A current subject in Approximation Theory is the approximation of functions by positive linear operators. Usually, two types of positive approximation processes are used – the discrete respectively continuous form. In the first case, various classes of operators are expressed by series. In this paper we are interested in these last sequences, denoted generically by ${(L_n)}_{n\ge 1}$. The structure of these operators is detailed in Section 2. Since a property often seen in linear positive operators is to reproduce the constants which are involved in preservation of the monomial $e_0$, $e_0(x)=1$, we assume that the following relation

occurs.

In recent years, it was studied the behavior of the expression

for given functions $f$, $g$ belonging to the domain of $L_n$ and $x$ fixed and compatible with the domain of the functions. The analysis of quantity (1.2) took different approaches, but usually on particular cases of linear (positive or non-positive) operators.

In the present paper we deal with the following two aspects.

Among the properties studied for linear approximation processes, the asymptotic behavior known as the so called Voronovskaja type formula plays an important role. Knowing a such Voronovskaja formula verified by $L_n$, in Section 3 we establish the asymptotic behavior of quantity defined by (1.2). Particular cases are shown.

In Section 4 we present a Chebyshev-Grüss-type inequality involving linear operators without supposing that they are positive. Also, we indicate sufficient conditions that ensure the statistical convergence of the expression given by (1.2).

Let $J\subset ℝ$ be an interval and its interior $Int(J)=(a,b)$, . Since a linear substitution maps a bounded interval $(a,b)$ into the interval $(0,1)$, without loss of generality, we will consider as $Int(J)$ one of the intervals $(0,1)$, $(0,\mathrm{\infty })={ℝ}_{+}^{\ast }$, $(-\mathrm{\infty },\mathrm{\infty })=ℝ$.

Set ${\mathbb{N}}_0=\{0\}\cup \mathbb{N}$. Let ${\mathrm{\Delta }}_n={(x_{n,k})}_{k\in I_n}$ be a net on the interval $J$, where $I_n\subseteq \mathbb{Z}$ is a set of indices consistent with $J$, this meaning $\{x_{n,k}:k\in I_n\}\subset J$. We denote by $B(J)$, respectively $C(J)$, the space of all real-valued bounded, respectively continuous, functions defined by $J$.

Linear operators of discrete type are often designed as follows

where $a_{n,k}\in C(J)$, $k\in I_n$, such that

and

The last condition ensures the fulfillment of relation (1.1). In order that $L_n$ to be also positive, we impose

In this case, ${(a_{n,k})}_{k\in I_n}$ forms a so called blending system. The operator $L_n$ being linear and positive, is continuous with its norm equal to 1, because of (1.1) takes place.

The domain of the operators $L_n$, $n\in \mathbb{N}$, denoted by $ℱ(J)$, consists of the set of all functions for which the right-hand side of the relation (2.1) is well defined. For example, due to (2.3), $e_0\in ℱ(J)$. Moreover, based on (2.2), we get $B(J)\subset ℱ(J)$,

Regarding the asymptotic behavior of linear (not necessarily positive) operators $L_n$, we make the following assumption. A real constant $\lambda >0$ and a second order differential operator $\varphi$ exist such that

for some $x\in J$, where

$\alpha$, $\beta$ being continuous functions defined on $J$.

In order to take place relation (2.5), function $f$ has to be differentiable in some neighborhood of $x$ and has second derivative $f^{\prime \prime }(x)$. This is a Voronovskaja type result given in its local version.

Here we give some examples of formula described by (2.5) associated with well known linear positive operators. The cases $J=[0,1]$, $J={ℝ}_{+}$, $J=ℝ$ are illustrated.

1. Bernstein operators. $J=[0,1]$, $I_n=\{0,1,\mathrm{\dots },n\}$, $x_{n,k}={\displaystyle \frac{k}{n}}$,

Set $C^2([0,1])$ the space twice continuously differentiable functions defined on $[0,1]$. It takes place [17]

The above convergence is uniform on $[0,1]$. Here $\lambda =1$, $\alpha (x)=0$ and $\beta (x)=x(1-x)/2$.

We notice that Voronovskaja gave the first asymptotic formula for the pointwise approximation of continuous functions which have a second derivative at a point $x\in J=[0,1]$.

2. Jain operators. $J=[0,\mathrm{\infty })$, $I_n={\mathbb{N}}_0$, $x_{n,k}={\displaystyle \frac{k}{n}}$,

where

whenever the sum in (2.7) is convergent. Jain [11] showed that for $\alpha >0$ and ,

Note that this is a probability distribution which is called generalized Poisson distribution.

The special case $\beta =0$, Jain operators reduce to the classical Szász-Mirakjan operators [15], [12].

In (2.7) the parameter $\beta$ may depend on a natural number $n$, say $\beta :={\beta }_n$. If , $\underset{n\to \mathrm{\infty }}{lim}{\beta }_n=0$, and a suitable condition on the growth of $f$ is satisfied, then Farcaş [5] proved

Here $\lambda =1$, $\alpha (x)=0$, $\beta (x)=x/2$.

We point out that the same Voronovskaja formula is verified by genuine Szász-Mirakjan operators.

3. Sampling operators. $J=ℝ$, $I_n=\mathbb{Z}$, $x_{n,k}={\displaystyle \frac{k}{n}}$.

We consider a compactly supported function $\phi \in C(ℝ)$ satisfying the following assumptions

for a given constant $C\in ℝ$ and every $x\in ℝ$. Following the results obtained by Butzer, Stens, Ries, see e.g., [3], [13], the sampling operators are defined by

Since $\phi$ has a compact support, for each fixed $n\in \mathbb{N}$ and $x\in ℝ$, the above sum has a finite number of terms. Consequently, we can associate $G_n$ operators to any real-valued function $f$ defined on the whole real line.

Set $L^{\mathrm{\infty }}(ℝ)$, the space of all real-valued functions defined on $ℝ$, essentially bounded and measurable. For any function $f\in L^{\mathrm{\infty }}(ℝ)$ and at every point $x\in ℝ$ in which $f^{\prime \prime }(x)$ exists, one has (see [2, Corollary 1])

where $C$ is defined at (2.8). In this example $\lambda =2$, $\alpha (x)=0$ and $\beta (x)=C/2$.

It is worthy to note that in the paper of Bardaro and Mantellini [2] is given a Voronovskaja type formula of a general class of discrete operators. A distinct section represents a survey on various asymptotic formulas for many classical operators which can be deduced as special cases of the original outcome established in the quoted paper.

We are only concerned with sequences ${(L_n)}_{n\ge 1}$ of linear operators forming an approximation process. This means that a subspace $𝒜(J)\subset ℱ(J)$ exists such that

The above convergence is generally pointwise.

About Voronovskaja’s formula for quantity defined by (1.2), our result will be read as follows.

Note the following: although Voronovskaja’s formula of $L_n$ operators depends on function $\alpha$, the limit in (3.2) does not contain $\alpha$.

Returning to the examples presented in the previous section, under the assumption that the requirements of Theorem 3.1 are fulfilled, the identity given by (3.2) takes the following particular forms.

$1^{\circ }$ Bernstein operators

$2^{\circ }$ Jain operators

In the special case ${\beta }_n=0$, these operators turn to so called Szász-Mirakjan operators and be noticed that the second member of the identity (3.2) remains unchanged.

$3^{\circ }$ Sampling operators

where $C$ is defined by (2.8).

The roots of inequality on which we investigate goes back to the classical works of Chebyshev [4] and Grüss [10].

The first author proved

$f,g\in C^1([a,b])$ and $\parallel \cdot \parallel$ is the sup-norm on the Banach space $C([a,b])$.

The second author extended (4.1) to Riemann integrable functions defined on $[a,b]$, say $f$ and $g$, obtaining

where $M_h=\underset{t\in [a,b]}{sup}|h(t)|$, $m_h=\underset{t\in [a,b]}{inf}|h(t)|$ for $h\in \{f,g\}$.

Recently, in many papers, modeling these inequalities for non-multiplicative linear operators have been highlighted. We mention as pioneering reference papers [1] and [14]. In these articles the authors have explored in depth linear (positive) operators acting on continuous functions defined on a compact metric space. Among subsequent papers we also recall [7], [8].

Under the assumption of Theorem 4.1, identity (4.2) implies the following relation

As we stated in Section 2, $B(J)\subset ℱ(J)$. Also, for any $h\in B(J)$ the real constants $m(h)$, $M(h)$ exist such that $m(h)\le h(t)\le M(h)$, $t\in J$. Consequently

and taking in view (4.4), we get

It is worth mentioning, by using the notion of oscillation associated to a certain functional, a similar result was obtained in [9, Theorem 8].

We return to quantity (1.2). If $L_n$, $n\in \mathbb{N}$, are linear positive operators on the space $C(J)$, $J=[0,1]$, and (1.1) holds, then there is a Borel probability measure $\mu$ on $[0,1]$ such that $L_{n}h={\displaystyle {\int }_0^1}h𝑑\mu$, for any $h\in C([0,1])$. We can write

This quantity represents the variance of the function $h$ with respect to Borel measure $\mu$. Set

Our goal is to analyze the expression (1.2) in terms of the statistical convergence.

We pause to collect some notation and definitions.

The density of a set $U\subset \mathbb{N}$ is defined by

provided the limit exists, where ${\chi }_U$ is the characteristic function of $U$. A sequence of real numbers $x={(x_k)}_{k\ge 1}$ is statistically convergent to a real number $l$, denoted $st-\underset{k}{lim}{x}_k=l$, if, for every $\epsilon >0$,

holds. In Approximation Theory by linear positive operators, the statistical convergence has been examined for the first time in 2002 by A.D. Gadjiev and C. Orhan [6] who gave the Bohman-Korovkin criterion via statistical convergence.

Throughout this theorem, $\parallel \cdot \parallel$ represents the uniform norm of the space $C([0,1])$. The key relation used in the proof was established by Uchiyama [16, Eq. (3)], namely

For an arbitrary fixed $\epsilon >0$, we define the following two sets

On the basis of (4.8), we have $M_1(\epsilon )\subseteq M_2(\epsilon )$. Consequently, we can write

see (4.7).

This way $0\le \delta (M_1(\epsilon ))\le 0$ for any $\epsilon >0$, which ensures the relation (4.6). $\mathrm{\square }$