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

the convergence being uniform over an interval $[a,b]$, where . 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

where $n\in \mathbb{N}$ is fixed and $\mathrm{\Gamma }$ 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 $C(ℝ)$, the space of real valued continuous functions on $ℝ$, and in the weighted space $B_{{\rho }_m}(ℝ)$, where

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 $L_p(ℝ)$, , 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 $L_2(ℝ)$ 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.

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

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

The sequences ${(a_n)}_{n\ge 1}$ and ${({\tau }_{p,n})}_{n\ge 1}$ for $p\ge 2$, are strictly decreasing. Moreover, for $p\ge 2$,

We consider the linear and positive operators defined as follows

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

Further, we focus on the study of approximation properties in unweighted and weighted $L_p$ spaces, .

As usual, we denote the norm of the $L_p$ space, , by $\parallel \cdot {\parallel }_{L_p}$.

The above property says that our operators are non-expansive on the spaces $L_p(ℝ)$.

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

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

Actually, $\omega {(\cdot ;\delta )}_p$ meets the conditions that ensure it is a seminorm on $L_p(ℝ)$, $p\ge 1$.

We deduce

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

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

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. $\mathrm{\square }$

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

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

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

Using again Minkowski’s inequality, we can write

Appealing to the inequality

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

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

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

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

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

for any function $f\in L_{p,\nu }(ℝ)$. 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 $\delta \ge 0$ and any positive integer $m$,

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

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

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

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

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

This inequality can be written

For $\delta >0$, this relation implies

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

We used again inequalities shown in (3.8).

Taking $\delta \le 1$ we can write

Thus, we have

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

That is, for a given $\epsilon >0$, there exists a rank $n_0\in \mathbb{N}$ such that

whenever $n\ge n_0$ and $\delta >0$. Thus, we get

for $n\ge n_0$.

Applying the Minkowski’s inequality yields

From (3.13) and (3.14) it follows that

for $n\ge n_0$ and $\delta >0$.

Since ${\phi }_n$, $n\in \mathbb{N}$, are continuous functions on a compact interval, based on Cantor’s theorem, they are uniformly continuous. Consequently, for $0\le h\le \delta$ and $n\ge n_0$ we can write

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

and, from (3.15) we get

Returning at (3), the following inequality is true for , which leads us to the statement (3.11). $\mathrm{\square }$

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

and the relation (3.16) follows. $\mathrm{\square }$

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

Using the inequality

we have