As the Bernstein operators represent the most investigated linear positive approximation process of functions defined on a compact interval, the Szász operators and their generalizations are the most studied approximation operators for functions defined on an unbounded interval. For the complete information of the readers, we will present the first expressions of these operators keeping the same notations used in the cited works.

In 1950, for the infinite interval $(0,\mathrm{\infty })$ Szász [32, Eq. (2)] defined the transform

proving the following main result: suppose that $f(x)$ is bounded in every finite interval; if $f(x)=𝒪(x^k)$ for some $k>0$ as $x\to \mathrm{\infty }$ and if $f(x)$ is continuous at a point $\xi$, then $P(u;f)$ converges uniformly to $f(x)$ at $x=\xi$ ([32, Theorem 1]).

As mentioned in [12, p. 547], the Polish mathematician Mark Kac pointed out to Butzer that he considered the above transform several years ago and made use of it in his lectures, but never published his result. In fact, the definition of this approximation process has been carried out over time by several mathematicians. Thus, in 1941 Mirakyan [27] considered a finite sum. The above series was also studied in 1944 by Favard [17].

The domain of definition of operators was considered in papers that appeared in different forms. For example, Sikkema [31, Application II] denoted $H^{(s)}(\xi )$ the set of all real functions $f(x)$ which are defined on the whole of the real $x$-axis and possess the following three properties: $f(x)$ is $s$ times $(s\ge 1)$ differentiable at $x=\xi$, $f(x)$ is bounded on every finite interval of the $x$-axis and $f(x)=𝒪({|x|}^s)$ if $|x|\to \mathrm{\infty }$. Further, Sikkema considered the operators $S_{n,p}:H^{(q)}(\xi )\to C([0,b])$, $\xi \in [0,b]$, where $b$ denotes an arbitrary positive number and $p$ is non-negative

In case $p=0$ we obtain the Mirakjan operators [27] which also have been considered by Szász in 1950. Case $p>0$ occurs in a paper by Schurer [30].

Altomare and Campiti in their monograph [6, Eqs. (5.3.54)-(5.3.57)] considered the Banach lattice

which becomes the domain for the $n$-th Szász operator, $S_{n}f\equiv P(n;f)$, $n\ge 1$, proving that for any $f\in E_2$ the series is absolutely convergent. Moreover, every $S_n$ maps $C_B({ℝ}_{+})$, respectively $C_0({ℝ}_{+})$, into itself. Here $C_B({ℝ}_{+})=C({ℝ}_{+})\cap B({ℝ}_{+})$ and $C_0({ℝ}_{+})$ is the space of all functions $f\in C({ℝ}_{+})$ which vanish at infinity. $B({ℝ}_{+})$ stands for the space of all real valued bounded functions defined on ${ℝ}_{+}$.

In our opinion, the most appropriate domain for operators is the space $E$ representing the class of all locally integrable functions of exponential type on ${ℝ}_{+}$ with the property $|f(t)|\le M{e}^{At}$ $(t\ge 0)$ for some constants $M,A>0$.

The Szász–Mirakjan operators associate to each function $f\in E$ the series

where

Results for approximating $L_p$ functions may have potential applications in machine learning generalization analysis. Moreover, problems for approximating functions in higher dimensions are more and more important in various applications of big data.

Butzer [12, Section 3] published, for the first time, an integral generalization in the Kantorovich sense of the genuine Szász operators. Using the $P_u^f$ notation for Szász operators, Butzer considered a function $f$ Lebesgue integrable over the interval $0\le x\le R$ for every $R>0$ and $F(x)=𝒪(x^k)$ as $x\to \mathrm{\infty }$, for some $k>0$, where

The new operators have the following form

which can be written as

where

for , $\nu =0,1,2,\mathrm{\dots }$; $K_u(x;0)=0$, $x\in {ℝ}_{+}$.

The main approximation property established by Butzer [12, Theorem 2] is read as follows: if $f(x)\in L(0,\mathrm{\infty })$, then

at every point where $f(x)=F^{\prime }(x)$, i.e. almost everywhere in $(0,\mathrm{\infty })$.

These integral operators were studied in different functional spaces, for example in $L^p(0,\mathrm{\infty })$, , spaces the investigation has been achieved by Totik [33].

It is worth emphasizing that direct and converse results for exponential-type operators (including Szász operators) and their Kantorovich analogue are analyzed in the monograph [16, Section 9.3].

The Kantorovich-type variant $W_u^f$ of Szász operators can be rewritten in the form

where $p_{n,\nu }(x)=e^{-nx}{(nx)}^{\nu }/\nu !$ $(\nu =0,1,\mathrm{\dots })$.

Recently (2022), Sabancigil, Kara, and Mahmudov [29] introduced, for $\mathrm{\ell }\in \mathbb{N}$, an $\mathrm{\ell }$-th order Kantorovich-type Szász–Mirakjan operator

Since $K_n^{\mathrm{\ell }=1}$ does not coincide with $K_n$ we define, for $\alpha \ge 0$,

such that $K_n^{\mathrm{\ell }}=K_n^{\mathrm{\ell },\mathrm{\ell }}$. A natural generalization of the operators $K_n$ are $K_n^{\mathrm{\ell },0}$ and $K_n^{\mathrm{\ell },\mathrm{\ell }-1}$. Note that in the special case $\mathrm{\ell }=1,$ these operators both reduce to the Kantorovich-type variant $K_n^{1,0}=K_n$.

It is easy to see that $K_n^{\mathrm{\ell },\alpha }$ preserves constant functions, i.e., $K_n^{\mathrm{\ell },\alpha }{e}_0=e_0$, where $e_r$ denote the monomials given by $e_r\left(x\right)=x^r$ $\left(r=0,1,2,\mathrm{\dots }\right)$. Since

we obtain

A direct computation yields

Thus, the Popoviciu–Bohman–Korovkin (see [28, 9, 25]) theorem implies

for all bounded continuous functions $f\in E$. Later on, we shall see that the approximation property is valid for all $f\in E$.

The power and, at the same time, the simpleness of the Popoviciu–Bohman–Korovkin criterion turned it into the main tool for studying linear positive approximation processes. Unfortunately, for operators acting on spaces of functions defined on unbounded intervals, the uniform convergence is not guaranteed by this theorem. This is true even if we restrict at a compact interval the functions obtained as images. An example in this direction was given by Ditzian [15]. To obtain the extension of this theorem in the case of non-compact intervals, additional conditions are required. We highlight that these conditions are not uniquely determined, they are varied depending on the considered function spaces. We propose a very short foray into this domain of research in order to point out some achievements. A pioneering activity in this direction was carried out by Boyanov and Veselinov [10]. In their approach they took into account functions that have a finite limit at infinity. We also note the significant results of Gadžhiev [19] who studied this subject in weighted spaces.

The approximation property of Szász transform $S_{n}f$ was settled in polynomial weight spaces by Becker [8] and in exponential weight spaces by Becker, Kucharski and Nessel [7]. To obtain results at the uniform convergence of the $S_{n}f$, $n\ge 1$, operators, Totik worked in the $C_B({ℝ}_{+})$ space involving modulus of continuity, see [34, Theorems 1-3]. By using the representation of the operators in terms of appropriate stochastic processes, J. de la Cal and Cárcamo [14] obtained new results on uniform approximation for families of operators of probabilistic type over non-compact interval. The same concerns to establish sufficient conditions of uniform convergence appeared, for example, in [11] and [4]. Among the particular cases presented in these two papers are the Szász operators.

We also mention that in more abstract context of metric spaces, Altomare [5] presented general results of Korovkin-type establishing conditions which ensure the uniform convergence.

In this paper, we derive the complete asymptotic expansion for the sequence of operators $K_n^{\mathrm{\ell },\alpha }$ in the form

provided that $f$ admits derivatives of sufficiently high order at $x>0$. Formula $\left(\text{<a href="\#S2.E1" title="In 2 Preliminaries ‣ Asymptotic properties of Kantorovich-type Szász–Mirakjan operators of higher order" class="ltx\_ref ltx\_font\_italic"><span class="ltx\_text ltx\_ref\_tag">1</span></a>}\right)$ means that, for all $q=1,2,\mathrm{\dots }$, there holds

The coefficients $a_k(f,\mathrm{\ell },\alpha ;x)$, independent of $n$, will be given in an explicit form. It turns out that Stirling numbers of the second kind play an important role. As a special case we obtain the complete asymptotic expansion for the sequence of Szász–Mirakjan–Kantorovich operators $K_n$.

Asymptotic expansions for the Bernstein-Kantorovich operators are derived in [2]. Asymptotic expansions for the Szász–Mirakjan operators and their generalizations can be found in [1].

For $q\in \mathbb{N}$ and $x\in (0,\mathrm{\infty })$, let $K[q;x]$ be the class of all functions $f\in E$ which are $q$ times differentiable at $x$. The following theorem presents as our main result the complete asymptotic expansion for the operators $K_n^{\mathrm{\ell },\alpha }$.

Here and in the following, the quantities $\left\{\genfrac{}{}{0.0pt}{}{\mathrm{\ell }}{j}\right\}$ denote the Stirling numbers of the second kind defined through

where $z^{\underset{¯}{0}}=1$, $z^{\underset{¯}{j}}=z\left(z-1\right)\mathrm{\cdots }\left(z-j+1\right)$, $j\in \mathbb{N}$, are the falling factorials (we follow the convention to define $\left\{\genfrac{}{}{0.0pt}{}{\mathrm{\ell }}{j}\right\}=0$, for all negative integers $j$).

In the case $q=1$, an immediate consequence of Theorem 1 is the following Voronovskaja-type formula.

In the case $\alpha =\mathrm{\ell }$, we have, for $f\in K[2;x]$, the Voronovskaja-type formula

which was derived in [29, Theorem 10], for bounded functions having bounded first and second derivatives on $[0,\mathrm{\infty })$.

In the special case $\alpha =0$ we obtain the following result for the Szász–Mirakjan-Kantorovich operators $K_n^{\mathrm{\ell },0}$.

We emphasize the fact that $a_k(f,\mathrm{\ell },\alpha =0;x)$ contains only derivatives $f^{\left(s\right)}\left(x\right)$ of orders $s=k,\mathrm{\dots },2k$.

We give the series explicitly, for $q=3$:

In order to prove our main result, we shall need some auxiliary results. Recall that throughout the paper $e_r$ denote the monomials $e_r\left(t\right)=t^r$ $\left(r=0,1,2,\mathrm{\dots }\right)$ and, for each real $x$, we put ${\psi }_x=e_1-x{e}_0$. The proof of Theorem 1 is based on several lemmas, which are gathered in this section.

We recall some known facts about Stirling numbers which will be useful in the sequel. The Stirling numbers of the second kind possess the representation

(see [13, page 226, Ex. 16]). The coefficients ${\sigma }_2(i,i-m)$, called associated Stirling numbers of the second kind, are independent of $r$. Furthermore, we make use of the following formula for iterated integrals (see [23, page 202, Eq. (4)]).

Hence, the second central moment satisfies $\left(K_n^{\mathrm{\ell },\alpha }{\psi }_x^2\right)\left(x\right)=O(1/n)\text{ as }n\to \mathrm{\infty }.$ More generally, we show

A direct consequence is the representation

provided that $n>\alpha$.