The classical Korovkin theory is mainly connected with the approximation of continuous functions by linear operators having as essential ingredient the property of positivity. Considering such operators, say $L_n$ $(n\in \mathbb{N})$, they have the approximation property on the normed space $(S,\parallel \cdot {\parallel }_S)$ if $\underset{n\to \mathrm{\infty }}{lim}{\Vert L_{n}f-f\Vert }_S=0$ for each $f\in S$.

In what follows, we consider linear positive operators of discrete type which approximate functions defined on ${ℝ}_{+}=[0,\mathrm{\infty })$. In order to construct the proposed class of operators, we use a strictly decreasing positive sequence ${({\lambda }_n)}_{n\ge 1}$ with the property

The operators investigated in this paper are designed as follows

where $f$ is continuous function on ${ℝ}_{+}$ satisfying a certain growth condition. We also require

to be fulfilled.

The construction given at (2) is not new. With a slight modification, it appears in [7, Eq. (1.2)]. In this mentioned paper, K. Balázs studied the special case ${\lambda }_n=n^{-1/3}$. In time, another special case has been studied deeper

and for these particular operators we use the notation $R_n^{[\beta ]}$. We point out some of the most outstanding results concerning these operators. Balázs and Szabados [8] gave weighted estimated and investigated the uniform convergence of $R_n^{[\beta ]}$. V. Totik [19] settled the saturation properties of $R_n^{[\beta ]}$, proved a general convergence theorem for $R_n^{[\beta ]}$ – like rational functions and obtained a Voronovskaja-type result. Biancamaria Della Vecchia [9] proved some preservation properties and asymptotic relations for $R_n^{[\beta ]}$. Further on, Ulrich Abel and B. Della Vecchia [4] obtained the complete asymptotic expansion for $R_n^{[\beta ]}$, all coefficients being calculated explicitly. An integral generalization in Kantorovich sense was achieved in [5].

In Quantum Calculus, the $q$-analogues of $R_n^{[\beta ]}$ operators were studied by O. Doğru [11]. The rational complex $R_n^{[\beta ]}$ operators, , appear in the monograph of S.G. Gal [13]. See, respectively, the papers of Ispir and Yildiz Ozkan [14], Mahmudov [17].

Returning at general form indicated by relation (2), in the recent paper [3] the authors showed that $L_n$, $n\in \mathbb{N}$, operators enjoy the variation detracting property. At this moment, we want to point out some features of $L_n$, $n\in \mathbb{N}$, operators. They are built using a finite sum. Other classes of discrete operators are expressed by series, making them harder to use in approximation of functions, see for example, Baskakov operators or Favard-Szász-Mirakjan operators. For each $n\in \mathbb{N}$, $L_n$ uses an equidistant net ${\mathrm{\Delta }}_n={(k/(n{\lambda }_n))}_{k=\overline{0,n}}$ with a flexible step, that is ${(n{\lambda }_n)}^{-1}$. It changes depending on an arbitrary sequence ${({\lambda }_n)}_{n\ge 1}$ satisfying the conditions (1) and (3).

Regarding the definition of $L_n$, $n\in \mathbb{N}$, operators, it is fair to mention that a more general form was proposed a long time ago by Abdulofizov [1]. The author considered the linear operators $S_n$, $n\in \mathbb{N}$, defined as follows

where

Instead of the net used in (2), here appears a net of more general nodes. Estimations of $|(S_{n}f)(x)-f(x)|$ under various assumptions concerning $f$ have been proved by Abdulofizov. Modifications of Bernstein operators by using nodes of the form ${({\xi }_{k,n})}_{1\le k\le n}$ can also be found in [2].

The approximation properties highlighted in our paper are different from those obtained in [1]. More precisely, we establish two variants of local upper bounds of the error of approximation, one of them including the distance $d(x,E)$ between a certain $x>0$ and a subset $E\subset {ℝ}_{+}$. Also, we prove that our class of operators does not always form an approximation process in some weighted spaces. Further on, by using a weighted modulus of smoothness defined in 2004, we give both a local and global estimation of the rate of convergence. It is known that a more general class of sequences enjoys fewer specific properties than special cases of that. Particular cases narrow the approach but sometimes provide better approximation variants.

Set ${\mathbb{N}}_0=\{0\}\cup \mathbb{N}$. For every $m\in {\mathbb{N}}_0$ we shall denote by $e_m$ the monomial defined as $e_0(t)=1$ and $e_m(t)=t^m$ $(m\ge 1)$, where $t\in {ℝ}_{+}$.

(ii) $(L_{n}f)(0)=f(0)$, accordingly the operators interpolate functions in $x=0$.

We introduce the $s$-th order central moment of the operator $L_n$, $s\in {\mathbb{N}}_0$, that is ${\mu }_{n,s}(x)=(L_n{\phi }_{x,s})(x)$, where

In view of the identities (5)-(7), by a simple computation, for any $x\ge 0$, we get

We mention that in the particular case (4), for $s\in \mathbb{N}$ and for all integers $n>x^{1/(1-\beta )}$, the moments $R_n^{[\beta ]}{e}_s$ and the central moments $R_n^{[\beta ]}({(\cdot -x)}^s,\cdot )$ have been determined in the paper [4, Lemma 3.1, Lemma 3.3].

In the sequel we denote by $C_B({ℝ}_{+})$ the Banach lattice of all real-valued bounded and continuous functions on ${ℝ}_{+}$ endowed with the natural order and the sup-norm $\parallel \cdot {\parallel }_{\mathrm{\infty }}$,

If $K$ is a compact interval (in our case $K\subset {ℝ}_{+}$), the same norm is valid in the space $C(K)$.

We also recall the notion of modulus of smoothness associated with any bounded function $h$ defined on ${ℝ}_{+}$. For any $\delta \ge 0$, it is given as follows

If $h\in C({ℝ}_{+})$ and supremum in (2) is taken only for $(x^{\prime },x^{\prime \prime })\in K\times K$, $K\subset {ℝ}_{+}$ compact interval, we can use the writing option ${\omega }_K(h;\cdot )$.

The significant properties of $\omega (h;\cdot )$ are presented, e.g., in [16, pp. 43-46]. Among these, we emphasize that ${\omega }_h$ is a non-decreasing function and

Modulus of smoothness is continuous at $\delta =0$, i.e.

if and only if $h$ is uniformly continuous on its domain, [10, page 40] can be consulted.

At the end of this section we present a probabilistic approach of the studied sequence. To accomplish this, we use a classical scheme that can be found, for example, in [6, Section 5.2]. We are fixing a probability space $(\mathrm{\Omega },ℱ,P)$ and a random scheme on ${ℝ}_{+}$, $Z:\mathbb{N}\times {ℝ}_{+}\to {ℳ}_2(\mathrm{\Omega })$, where ${ℳ}_2(\mathrm{\Omega })$ is the space of all real square-integrable random variables on $\mathrm{\Omega }$. $Z(n,x)$ is discretely distributed taking the values $k/(n{\lambda }_n)$, $0\le k\le n$, such that

Consequently, for $f\in C_B({ℝ}_{+})$ we get

which leads us to relation (2). In the above $E$ is the expected value operator and $P_{Z(n,x)}$ represents the distribution of $Z(n,x)$ with respect to $P$. The above identities imply $\Vert L_{n}f\Vert \le \Vert f\Vert$.

We show that the sequence ${(L_n)}_{n\ge 1}$ is an approximation process in a certain space, more precisely it takes place

The rate of convergence will be established by using modulus of smoothness defined by (2).

Choosing $\mathrm{\Lambda }=L_n$ and taking in view (5) and (9), relation (17) becomes

On the other hand, from (9) we get

Taking in (18) $\delta =\sqrt{{\lambda }_n^{\ast }}$, the proof is ended. $\mathrm{\square }$

(ii) In the light of the definition of ${\lambda }_n^{\ast }$ and relations (1), (3), we infer the following two possibilities, preserving the condition that the sequence ${({\lambda }_n)}_{n\ge 1}$ to be strictly decreasing positive.

$\alpha )$ ${\lambda }_n\le n^{-1/3}$ and $\underset{n\to \mathrm{\infty }}{lim}n{\lambda }_n=\mathrm{\infty }$. In this case ${\lambda }_n^{\ast }={(n{\lambda }_n)}^{-1}$.

$\beta )$ ${\lambda }_n>n^{-1/3}$ and $\underset{n\to \mathrm{\infty }}{lim}{\lambda }_n=0$. Now, we have ${\lambda }_n^{\ast }={\lambda }_n^2$.

It is noted that in the above, the numerical sequence ${(n^{-1/3})}_{n\ge 1}$ appears. This sequence was used in the first study dedicated to Bernstein type rational functions, see K. Balázs’ work [7].

In the next step we prove a relation between the local smoothness of functions and the local approximation. For the completeness of the information, we recall that a continuous function $f:{ℝ}_{+}\to ℝ$ is locally $\mathrm{Lip}\alpha$ on $E$ (, $E\subset {ℝ}_{+}$) if it satisfies the condition

where $M_f$ is a constant depending only on $f$.

Also, the distance between $x\in {ℝ}_{+}$ and $E$ is denoted by $d(x,E)$ and defined as follows

Unfortunately this class of operators is not an approximation process in some weighted spaces. We present such a negative result, indicating first the general frame of our study. Let $\phi$ be a continuous and strictly increasing function defined on ${ℝ}_{+}$ such that $\underset{x\to \mathrm{\infty }}{lim}\phi (x)=\mathrm{\infty }$. Set $\rho =1+{\phi }^2$. We consider the weighted space

$M_f$ being a positive constant depending only on $f$, and the following subspaces of $B_{\rho }({ℝ}_{+})$

The usual norm of these spaces is $\parallel \cdot {\parallel }_{\rho }$ defined by

For any $f\in C_{\rho }({ℝ}_{+})$, based on (23), we have

Hence, the operator $L_n$ maps $C_{\rho }({ℝ}_{+})$ into $C_{\rho }({ℝ}_{+})\subset B_{\rho }({ℝ}_{+})$ if and only if

where $M$ is a constant.

We recall the following result due to Gadzhiev [12, Theorem 2] which says: a sequence ${(A_n)}_{n\ge 1}$ of linear and positive operators acting from $C_{\rho }({ℝ}_{+})$ to $B_{\rho }({ℝ}_{+})$ satisfies

if and only if

We mention that in [12] the result was presented without ”only if” and considering the set $ℝ$. Clearly, to insert ”only if” is trivial and the use of the set ${ℝ}_{+}$ does not change the result.

As it seen above, the weight function $\rho$ not only characterizes the growth of $f$ at infinity but also defines the test functions in Korovkin type theorem.

Choosing a very simple weight namely $\phi :=e_1$, based on (5) and (7), we find that for $L_n$, $n\ge 1$, relation (24) is fulfilled. For example, we can consider $M=1+{(\underset{n\ge 1}{inf}(n{\lambda }_n))}^{-1}$. Trying to apply Gadzhiev’s result for our operators, we notice that (26) does not hold for $\nu =2$. Indeed,

consequently $\underset{n\to \mathrm{\infty }}{lim}{\Vert L_n{e}_2-e_2\Vert }_{\rho }\ne 0$.

We can enunciate

It is worth mentioning that for this particular weight we could use the fact that $\{e_0,e_1,e_2\}$ is a strict Korovkin subset in $C_{\rho }^0({ℝ}_{+})$, see the monograph [6, Proposition 4.2.5, statement (6)].

For a more general weight, i.e. $\phi (x)=x^{m/2}$, $m\ge 2$ fixed, we give estimates of the error $|(L_{n}f)(x)-f(x)|$, $n\in \mathbb{N}$, for functions $f$ belonging to $C_{\rho }^0({ℝ}_{+})$. Clearly, this does not mean that the sequence is an approximation process on the mentioned space, the evaluation being pointwise. For our purpose we use a weighted modulus of smoothness defined as follows

Obviously, ${\mathrm{\Omega }}_m(f;\delta )\le 2{\Vert f\Vert }_{\rho }$, $\delta >0$, $f\in B_{\rho }({ℝ}_{+})$, and it possesses the following main properties ([15])

Set ${\theta }_{x,m}(t)=1+{(2x+t)}^m$. We recall $|{\phi }_{x,1}(t)|=|t-x|$, see (8). By using definition of ${\mathrm{\Omega }}_m(f;\cdot )$ and properties (28), we can write

Knowing that the operators are linear positive therefore monotone, and applying Cauchy inequality, we get

Since $L_n{e}_m\le C_1(1+e_m)$, $C_1$ a certain constant depending on $m$, there are positive constants $C_2$ and $C_3$ depending also on $m$, such that

On the other hand, based on (9), we deduce

Returning at (30) and choosing $\delta =\sqrt{{\lambda }_n^{\ast }}$, the result follows. $\mathrm{\square }$