A special branch of Approximation Theory is the approximation of functions by using linear sequences of operators, say ${(L_n)}_{n\ge 1}$, the essential feature being that of positivity. The cornerstone of this approach is Bohman-Korovkin criterion [5], [15]. This criterion says: if ${(L_n{e}_k)}_{n\ge 1}$ converges to $e_k$ uniformly on $[a,b]$, $k\in \{0,1,2\}$, for the test functions $e_0(x)=1$, $e_1(x)=x$, $e_2(x)=x^2$, then ${(L_{n}f)}_{n\ge 1}$ converges to $f$ uniformly on $[a,b]$ for each $f\in C([a,b])$. Usually, for continuous functions defined on an unbounded interval only pointwise convergence occurs. It is one of the reasons why the study of such operators is more productive.

We recall that two types of positive approximation processes are used – the discrete, respectively continuous form. In what follows we consider as a starting point a linear positive process of discrete type which approximate functions defined on unbounded intervals. Since a linear substitution maps the interior of any unbounded interval $J$ onto $(0,\mathrm{\infty })$ or $ℝ$, we have only two options. In our study we consider the benchmark interval $J=[0,\mathrm{\infty })={ℝ}_{+}$ as it exhibits the problems caused by a finite endpoint and by the nonboundedness of the interval. Operators in questions are often designed as follows

where $a_k(n;\cdot ):{ℝ}_{+}\to {ℝ}_{+}$ are continuous functions for each $n\in \mathbb{N}$, $k\in \{0\}\cup \mathbb{N}={\mathbb{N}}_0$, and ${\mathrm{\Delta }}_n={(x_{n,k})}_{k\ge 0}$ is a net on ${ℝ}_{+}$. Here $f$ belongs to a function space for which the right-hand side of the relation (1.1) is well defined.

The most common case for classical operators is the equidistant network $x_{n,k}=k/n$, $k\ge 0$. We mention that in (1.1) could be a finite sum. For example, if we take

and we choose the net ${\mathrm{\Delta }}_n={(k/(n-k+1))}_{k=\overline{0,n}}$, then $L_n$, $n\in \mathbb{N}$, become Bleiman-Butzer-Hahn operators [4].

The disadvantage of the positive linear approximation processes is definitely determined by the fact that they have a low rate of convergence. For some spaces of functions considered, the error of approximation $|(L_{n}f)(x)-f(x)|$ is assessed by using the modulus of continuity $\omega$ of $f:I\to ℝ$ with argument $\delta$ defined by

We are considering functions for which the supremum is finite, for example bounded functions, uniformly continuous functions, Lipschitz functions.

For a wide majority of linear positive operators, the error of approximation is described by using $\omega (f;1/\sqrt{n})$ [8, page 40], consequently the order of approximation is $𝒪(n^{-1/2})$. Is it possible to construct classes of operators having a faster rate of convergence? Recently, such classes of operators acting on certain function spaces have been obtained.

Starting from (1.1) we consider a general class of discrete operators with the property that the order of approximation is improved, meaning that becomes arbitrarily small. Also, we prove that the same phenomenon holds for quantitative estimates in Voronovskaya type theorems. The next section consists in associating to this new process an integral generalization of Kantorovich type, obtaining for it the same kind of results. Particular sequences are identified in our construction, such that the result established can be applied to several cases.

Let $\lambda ={({\lambda }_n)}_{n\ge 1}$ be a strictly decreasing sequence of positive numbers with the property

We modify the operators defined by (1.1) replacing $n$ with ${\lambda }_n^{-1}$. Also we consider the equidistant network having the nodes $x_{n,k}=k{\lambda }_n$. The discrete class of operators is given as follows

where $a_k({\lambda }_n;\cdot )$ are non-negative functions belonging to $C({ℝ}_{+})$ and

We are working under assumption that the monomials $e_j$, $0\le j\le 2$, belong to the above domain.

Clearly, $L_n^{⟨\lambda ⟩}$, $n\in \mathbb{N}$, are linear and positive operators. At this point we introduce the first and the second central moments of $L_n^{⟨\lambda ⟩}$ operators, i.e.,

and $i\in \{1,2\}$. Concerning the operators defined by (2.2), we consider that for each $x\in {ℝ}_{+}$ and $n\in \mathbb{N}$ they satisfy the following properties

where $l_j\in C({ℝ}_{+})$, $j\in \{1,2,3\}$.

Taking into account Bohman-Korovkin theorem, ${(L_n^{⟨\lambda ⟩})}_{n\ge 1}$ is an approximation process on the space $C(K)$, endowed with the uniform norm, $K\subseteq {ℝ}_{+}$ compact, if

uniformly on $K$.

Based on the requirements (2.4), (2.5), (2.6), the above limits are achieved. On the other hand, our hypothesis package means much more. The real problem that arises here is whether there are classes of ${(L_n^{⟨\lambda ⟩})}_{n\ge 1}$ operators which meet all assumptions. We certify this fact presenting two examples. We also indicate successive steps taken to obtain them.

By using two sequences of non-negative real numbers ${({\alpha }_n)}_n$, ${({\beta }_n)}_n$ with the property that ${({\beta }_n/{\alpha }_n)}_n$ is strictly decreasing having null limit, Cetin and Ispir [6] introduced and studied a notable generalization of Szász operators attached to analytic functions $f$ of exponential growth in a compact disk of the complex plane, ,

For the particular case $z\in {ℝ}_{+}$, ${\alpha }_n=n$, ${\beta }_n=1$, these operators turn into genuine Szász operators [20].

In [10] Gal considered the Ismail’s kind generalization (2.7), this time the operators being applied to certain real-valued functions defined on ${ℝ}_{+}$. More precisely, the following operators are studied

$x\in {ℝ}_{+}$, under the hypothesis on $A,H$ and $p_k$ specified at (2.8).

Setting ${\lambda }_n:={\beta }_n/{\alpha }_n$, condition (2.1) is satisfied and we observe that these operators are of the form (2.2) with

For ${\lambda }_n=1/n$ we reobtain $T_n$ operators which have the form indicated in (1.1). Following [10, Lemma 2.1], the author proved

Comparing these identities to those describing by relations (2.4), (2.5), (2.6), we deduce

Recently, Gal and Opriş [11] introduced a modified Baskakov type operator. In the definition of ${\phi }_n(x)={(1+x)}^{-n}$, $n\in \mathbb{N}$, the authors replaced $1/n$ by ${\lambda }_n$, consequently $\lambda ={({\lambda }_n)}_{n\ge 1}$ satisfies relation (2.1). The new operators are expressed for $x\ge 0$ as follows

Regarding these operators, the authors proved [11, Corollary 2] the following relations

Comparing these relations with the requirements (2.4), (2.5), (2.6), we conclude $l_1(x)=l_3(x)=0$ and $l_2(x)=x(1+x)$, $x\ge 0$.

The space of real valued uniformly continuous functions defined on ${ℝ}_{+}$ is denoted by $UC({ℝ}_{+})$.

Even if this result was proved for continuous functions defined on compact intervals, the reasonings are the same for functions belonging to $UC({ℝ}_{+})$. In (2) we choose $\mathrm{\Lambda }=L_n^{⟨\lambda ⟩}$. Taking into account that $L_n^{⟨\lambda ⟩}$ operators reproduce the constants and relation (2.6) holds, we can write

Taking $\delta =\sqrt{{\lambda }_n}$, we get inequality (2.12) and the proof is ended. $\mathrm{\square }$

Returning to the two examples, relation (2.12) becomes as follows:

(i) For Szász-type operators (2.9),

where ${\lambda }_n={\beta }_n/{\alpha }_n$.

The result coincides with that obtained in [10, Theorem 2.2].

(ii) For Baskakov type operators (2),

The result is similar with that obtained in [11, Corollary 2.1(ii)].

In what follows, we will show that the operators of the form $L_n^{⟨\lambda ⟩}$, $n\in \mathbb{N}$, defined by (2.2), may give an arbitrary small order of approximation for quantitative estimates in Voronovskaya type theorems too. Let us recall that Voronovskaya type results for linear positive operators applied to functions defined on $[0,+\mathrm{\infty })$ were obtained, for example, in [12] for operators reproducing $e_0$ and $e_1$ and in [2] for operators reproducing only $e_0$. For the simplicity of presentation, we will use below the results in [12] applied to the above particular cases.

For this purpose, firstly we need the following notations: for $m\in \mathbb{N}$ with $m\ge 2$ and $k\in \mathbb{N}\cup \{0\}$, the Pǎltǎnea [18] modulus of continuity is given by

where ${\phi }_m(x)={\displaystyle \frac{\sqrt{x}}{1+x^m}}$, for all $x\in [0,+\mathrm{\infty })$,

The above two theorems are new.

Starting from the operators $L_n^{⟨\lambda ⟩}$, $n\in \mathbb{N}$, designed as in (2.2) which verify the assumptions (2.4), (2.5), (2.6), we associate them the following integral generalization of Kantorovich type

The construction involves the mean values of the approximating function $f$ on the subintervals $[k{\lambda }_n,(k+1){\lambda }_n]$, $k\in {\mathbb{N}}_0$.

In the above $f$ must be locally integrable function on ${ℝ}_{+}$ such that the antiderivative of $f$ belongs to the domain $𝒟$ defined at (2.3). Our aim is to show that this linear and positive integral process can approximate certain classes of functions with an arbitrary good order of approximation on each compact interval included in ${ℝ}_{+}$.

By using Lemma 3.1 we can express the first and the second central moments of $K_n^{⟨\lambda ⟩}$ operators with the help of corresponding moments of $L_n^{⟨\lambda ⟩}$. For each $x\ge 0$ we get

A question arises: are well defined the operators $L_n^{⟨\lambda ⟩}$ and $K_n^{⟨\lambda ⟩}$ for functions of $UC({ℝ}_{+})$? Since $f$ belongs to this class, its growth is linear, i.e., the non-negative real constants $c_1$ and $c_2$ exist such that $|f(x)|\le c_{1}x+c_2$, $x\ge 0$. Consequently, the answer is positive.

It is known that the function $\omega$ defined by (1.2) is continuous at $\delta =0$, i.e.

if and only if $f$ is uniformly continuous on its domain, see [8, page 40]. Examining relations (3.4) and (2.1), the following pointwise convergence

takes place. These operators can also uniformly approximate any function $f\in UC({ℝ}_{+})$ on each compact, say $K$, included in ${ℝ}_{+}$. Relative to the functions $l_i\in C({ℝ}_{+})$, $i\in \{1,2,3\}$, introduced by relations (2.4)-(2.6), we denote

Relation (3.4) implies

therefore $\underset{n\to \mathrm{\infty }}{lim}{K}_n^{⟨\lambda ⟩}f(x)=f(x)$, uniformly on the compact $K$.

We recall $f:{ℝ}_{+}\to ℝ$ is Lipschitz continuous if there exists a real constant $M\ge 0$ such that $|f(x)-f(y)|\le M|x-y|$ for any $(x,y)\in {ℝ}_{+}\times {ℝ}_{+}$. We denote this set by $Li{p}_{M}1$. One hand, any Lipschitz continuous map is uniformly continuous; on the other hand, $f\in Li{p}_{M}1$ if and only if ${\omega }_f(\delta )\le M\delta$. Taking in view these statements, we enunciate