{n}% dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|\displaystyle\frac{f(x-y+\tau_{2,n})}{1+x^{2m}}\right\|^{p}dx\right)^{1% /p}(1-y^{2\alpha})^{n}dy.$

Appealing to the inequality

1+(x-y+\tau_{2,n})^{2m}\leq 2^{2m-1}(1+x^{2m})(1+(\tau_{2,n}-y)^{2m}),

true for any $m\in\mathbb{N}$ , we can continue to write

	$\displaystyle\\|L_{n}f\\|_{p,\nu}$	$\displaystyle\leq\\|f\\|_{p,\nu}\ \displaystyle\frac{2^{2m-1}}{a_{n}}\int_{0}^{1% }(1+(\tau_{2,n}-y)^{2m})(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\\|f\\|_{p,\nu}\ \displaystyle\frac{2^{2m-1}}{a_{n}}\int_{0}^{1% }\left(1+\sum_{j=0}^{2m}\displaystyle\binom{2m}{j}\tau_{2,n}^{j}\right)(1-y^{2% \alpha})^{n}dy.$

Considering the relations (2.1), we deduce $\tau_{2,n}<1/2$ and choosing in (3.3) the constant $M=(2^{2m}+3^{2m})/2$ , the proof of our lemma is finished. $\square$

Theorem 3.2.

Let $L_{n}$ , $n\in\mathbb{N}$ , be defined by (2.3). For every $f\in L_{p,\nu}(\mathbb{R})$

\lim\limits_{n\to\infty}\|L_{n}f-f\|_{p,\nu}=0

(3.4)

holds.

Proof. The demonstration is based on a Korovkin type theorem for a general weighted space $L_{p,\omega}(\mathbb{R})$ established in [6, Theorem 1]. The proved theorem says: Let $(\Lambda_{n})_{n\in\mathbb{N}}$ be a uniformly bounded sequence of positive linear operators from $L_{p,\omega}(\mathbb{R})$ into $L_{p,\omega}(\mathbb{R})$ , $1\leq p<\infty$ . We assume that

\lim\limits_{n\to\infty}\|\Lambda_{n}e_{i}-e_{i}\|_{p,\omega}=0,\ i=0,1,2,

(3.5)

where the weight $\omega$ is a positive continuous function on $\mathbb{R}$ satisfying the condition

\int_{\mathbb{R}}t^{2p}\omega(t)dt<\infty.

(3.6)

Then, for every $f\in L_{p,\omega}(\mathbb{R})$ , we have

\lim\limits_{n\to\infty}\|\Lambda_{n}f-f\|_{p,\omega}=0.

We consider a particular case of the weight $\omega$ , namely $\omega:=\nu$ . Since $p\geq 1$ and $m>1$ , this weight function satisfies condition (3.6). Also, Lemma 3.2 ensures that our operators $L_{n}\equiv\Lambda_{n}$ , $n\in\mathbb{N}$ , are uniformly bounded. Using (2.4), the conditions of (3.5) are obviously fulfilled for $i=0$ and $i=1$ . Knowing the expression of the $L_{n}e_{2}$ function, we get

\|L_{n}e_{2}-e_{2}\|_{p,\nu}=|\tau_{3,n}-\tau_{2,n}^{2}|\ \|e_{0}\|_{p,\nu}.

Taking in view the relation (2.2), condition (3.5) also takes place for $i=3$ . The proof of the identity (3.4) is ended. $\square$

Moduli of smoothness have become standard instrument to estimate errors of approximation. Yuksel and Ispir [20] proposed a weighted modulus of the following form

(f,\delta)\to\sup_{\begin{subarray}{c}x\in\mathbb{R}\\ 0\leq h\leq\delta\end{subarray}}\displaystyle\frac{|f(x+h)-f(x)|}{1+(x+h)^{2m}% },\ \delta\geq 0,

for any function $f$ satisfying a polynomial growth, more precisely

|f(x)|\leq M_{f}(1+x^{2m}),\ x\in\mathbb{R},

$M_{f}$ being a positive constant depending only on $f$ . Starting from this construction, we define a weighted modulus of smoothness $\omega_{p,m}$ as follows

\omega_{p,m}(f;\delta)=\sup_{0\leq h\leq\delta}\left(\int_{\mathbb{R}}\left|% \displaystyle\frac{f(x+h)-f(x)}{1+(|x|+h)^{2m}}\right|^{p}dx\right)^{1/p},\ % \delta\geq 0,

(3.7)

for any function $f\in L_{p,\nu}(\mathbb{R})$ . We show that this modulus satisfies some classical properties of $L_{p}$ -modulus. Clearly, $\omega_{p,m}(f;0)=0$ and it is a monotone increasing function with respect to $\delta$ .

Also, for any $\delta\geq 0$ and any positive integer $m$ ,

\omega_{p,m}(f;\delta)\leq\|f\|_{p,\nu}(1+2^{2m-1}(1+\delta^{2m})),

takes place. The proof is simple, based on Minkowski’s inequality and on the following increase

\displaystyle\frac{1+x^{2m}}{1+(|x|+h)^{2m}}\leq 2^{2m-1}(1+\delta^{2m})\mbox{% for }x\in\mathbb{R}\mbox{ and }0\leq h\leq\delta.

(3.8)

Lemma 3.3.

Let $\omega_{p,m}(f;\cdot)$ be defined by (3.7). For any non-negative real numbers $\delta$ and $c$ the following relation

\omega_{p,m}(f;c\delta)\leq(c+1)\omega_{p,m}(f;\delta),\ f\in L_{p,\nu}(% \mathbb{R}),

(3.9)

holds.

Proof. For $\delta=0$ or $c=0$ the statement is evident. Further, we consider $\delta>0$ and $c>0$ .

At first step, for any positive integer $n$ we can write

$\displaystyle\omega_{p,m}(f;n\delta)$	$\displaystyle=\sup_{0\leq h\leq n\delta}\left(\int_{\mathbb{R}}\left\|% \displaystyle\frac{f(x+h)-f(x)}{1+(\|x\|+h)^{2m}}\right\|^{p}dx\right)^{1/p}$
	$\displaystyle=\sup_{0\leq h^{\prime}\leq\delta}\left\\|\sum_{k=1}^{n}% \displaystyle\frac{f(\cdot+kh^{\prime})-f(\cdot+(k-1)h^{\prime})}{1+(\|e_{1}\|+% nh^{\prime})^{2m}}\right\\|_{p}$
	$\displaystyle\leq\omega_{p,m}(f;\delta)\sum_{k=1}^{n}\displaystyle\frac{1+(\|x\|% +h^{\prime}+(k-1)h^{\prime})^{2m}}{1+(\|x\|+nh^{\prime})^{2m}}$
	$\displaystyle\leq n\omega_{p,m}(f;\delta).$	(3.10)

In the above we replaced $h$ with $nh^{\prime}$ . Further, for an arbitrary $c>0$ , using both the monotonicity of $\omega_{p,m}(f;\cdot)$ and inequality established at (3), we get

\omega_{p,m}(f;c\delta)\leq\omega_{p,m}(f;([c]+1)\delta)\leq([c]+1)\omega_{p,m% }(f;\delta)\leq(c+1)\omega_{p,m}(f;\delta),

where $[c]$ indicates the integer part of $c$ . We arrived at (3.9). $\square$

Lemma 3.4.

Let $\omega_{p,m}(f;\cdot)$ be defined by (3.7). It takes place

\lim\limits_{\delta\to 0^{+}}\omega_{p,m}(f;\delta)=0,\ f\in L_{p,\mu}(\mathbb% {R}).

(3.11)

Proof. For a positive real number $a$ , let $\chi_{1}^{a}$ , $\chi_{2}^{a}$ , $\chi^{a}$ be the characteristic functions of the intervals $[a,\infty)$ , $(-\infty,a]$ , $[-a,a]$ , respectively.

Since $f\in L_{p,\nu}(\mathbb{R})$ , for each $\varepsilon>0$ there exists $a>0$ large enough such that

\left(\int_{-\infty}^{-a}\left|\displaystyle\frac{f(x)}{1+x^{2m}}\right|^{p}dx% \right)^{1/p}+\left(\int_{a}^{\infty}\left|\displaystyle\frac{f(x)}{1+x^{2m}}% \right|^{p}\right)^{1/p}<\displaystyle\frac{\varepsilon}{2^{4m+3}}.

This inequality can be written $\|f\chi_{2}^{-a}\|_{p,\nu}+\|f\chi_{1}^{a}\|_{p,\nu}<2^{-(4m+3)}\varepsilon.$

For $\delta>0$ , this relation implies

\|f\chi_{2}^{-(a+\delta)}\|_{p,\nu}+\|f\chi_{1}^{a+\delta}\|_{p,\nu}<% \displaystyle\frac{\varepsilon}{2^{4m+3}}.

Considering $0\leq h\leq\delta$ , the above inequality involves

\|f(\cdot+h)\chi_{2}^{-(a+\delta)}\|_{p,\nu}+\|f(\cdot+h)\chi_{1}^{a+\delta}\|% _{p,\nu}<2^{2m-1}(1+\delta^{2m})\displaystyle\frac{\varepsilon}{2^{4m+3}}.

We used again inequalities shown in (3.8).

Taking $\delta\leq 1$ we can write

\|f(\cdot+h)\chi_{2}^{-(a+\delta)}\|_{p,\nu}+\|f(\cdot+h)\chi_{1}^{a+\delta}\|% _{p,\nu}<\displaystyle\frac{\varepsilon}{2^{2m+3}}.

Thus, we have

$\displaystyle\!\!\!\!\omega_{p,m}(f;\delta)$	$\displaystyle\leq 2^{2m-1}(1+\delta^{2m})^{2}\sup_{0\leq h\leq\delta}\left\\|% \displaystyle\frac{(f(\cdot+h)-f)\chi^{a+\delta}}{1+h^{2m}}\right\\|_{p,\nu}$
	$\displaystyle+2^{2m-1}(1+\delta^{2m})^{2}\displaystyle\frac{\varepsilon}{2^{2m% +2}}$
	$\displaystyle\leq 2^{2m+1}\sup_{0\leq h\leq\delta}\left\\|\displaystyle\frac{(f% (\cdot+h)-f)\chi^{a+\delta}}{1+h^{2m}}\right\\|_{p,\nu}+\displaystyle\frac{% \varepsilon}{2},\ 0<\delta\leq 1.$	(3.12)

By the Weierstrass theorem, there exists a sequence $(\varphi_{n})_{n\geq 1}$ , where $\varphi_{n}\in C([-a-2\delta,a+2\delta])$ , such that

\lim\limits_{n\to\infty}\|(f-\varphi_{n})\chi^{a+2\delta}\|_{p,\nu}=0.

That is, for a given $\varepsilon>0$ , there exists a rank $n_{0}\in\mathbb{N}$ such that

\|(f-\varphi_{n})\chi^{a+2\delta}\|_{p,\nu}<\displaystyle\frac{\varepsilon}{2^% {4m+4}},

(3.13)

whenever $n\geq n_{0}$ and $\delta>0$ . Thus, we get

\|(f(\cdot+h)-\varphi_{n}(\cdot+h))\chi^{a+\delta}\|_{p,\nu}\leq 2^{2m}\|(f-% \varphi_{n})\chi^{a+2\delta}\|_{p,\nu}<\displaystyle\frac{\varepsilon}{2^{2m+4}}

(3.14)

for $n\geq n_{0}$ .

Applying the Minkowski’s inequality yields

	$\displaystyle\left\\|\displaystyle\frac{(f(\cdot+h)-f)\chi^{a+\delta}}{1+h^{2m}% }\right\\|_{p,\nu}$	$\displaystyle\!\leq\\|(f(\cdot+h)-\varphi_{n}(\cdot+h))\chi^{a+\delta}\\|_{p,\nu}$
		$\displaystyle+\\|(\varphi_{n}(\cdot+h)-\varphi_{n})\chi^{a+\delta}\\|_{p,\nu}\!+% \!\\|(\varphi_{n}\!-\!f)\chi^{a+\delta}\\|_{p,\nu}.$

From (3.13) and (3.14) it follows that

\sup_{0\leq h\leq\delta}\left\|\displaystyle\frac{(f(\cdot+h)-f)\chi^{a+\delta% }}{1+h^{2m}}\right\|_{p,\nu}<\displaystyle\frac{\varepsilon}{2^{2m+3}}+\sup_{0% \leq h\leq\delta}\|(\varphi_{n}(\cdot+h)-\varphi_{n})\chi^{a+\delta}\|_{p,\nu}

(3.15)

for $n\geq n_{0}$ and $\delta>0$ .

Since $\varphi_{n}$ , $n\in\mathbb{N}$ , are continuous functions on a compact interval, based on Cantor’s theorem, they are uniformly continuous. Consequently, for $0\leq h\leq\delta$ and $n\geq n_{0}$ we can write

|\varphi_{n}(x+h)-\varphi_{n}(x)|\leq\displaystyle\frac{\varepsilon}{2^{2m+3}% \|\chi^{a+\delta}\|_{p,\nu}},

where $x\in[-a-\delta,a+\delta]$ . Rhus, we obtain

\sup_{0\leq h\leq\delta}\|(\varphi_{n}(\cdot+h)-\varphi_{n})\chi^{a+\delta}\|_% {p,\nu}\leq\displaystyle\frac{\varepsilon}{2^{2m+3}}

and, from (3.15) we get

\sup_{0\leq h\leq\delta}\left\|\displaystyle\frac{(f(\cdot+h)-f)\chi^{a+\delta% }}{1+h^{2m}}\right\|_{p,\nu}\leq\displaystyle\frac{\varepsilon}{2^{2m+2}}.

Returning at (3), the following inequality $\omega_{p,m}(f;\delta)<\varepsilon$ is true for $0<\delta<1$ , which leads us to the statement (3.11). $\square$

Theorem 3.3.

Let $L_{n}$ , $n\in\mathbb{N}$ , be defined by (2.3). For every $f\in L_{p,\nu}(\mathbb{R})$

\|L_{n}f-f\|_{p,\nu}\leq 2\omega_{p,m}\left(f;\sqrt{\tau_{3,n}-\tau_{2,n}^{2}}\right)

(3.16)

holds, where $\omega_{p,m}$ is given by (3.7) and $\tau_{2,n}$ , $\tau_{3,n}$ are defined at (2.1).

Proof. Starting from relation (3.2), we have

	$\displaystyle\\|L_{n}f\!-\!f\\|_{p,\nu}$	$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int% _{0}^{1}\displaystyle\frac{f(x-y+\tau_{2,n})-f(x)}{1+x^{2m}}(1-y^{2\alpha})^{n% }dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|\displaystyle\frac{f(x-y+\tau_{2,n})-f(x)}{1+x^{2m}}\right\|^{p}dx% \right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\omega_{p,m}(f;\|\tau_% {2,n}-y\|)(1-y^{2\alpha})^{n}dy.$

From this point the proof runs similarly to that of Theorem 3.1.

By using (3.9) and taking $c=|\tau_{2,n}-y|/\delta$ , $\delta>0$ , we get

	$\displaystyle\\|L_{n}f-f\\|_{p,\nu}$	$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\omega_{p,m}(f;\delta)\int_{0}^{1% }\left(1+\displaystyle\frac{\|\tau_{2,n}-y\|}{\delta}\right)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\omega_{p,m}(f;\delta)\left(1+\displaystyle\frac{1}{\delta}% \left(\displaystyle\frac{1}{a_{n}}\int_{0}^{1}(\tau_{2,n}-y)^{2}(1-y^{2\alpha}% )^{n}dy\right)^{1/2}\right)$
		$\displaystyle\leq\omega_{p,m}(f;\delta)\left(1+\displaystyle\frac{1}{\delta}% \sqrt{\tau_{3,n}-\tau_{2,n}^{2}}\right)$

and the relation (3.16) follows. $\square$

In the final part of this section we make a stop over global smoothness preservation property.

Theorem 3.4.

Let $L_{n}$ , $n\in\mathbb{N}$ , be defined by (2.3) and let $\omega_{p,m}$ be given by (3.7). For every $f\in L_{p,\nu}(\mathbb{R})$ and $\delta>0$

\omega_{p,m}(L_{n}f,\delta)\leq 2^{2m-1}\left(1+\sqrt{\tau_{3,n}-\tau_{2,n}^{2% }}\right)\omega_{p,m}(f;\delta).

(3.17)

Proof. By identity (3.2) we get

	$\displaystyle J_{h}$	$\displaystyle:=\left(\int_{\mathbb{R}}\left\|\displaystyle\frac{L_{n}(f;x+h)-L_% {n}(f;x)}{1+(\|x\|+h)^{2m}}\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\!\displaystyle\frac{1}{a_{n}}\!\left(\int_{\mathbb{R}}\left\|% \int_{0}^{1}\displaystyle\frac{f(x\!-\!y+\!\tau_{2,n}\!+\!h)\!-\!f(x\!-y+\tau_% {2,n})}{1+(\|x\|+h)^{2m}}(1\!-\!y^{2\alpha})^{n}dy\right\|^{p}dx\right)^{1/p}.$

Using the inequality

1+(|x-y+\tau_{2,n}|+h)^{2m}\leq 2^{2m-1}(1+(|x|+h)^{2m})(1+(\tau_{2,n}-y)^{2m}),

we have

	$\displaystyle J_{h}$	$\displaystyle\leq\displaystyle\frac{2^{2m-1}}{a_{n}}\int_{0}^{1}\left(\int_{% \mathbb{R}}\left\|\displaystyle\frac{f(x-y+\tau_{2,n}+h)-f(x-y+\tau_{2,n})}{1+(% \|x-y+\tau_{2,n}\|+h)^{2m}}\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\times(1+\|\tau_{2,n}-y\|)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{2^{2m-1}}{a_{n}}\omega_{p,m}(f;\delta)\int% _{0}^{1}(1+\|\tau_{2,n}-y\|)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq 2^{2m-1}\omega_{p,m}(f;\delta)\left(1+\sqrt{(L_{n}\varphi_{x% }^{2})(x)}\right).$

In the above $L_{n}\varphi_{x}^{2}$ represents the second order central moment of our operators. Based on (2.4), we get $(L_{n}\varphi_{x}^{2})(x)=\tau_{3,n}-\tau_{2,n}^{2}$ . Taking $\sup\limits_{0\leq h\leq\delta}J_{h}$ , we arrive at relation (3.17) and the proof is completed. $\square$

4. Multidimensional Landau type operators

For a positive integer $p\geq 2$ we consider the Euclidean space $\mathbb{R}^{p}$ . During this section we use the following notations: ${\boldsymbol{x}}=(x_{i})_{1\leq i\leq p}\in\mathbb{R}^{p}$ , ${\boldsymbol{n}}=(n_{i})_{1\leq i\leq p}\in\mathbb{N}^{p}$ , $D=[0,1]^{p}$ . According to the model offered by the relation (2.1) we consider the real numbers $\tau_{k,n_{i}}=B(n_{i}+1,k\lambda)/B(n_{i}+1,\lambda)$ , $k\in\mathbb{N}$ , and the vector ${\boldsymbol{\tau}}_{k,{\boldsymbol{n}}}=(\tau_{k,n_{i}})_{1\leq i\leq p}$ . Set

A_{n}=\prod_{i=1}^{p}a_{n_{i}},

where $a_{n_{i}}$ is defined as in formula (2.1). A variant of the Landau $p$ -dimensional operators which generalize those introduced by (2.3), can be described as follows

(\widetilde{L}_{{\boldsymbol{n}}}f)({\boldsymbol{x}})=\displaystyle\frac{1}{A_% {n}}\int_{D}\ldots\int f({\boldsymbol{x}}-{\boldsymbol{y}}+{\boldsymbol{\tau}}% _{2,{\boldsymbol{n}}})\prod_{j=1}^{p}(1-y_{j}^{2\alpha})^{n_{j}}dy_{1}\ldots dy% _{p},

(4.1)

where $f$ is a real-valued function defined on $\mathbb{R}^{p}$ such that the right hand side exists and is finite. We turn our attention to establishing approximation properties of these operators for the class of continuous functions. To achieve this, we introduce the following $p+2$ reference functions. Let $X\subseteq\mathbb{R}^{p}$ . The constant function on $X$ of constant value 1 is denoted by ${\boldsymbol{1}}$ . For each $j=1,\ldots,p$ we shall denote by $pr_{j}:X\to\mathbb{R}$ the $j$ -th canonical projection which is defined by

pr_{j}({\boldsymbol{x}})=x_{j},\mbox{ for every }{\boldsymbol{x}}=(x_{i})_{1% \leq i\leq p}\in X.

Set $\pi_{p}=\displaystyle\sum_{j=1}^{p}pr_{j}^{2}$ , $\pi_{p}({\boldsymbol{x}})$ representing the Euclidean inner product of ${\boldsymbol{x}}$ with itself.

$C(X)$ stands for the space of all real-valued continuous functions on $X$ endowed with the norm of the uniform convergence $\|\cdot\|$ ,

\|f\|=\sup_{{\boldsymbol{x}}\in X}|f({\boldsymbol{x}})|.

(4.2)

Theorem 4.1.

Let the operators $\widetilde{L}_{{\boldsymbol{n}}}$ be defined by (4.1). If

\|\widetilde{L}_{{\boldsymbol{n}}}{\boldsymbol{1}}-1\|\to 0,

(4.3)

\|\widetilde{L}_{{\boldsymbol{n}}}pr_{j}-pr_{j}\|\to 0,\ j=1,\ldots,p,

(4.4)

\|\widetilde{L}_{{\boldsymbol{n}}}\pi_{p}-\pi_{p}\|\to 0,

(4.5)

then $\|\widetilde{L}_{{\boldsymbol{n}}}f-f\|\to 0$ for any $f\in C(X)$ , where $X$ is a convex compact included in $\mathbb{R}^{p}$ and the norm is defined by (4.2).

Proof. Clearly, the operators are linear and positive. We use the multivariate Korovkin theorem. In this line we recall that

\mathcal{K}=\{{\boldsymbol{1}},pr_{1},\ldots,pr_{p},\pi_{p}\}

(4.6)

is a Korovkin system of test functions in $C(X)$ , see, e.g., [2, Eq. (4.4.22)]. For two dimensional case this result was first established by Volkov [19]. The $p$ -dimensional analogue of Korovkin’s theorem was proved by Shashkin [15]. Based on the definition of our operators, we have

(\widetilde{L}_{{\boldsymbol{n}}}{\boldsymbol{1}})({\boldsymbol{x}})=1,

consequently the identity (4.3) takes place. Further, using (4.1) along with the formulas (2.3) and (2.4), for each $j=1,\ldots,p$ , we obtain

	$\displaystyle(\widetilde{L}_{{\boldsymbol{n}}}pr_{j})({\boldsymbol{x}})$	$\displaystyle\!=\displaystyle\frac{1}{A_{n}}\int_{0}^{1}(x_{j}\!-\!y_{j}\!+\!% \tau_{n,j})\left(\prod_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{p}\int_{0}^{1}(1\!-\!y_{k}^{2\alpha})^{n_{k}}dy_{k}% \right)(1\!-\!y_{j}^{2\alpha})^{n_{j}}dy_{j}$
		$\displaystyle=\displaystyle\frac{1}{a_{n_{j}}}\int_{0}^{1}(x_{j}-y_{j}+\tau_{n% ,j})(1-y_{j}^{2\alpha})^{n_{j}}dy_{j}=(L_{n}e_{1})(x_{j})=x_{j},$

which ensures the fulfillment of the conditions required at (4.4).

Following a similar route to the one presented above, we can write

	$\displaystyle\quad\ (\widetilde{L}_{{\boldsymbol{n}}}\pi_{p})({\boldsymbol{x}}% )=\sum_{j=1}^{p}(\widetilde{L}_{{\boldsymbol{n}}}pr_{j}^{2})({\boldsymbol{x}})$
	$\displaystyle=\sum_{j=1}^{p}\displaystyle\frac{1}{A_{n}}\int_{0}^{1}(x_{j}-y_{% j}+\tau_{n,j})^{2}\left(\prod_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{p}\int_{0}^{1}(1-y_{k}^{2\alpha})^{n_{k}}dy_{k}\right)% (1-y_{j}^{2\alpha})^{n_{j}}dy_{j}$
	$\displaystyle=\sum_{j=1}^{p}\displaystyle\frac{1}{a_{n_{j}}}\int_{0}^{1}(x_{j}% -y_{j}+\tau_{n,j})^{2}(1-y_{j}^{2\alpha})^{n_{j}}dy_{j}=\sum_{j=1}^{p}(L_{n}e_% {2})(x_{j})$
	$\displaystyle=\pi_{p}({\boldsymbol{x}})+\sum_{j=1}^{p}(\tau_{3,n_{j}}-\tau_{2,% n_{j}}^{2}).$

We deduce $\|\widetilde{L}_{{\boldsymbol{n}}}\pi_{p}-\pi_{p}\|\leq\displaystyle\sum_{j=1}% ^{p}|\tau_{3,n_{j}}-\tau_{2,n_{j}}^{2}|$ . The result established at (2.2) guarantees that relation (4.5) is achieved. The proof is completed. $\square$

We present an estimate of the error of approximation in terms of the corresponding quantities for the test functions referred to in (4.6). For linear positive operators univariate case, a general result was establishing by Shisha and Mond [16]. For $p$ -dimensional case, the general result was obtained by Censor [3]. Such estimates involve moduli of smoothness. For real multivariate functions we consider the following type of modulus [3, Eq. (1)]

\omega(f;\delta)=\max_{\begin{subarray}{c}{\boldsymbol{t}},{\boldsymbol{x}}\in X% \\ d({\boldsymbol{t}},{\boldsymbol{x}})\leq\delta\end{subarray}}|f({\boldsymbol{t% }})-f({\boldsymbol{x}})|,\ \delta\geq 0,

(4.7)

where $X\subseteq\mathbb{R}^{p}$ is a convex compact, $d(\cdot,\cdot)$ stands for the Euclidean distance in $\mathbb{R}^{p}$ and $f$ is a real valued function continuous on $X$ .

Theorem 4.2.

Let the operators $\widetilde{L}_{{\boldsymbol{n}}}$ be defined by (4.1). The following relation

\|\widetilde{L}_{{\boldsymbol{n}}}f-f\|\leq 2\omega\left(f;\sqrt{d({% \boldsymbol{\tau}}_{3,{\boldsymbol{n}}},{\boldsymbol{\tau}}_{2,{\boldsymbol{n}% }}^{2})}\right)

(4.8)

holds, where $\omega$ is given at (4.7) and for $s=2$ , $s=3$ ,

{\boldsymbol{\tau}}_{s,{\boldsymbol{n}}}=(\tau_{s,n_{i}})_{1\leq i\leq p}\in% \mathbb{R}^{p}.

Proof. Using the notions and the notations specified above, the proof of formula (4.8) is based on [3, Theorem 1] which says: if $\Lambda_{n}$ is positive linear operator on $C(X)$ such that $\Lambda_{n}{\boldsymbol{1}}=1$ , then

\|\Lambda_{n}f-f\|\leq 2\omega(f;\mu_{n})

holds, where $\mu_{n}^{2}=\left\|\Lambda_{n}\left(\displaystyle\sum_{k=1}^{p}(\xi_{k}-x_{k})% ^{2};x_{1},\ldots,x_{p}\right)\right\|$ . Here $\Lambda_{n}$ operates on a function of $\xi_{1},\ldots,\xi_{p}$ and the resulting function is evaluated at the point $(x_{1},\ldots,x_{p})$ .

In our case we have

	$\displaystyle\widetilde{L}_{{\boldsymbol{n}}}\left(\sum_{k=1}^{p}(pr_{k}-x_{k}% )^{2};{\boldsymbol{x}}\right)$	$\displaystyle=\sum_{k=1}^{p}\widetilde{L}_{{\boldsymbol{n}}}(pr_{k}^{2}-2x_{k}% pr_{k}+x_{k}^{2}{\boldsymbol{1}};{\boldsymbol{x}})$
		$\displaystyle=\sum_{k=1}^{p}(\tau_{3,n_{k}}-\tau_{2,n_{k}}^{2})^{2}=d(\tau_{3,% {\boldsymbol{n}}},\tau_{2,{\boldsymbol{n}}}^{2}),$

see the computations accomplished in the proof of Theorem 4.1. The components of the vectors ${\boldsymbol{\tau}}_{s,{\boldsymbol{n}}}$ , $s\in\{2,3\}$ , are explicitly defined at (2.1). Formula (4.8) follows. $\square$

References

[1] Agratini, O., Gal, S.G., On Landau type appproximation operators, Mediterr. J. Math. (sent to publication).
[2] Altomare, F., Campiti, M., Korovkin-type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, Vol. 17, Walter de Gruyter & Co., Berlin, New York, 1994.
[3] Censor, E., Quantitative results for positive linear approximation operators, J. Approx. Theory, 4(1971), 442-450.
[4] Chen, Zhanben, Shih, Tsimin, A new class of generalized Landau linear positive operator sequence and its properties of approximation, Chinese Quart. J. Math., 13(1998), no. 1, 29-43.
[5] De Vore, R.A., Lorentz, G.G., Constructive Approximation, A Series of Comprehensive Studies in Mathematics, Vol. 303, Springer Verlag, Berlin, Heidelberg, 1993.
[6] Gadjiev, A.D., Aral, A., Weighted $L_{p}$ -approximation with positive linear operators on unbounded sets, Appl. Math. Lett., 20(2007), Issue 10, 1046-1051.
[7] Gal, S.G., Iancu, I., Quantitative approximation by nonlinear convolution operators of Landau-Choquet type, Carpathian J. Math., 37(2021), Issue 1 (to appear).
[8] Gao, J.B., Approximation properties of a kind of generalized discrete Landau operator, (Chinese), J. Huazhong Univ. Sci. Tech., 12(1984), no. 5, 1-4.
[9] Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities, Cambridge Mathematical Library, Cambridge University Press, 1988.
[10] Jackson, D., A proof of Weierstrass theorem, Amer. Math. Monthly, 41(1934), no. 5, 309-312.
[11] Landau, E., Über die Approximation einer stetingen Funktion durch eine ganze rationale Funktion, Rend. Circ. Mat. Palermo, 25(1908), 337-345.
[12] Mamedov, R.G., Approximation of functions by generalized linear Landau operators, (Russian), Dokl. Akad. Nauk SSSR, 139(1961), no. 1, 28-30.
[13] Mamedov, R.G., On the order and on the asymptotic value of the approximation of functions by generalized linear Landau operators, (Russian), Akad. Nauk Azerbadzan SSR, Trudy Inst. Mat. Meh., 2(10)(1963), 49-65.
[14] Pendina, T.P., Iterations of positive linear operators of exponential type and of Landau polynomials, (Russian), In: Geometric Problems of the Theory of Functions and Sets (Russian), Kalinin. Gos. Univ., Kalinin, 1987, 105-111.
[15] Shashkin, Yu.A., Korovkin systems in spaces of continuous functions, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 26(1962), Issue 4, 495-512; translated in Amer. Math. Soc. Transl., Series 2, 54(1996), 125-144.
[16] Shisha, O., Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60(1968), 1196-1200.
[17] Sikkema, P.C., Approximation formulae of Voronovskaya type for certain convolution operators, J. Approx. Theory, 26(1979), no. 1, 26-45.
[18] Veselinov, V., Certain estimates to the approximation of functions by de la Vallée-Poussin and Landau operators, (Russian), Ann. Univ. Sofia Fac. Math., 66(1974), 153-158.
[19] Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 115(1957), 17-19.
[20] Yuksel, I., Ispir, N., Weighted Approximation by a Certain Family of Summation Integral-Type Operators, Comput. Math. Appl., 52(2006), 1463-1470.

[1] Agratini, O., Gal, S.G., On Landau type appproximation operators. Mediterr. J. Math. (sent to publication)
[2]Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, vol. 17. Walter de Gruyter & Co., Berlin, New York (1994), Book Google Scholar
[3] Censor, E., Quantitative results for positive linear approximation operators. J. Approx. Theory 4, 442–450 (1971), MathSciNet Article Google Scholar
[4] Chen, Zhanben, Shih, Tsimin, A new class of generalized Landau linear positive operator sequence and its properties of approximation. Chin. Q. J. Math. 13(1), 29–43 (1998), Google Scholar
[5] De Vore, R.A., Lorentz, G.G., Constructive Approximation, A Series of Comprehensive Studies in Mathematics, vol. 303. Springer, Berlin, Heidelberg (1993) Google Scholar
[6] Gadjiev, A.D., Aral, A., Weighted Lp-approximation with positive linear operators on unbounded sets. Appl. Math. Lett. 20(10), 1046–1051 (2007), MathSciNet Article Google Scholar
[7] Gal, S.G., Iancu, I.: Quantitative approximation by nonlinear convolution operators of Landau-Choquet type. Carpathian J. Math. 37, Issue 1 (to appear) (2021)
[8] Gao, J.B., Approximation properties of a kind of generalized discrete Landau operator, (Chinese). J. Huazhong Univ. Sci. Tech. 12(5), 1–4 (1984), MathSciNet Google Scholar
[9] Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities. Cambridge University Press, Cambridge Mathematical Library (1988), MATH Google Scholar
[10] Jackson, D., A proof of Weierstrass theorem. Am. Math. Mon. 41(5), 309–312 (1934), MathSciNet Article Google Scholar
[11] Landau, E., Über die Approximation einer stetingen Funktion durch eine ganze rationale Funktion. Rend. Circ. Mat. Palermo 25, 337–345 (1908), Article Google Scholar
[12] Mamedov, R.G., Approximation of functions by generalized linear Landau operators, (Russian). Dokl. Akad. Nauk SSSR 139(1), 28–30 (1961), MathSciNet Google Scholar
[13] Mamedov, R.G., On the order and on the asymptotic value of the approximation of functions by generalized linear Landau operators, (Russian), Akad. Nauk Azerbadzan SSR, Trudy Inst. Mat. Meh., 2(10), 49–65 (1963)
[14] Pendina, T.P., Iterations of positive linear operators of exponential type and of Landau polynomials, (Russian). In: Geometric Problems of the Theory of Functions and Sets (Russian), Kalinin. Gos. Univ., Kalinin, 105–111, (1987)
[15] Shashkin, Y.A.: Korovkin systems in spaces of continuous functions, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 26, Issue 4, 495–512 (1962), translated in Am. Math. Soc. Transl., Series 2, 54, 125–144 (1996)
[16] Shisha, O., Mond, B., The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 60, 1196–1200 (1968), MathSciNet Article Google Scholar
[17] Sikkema, P.C., Approximation formulae of Voronovskaya type for certain convolution operators. J. Approx. Theory 26(1), 26–45 (1979), MathSciNet Article Google Scholar
[18] Veselinov, V., Certain estimates to the approximation of functions by de la Vallée–Poussin and Landau operators, (Russian). Ann. Univ. Sofia Fac. Math. 66, 153–158 (1974)Google Scholar
[19] Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR (N.S.), 115, 17–19 (1957) (Russian)
[20] Yuksel, I., Ispir, N., Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52, 1463–1470 (2006), MathSciNet Article Google Scholar

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	$\displaystyle\\|L_{n}f\\|_{L_{p}}$	$\displaystyle=\left(\int_{\mathbb{R}}\|(L_{n}f)(x)\|^{p}dx\right)^{1/p}$
		$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int_{0% }^{1}f(x-y+\tau_{2,n})(1-y^{2\alpha})^{n}dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\|f(x-y+\tau_{2,n}\|^{p}dx\right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\\|f\\|_{L_{p}}\ \displaystyle\frac{1}{a_{n}}\int_{0}^{1}(1-y^{% 2\alpha})^{n}dy$
		$\displaystyle=\\|f\\|_{L_{p}},$

	$\displaystyle\\|L_{n}f\!-\!f\\|_{L_{p}}$	$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int_{0% }^{1}(f(x-y+\tau_{2,n})-f(x))(1-y^{2\alpha})^{n}dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|f(x-y+\tau_{2,n})-f(x)\right\|^{p}dx\right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\omega(f;\|\tau_{2,n}-% y\|)_{p}(1-y^{2\alpha})^{n}dy.$

	$\displaystyle\\|L_{n}f\\|_{p,\nu}$	$\displaystyle=\left(\int_{\mathbb{R}}\left\|\displaystyle\frac{(L_{n}f)(x)}{1+x% ^{2m}}\right\|^{p}dx\right)^{1/p}$
		$\displaystyle=\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\displaystyle% \frac{1}{(1+x^{2m})^{p}}\left\|\int_{0}^{1}f(x-y+\tau_{2,n})(1-y^{2\alpha})^{n}% dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|\displaystyle\frac{f(x-y+\tau_{2,n})}{1+x^{2m}}\right\|^{p}dx\right)^{1% /p}(1-y^{2\alpha})^{n}dy.$

	$\displaystyle\\|L_{n}f\!-\!f\\|_{p,\nu}$	$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\left(\int_{\mathbb{R}}\left\|\int% _{0}^{1}\displaystyle\frac{f(x-y+\tau_{2,n})-f(x)}{1+x^{2m}}(1-y^{2\alpha})^{n% }dy\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\left(\int_{\mathbb{R% }}\left\|\displaystyle\frac{f(x-y+\tau_{2,n})-f(x)}{1+x^{2m}}\right\|^{p}dx% \right)^{1/p}(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{1}{a_{n}}\int_{0}^{1}\omega_{p,m}(f;\|\tau_% {2,n}-y\|)(1-y^{2\alpha})^{n}dy.$

	$\displaystyle J_{h}$	$\displaystyle\leq\displaystyle\frac{2^{2m-1}}{a_{n}}\int_{0}^{1}\left(\int_{% \mathbb{R}}\left\|\displaystyle\frac{f(x-y+\tau_{2,n}+h)-f(x-y+\tau_{2,n})}{1+(% \|x-y+\tau_{2,n}\|+h)^{2m}}\right\|^{p}dx\right)^{1/p}$
		$\displaystyle\times(1+\|\tau_{2,n}-y\|)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq\displaystyle\frac{2^{2m-1}}{a_{n}}\omega_{p,m}(f;\delta)\int% _{0}^{1}(1+\|\tau_{2,n}-y\|)(1-y^{2\alpha})^{n}dy$
		$\displaystyle\leq 2^{2m-1}\omega_{p,m}(f;\delta)\left(1+\sqrt{(L_{n}\varphi_{x% }^{2})(x)}\right).$

Approximation of some classes of functions by Landau type operators

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Approximation of some classes of functions by Landau type operators

Abstract.

1. Introduction

2. Preliminaries

3. Approximation in Lp spaces

Lemma 3.1.

Theorem 3.1.

Lemma 3.2.

Theorem 3.2.

Lemma 3.3.

Lemma 3.4.

Theorem 3.3.

Theorem 3.4.

4. Multidimensional Landau type operators

Theorem 4.1.

Theorem 4.2.

References

References

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