On the last five decades the interest of the study of positive approximation processes have emerged with growing evidence. In what follows we consider linear positive operators which approximate real valued functions defined on the unbounded interval ${ℝ}_{+}=[0,\mathrm{\infty })$. Among them, a special attention has been paid to operators which reproduce affine functions. Due to the linearity of the operators, this property is implied by the preservation of the test functions $e_0$ and $e_1$. Set $e_j$, $j\in {\mathbb{N}}_0=\{0\}\cup \mathbb{N}$, the monomial of degree $j$. The starting point is given by discrete operators of the following form

verifying the conditions

where ${\lambda }_{n,k}$ is a continuous function defined on ${ℝ}_{+}$ with non-negative real values for each $k\in {\mathbb{N}}_0$, ${(x_{n,k})}_{k\ge 0}$ represents a net on ${ℝ}_{+}$ and function $f$ is chosen such that the series from relation (1.1) is convergent. Because such operators are not suitable for approximating discontinuous functions, they are generalized to integral type operators. One of the usual techniques is known as Durrmeyer method [12] which leads us to an approximation process, say ${(L_n^{\ast })}_{n\ge 1}$, in spaces of integrable functions.

This type of integral generalization, studied in depth for the first time in 1981 by Derriennic [7], has been constant in the attention of many researches, among the most recent papers is that of Abel and Karsli [1].

For each $n\in \mathbb{N}$, let ${({\mu }_{n,k})}_{k\ge 1}$ be a sequence of continuous and positive functions defined on ${ℝ}_{+}$ such that the following relation is fulfilled

Setting $I_{n,k}(f)={\displaystyle {\int }_{{ℝ}_{+}}}{\mu }_{n,k}(t)f(t)𝑑t$, $k\in \mathbb{N}$, we introduce the operators

where $f\in ℱ({ℝ}_{+})$, this space consisting of all real valued functions $f$ defined on ${ℝ}_{+}$ with the property ${\lambda }_{n,k}f$ belongs to the Lebesgue space $L_1({ℝ}_{+})$ for each $k\in \mathbb{N}$ and the series from the right hand side of relation (1.5) is convergent. Denoting by $C_B({ℝ}_{+})$ the space of all continuous and bounded functions defined on ${ℝ}_{+}$, it is clear that $C_B({ℝ}_{+})\subset ℱ({ℝ}_{+})$.

It is obvious that $L_n^{\ast }$, $n\in \mathbb{N}$, operators are linear and positive. Moreover, relations (1.4) and (1.2) imply

This also gives a formula for the norm of the operators, $\Vert L_n^{\ast }\Vert =1$.

To motivate the consistency of our study, our first concern is to highlight families of such operators that have already been studied. Next, some approximation properties of our general operators are revealed.

We point out that, alternatively, by using a two-dimensional kernel, the operators can be rewritten as follows

where

${\delta }_x(t)$ being Dirac generalized function.

Finally we introduce the $j$-th central moment of $L_n^{\ast }$ operators $(j\in {\mathbb{N}}_0)$, i.e.,

This can be accomplished by assuming that $e_j\in ℱ({ℝ}_{+})$.

The section includes three examples of known operators defined following the structure given by the relation (1.5).

1. Operators using Baskakov-Szász type bases. Choosing in (1.5)

and

the resulting operators (denoted by $M_n$) have been introduced in 2001 by Purshottam Agrawal and Ali Mohammad [4, Eq. (1.1)] in order to approximate a class of continuous functions of exponential growth. Their study was deepened by both the authors mentioned above [5] and by Gupta V., Gupta M.K. [13].

In this case, the conditions (1.2) and (1.4) are verified. Also, considering the network $x_{n,k}=k/n$, $k\in {\mathbb{N}}_0$, the relation (1.3) is fulfilled.

2. Szász-Mirakjan-Păltănea operators. The integral version we referred appeared in [18].

It depends on two parameters $\alpha >0$, $\rho >0$. In this case we have

and the class of operators is designed as follows

Both relations (1.2), (1.4) and relation (1.3) with $x_{n,k}=k/n$, $k\in {\mathbb{N}}_0$, are satisfied. Păltănea continued the study of these operators in [19]. Among the most recent papers that deepen the study of ${(L_{\alpha }^{\rho })}_{\alpha >0}$ we mention [17].

3. Durrmeyer-Jain operators. Using a Poisson-type distribution with two parameters given by

for $\alpha >0$ and $\beta \in [0,1)$, Jain [14] introduced the following class of positive linear operators

where $f\in C({ℝ}_{+})$ whenever the above series is convergent. In [2] an extension in Durrmeyer sense of $P_n^{[\beta ]}$ has been introduced, which was named ${\mathrm{\Lambda }}_n^{[\beta ]}$. In this case, in (1.5) we choose as follows

Relations (1.2) and (1.4) take place but (1.3) fails. We get

For the special case $\beta =0$, $L_n^{\ast }\equiv {\mathrm{\Lambda }}_n^{[0]}$, $n\in \mathbb{N}$, turn into Phillips operators [20]. Among the first papers in which the approximation properties of these operators were studied, we mention [16].

Inspired by this fact, for certain results that we will establish in the next section, we will impose the following condition to be fulfilled

This additional condition ensures the identity

and the quantity is strictly positive for $x>0$.

Following the line of Ditzian and Totik [10, §1.2], we consider $\phi :{ℝ}_{+}\to {ℝ}_{+}^{\ast }$ an admissible weight function. In order to give an estimate of the approximation error, we use the weighted $K$-functional of second order $f\in C_B({ℝ}_{+})$ defined as follows

where $g^{\prime }\in A{C}_{loc}({ℝ}_{+})$ means that $g$ is differentiable and $g^{\prime }$ is absolutely continuous on every compact of ${ℝ}_{+}^{\ast }$, [10, Eq. 2.1.1].

In accordance with [10, Theorem 2.1.1], for some constants $M>0$ and $t_0$, the following inequality

takes place, where

From Theorem 3.2 we deduce

Examining relation (1.2), on the basis of Korovkin’s first theorem [15], we can state

For investigating other properties of our sequence, we need the following technical result.

Next, we study the approximation properties for functions that are not necessarily bounded. We start with functions that satisfy a Hölder type condition with $\beta \in (0,1]$ on $E\subseteq {ℝ}_{+}$, more precisely functions $f:{ℝ}_{+}\to ℝ$ which verify the relation

where $C_{\beta ,f}$ is a positive constant depending only on $\beta$ and $f$. Clearly, such functions belong to $ℱ({ℝ}_{+})$. For $\beta >1$ relation (3.3) leads us to constant functions on $E$, so we are not interested at this time.

We prove a relation between the local smoothness of functions and the local approximation.

Further, we discuss about the type of convergence of the sequence ${(L_n^{\ast }f)}_{n\ge 1}$ to the continuous function $f$ that it approximates. On compact intervals, if $f$ is continuous, the uniform convergence of ${(L_n^{\ast }f)}_{n\ge 1}$ to $f$ takes place, see Theorem 3.4. On unbounded intervals, if $f$ is continuous, usually only pointwise convergence occurs. We will indicate sufficient conditions which ensure the uniform convergence on unbounded intervals $J\subseteq {ℝ}_{+}$. For discrete approximation processes such an approach can be found, for example, in [3]. Similar results have been obtained for a long time ago by Jésus de la Cal and Javier Cárcamo [6] for families of operators of probabilistic type over non-compact intervals. The authors used representation of the operators in terms of appropriate stochastic processes.

Set $UC({ℝ}_{+})$ the space of all uniformly continuous real valued function on ${ℝ}_{+}$. Let $\tau :{ℝ}_{+}\to {ℝ}_{+}$ be a continuous one-to-one function such that $\tau (0)=0$. We define the following two functions

where $f\in C_B({ℝ}_{+})$.

Next we evaluate $B_x$. Clearly, $(L_n^{\ast }{\tau }_x)(x)>0$ for $x\ne 0$. The definition of modulus of smoothness and property (3.11) applied for

allow us to write

In the above we also used the monotonicity of the function $\omega (f^{\ast };\cdot )$ as well as the hypothesis specified at (3.9). Since $f^{\ast }$ is bounded on the domain, consequently $\omega (f^{\ast };\cdot )$ is well defined. Thus, we have

where

Using the inequalities (3.14) and (3.15), relation (3) implies

Relation ${\lambda }_{n,0}(0)=1$ corroborated with (1.2) implies ${\lambda }_{n,k}(0)=0$, $k\in \mathbb{N}$, consequently $L_n^{\ast }$ enjoys the interpolating property in $x=0$. Thus, relation (3.16) is true in the particular case $a=0$ as well.

Property (3.12) guarantees $\omega (f^{\ast };{\gamma }_{2,n})$ tends to zero as $n$ tends to infinity. Alongside (3.8), this statement leads to the completion of the proof.  $\mathrm{\square }$