The paper falls within the field of Approximation Theory motivated by the fact that over time the interest in the study of linear approximation processes has experienced

a wide development. This is justified by providing an easy-to-use method of modeling signals belonging to various spaces of functions.

The published researches focused on different aspects: revealing new properties of classical positive linear operators, integral extensions of discrete operators, generalizations of these operators by introducing some parameters, or by multidimensional extensions. For an overview of this domain, Altomare and Campiti’s monograph [4] can be consulted.

The Szász-Mirakjan operators [20], [14], are among the most intensively studied positive linear operators of discrete type. In 1969 Jakimovski and Leviatan [11] presented a generalization of this approximation process involving the Appell polynomials. We recall that the sequence $\left(P_{n}(x)\right)_{n\geq 0}$ of polynomials introduced by Paul Émile Appell [7] satisfies the fundamental relation

$P_{0}(x)$ being a non-zero constant.
This sequence has a generating function of the form

where $A(t)$ is an analytic function with a power series expansion

in the disk $|t|<R,R>1$, with $A(0)\neq 0$. The property given in (1) is equivalent to the existence of a scalar sequence $\left(b_{k}\right)_{k\geq 0}$ with $b_{0}\neq 0$, such that

where $D=\frac{d}{dx}$.
The class of these polynomials is comprehensive, containing as special cases a large number of classical sequences of polynomials. Examples, up to normalization, are the Bernoulli polynomials, the Euler polynomials, the Hermite polynomials. This is one of the reasons that paper [11] served as a starting point for numerous articles that appeared over time.

Set $\mathbb{R}_{+}=[0,\infty)$. Denoting by $E\left(\mathbb{R}_{+}\right)$the class of functions of exponential type which satisfy the property $|f(t)|\leq ce^{at},t\geq 0$, for some finite positive constants $c$ and $a$, Jakimovski and Leviatan defined the operators $L_{n}:E\left(\mathbb{R}_{+}\right)\rightarrow C\left(\mathbb{R}_{+}\right)$,

Wood [22, Theorem 2.1] proved that $L_{n},n\geq 1$, are positive operators in [ $0,\infty$ ) if and only if

where $a_{k}$ are the coefficients in (3).
In the particular case $A(t)=1$, from (2) we deduce $P_{n}(x)=x^{n}$ and the operators defined by (4) turn into classical Szász-Mirakjan operators.

The purpose of the present paper is to define an integral modification in Durrmeyer sense of a generalization of operators $L_{n},n\geq 1$. This construction involves a multiple Appell system on a space of dimension $d\in\mathbb{N},d\geq 2$. Our first challenge is the definition of Appell polynomials in $d$-dimensional space. Regarding the new class of operators introduced, we investigate its approximation properties, including the complete asymptotic expansion with the explicit indication of all coefficients, also with respect to simultaneous approximation, in a concise form.

Our obtained results are integrated into the landscape of recently published publications. For confirmation we cite some papers that aim at generalizations of Jakimovski-Leviatan operators. Varma [21] studied a bidimensional extension of discrete operators $L_{n},n\geq 1$. New one and two dimensional variants of these discrete operators have been studied in 2020 and 2023, see [16] and [15]. Due to the properties of Appell polynomials, they have been intensively used in different constructions of several general classes of operators. Thus, using the power summability method, a generalized class of Szasz operators was investigated in [8]. A one dimensional Durrmeyer extension of $L_{n},n\geq 1$, operators involving Brenke type polynomials was approached in [3]. A similar extension with the aid of two-dimensional Appell polynomials was defined and explored by Ansari, Mursaleen and Rahman [6]. The Kantorovich type modification of Jakimovski-Leviatan operators based on Appell polynomials of dimension two was obtained, for example, in [9] and [19]. To complete the information about the operators of Kantorovich type implying multiple Appell polynomials we mention paper [10] in which the authors establish quantitative Voronovskaja and Grüss Voronovskaja-type theorems.

Finally, we mention that there are many papers that contain the expression multiple Appell polynomials in the title, the authors referring to the two-dimensional case. Consequently, our research is a substantial extension.

First of all we introduce useful notations to define our operators. Let $d\in\mathbb{N}$ be fixed. We use the following multi-index notation: for $\boldsymbol{v}=\left(v_{1},\ldots,v_{d}\right)\in\mathbb{N}_{0}^{d}$ we set
$|\boldsymbol{\nu}|=\nu_{1}+\ldots+\nu_{d}$. For real $m$, the multinomial coefficient is defined as

where $\Gamma$ stands for Gamma function. We recall that the condition $\boldsymbol{v}\geq 0$ means that all coordinates of the vector $\boldsymbol{v}$ are greater than or equal to zero. Since $\mathbb{N}_{0}^{d}\subset\mathbb{R}^{d},\boldsymbol{v}+\boldsymbol{\mu}$ and $\boldsymbol{v}-\boldsymbol{\mu}$ represent usual operations in the vector space $\mathbb{R}^{d}$.

A multiple polynomial system $\left\{s_{\nu_{1},\ldots,\nu_{d}}(x)\right\}_{\nu_{1},\ldots,\nu_{d}=0}^{\infty}$ with $\operatorname{deg}\left(s_{\nu_{1},\ldots,\nu_{d}}\right)=|\boldsymbol{\nu}|$ is called a multiple Appell system if it has a generating function of the form

where the real analytic function $H$ has a series expansion on an open set $U\subset\mathbb{R}^{d}$ in a neighborhood of the origin, containing $(1,\ldots,1)$, given by

with $H(0,\ldots,0)=h_{0,\ldots,0}\neq 0$.
For $d=2$ we rediscover the definition that appeared in several papers, for example in [12, Eqs. (2.1),(2.2)], [21, Eqs. (1.4),(1.5)], [6, Eq. (1.5)] or [9, Eq. (1.1)].

For our purposes we will impose some restrictions on the Appell system introduced above. We consider throughout the paper that the following conditions are met

Further we define the discrete $d$-dimensional operators of Szász-JakimovskiLeviatan by the formula

whenever the right-hand side of (8) is finite.
In the special case $d=2$ we find the construction from [21, Eq. (1.6)], the author proving various approximation properties of the sequence. Replacing in (8) the point evaluation $f(|\boldsymbol{\nu}|/n)$ with the integral

we propose the following type of Durrmeyer operators

where $x\in\mathbb{R}_{+},n\in\mathbb{N}$. We denote by $D\left(\mathbb{R}_{+}\right)$the space of functions to which $f$ belongs such that (10) is well defined for all $n\in\mathbb{N}$.

If we consider the function $\exp_{a}(t)=e^{at},t\in\mathbb{R}_{+}$, where $a>0$, we get, for $n>a$,

Hence, $\left(\bar{S}_{n}^{H}f\right)\left(\exp_{a}\right)$ is well defined for all $n$, which are so large that $\left(\frac{n}{n-a},\ldots,\frac{n}{n-a}\right)\in U$.

Obviously, the operators are linear and due to the conditions imposed on the multiple Appell polynomials, see (7), they are also positive. This shows that $\bar{S}_{n}^{H}f$ is well defined for all $f\in E\left(\mathbb{R}_{+}\right)$, provided that $n$ is sufficiently large.

Remark 1 In the special case $H\left(t_{1},\ldots,t_{d}\right)=1$ for all $\boldsymbol{t}$, based on (5) we get

and this implies $s_{\nu_{1},\ldots,\nu_{d}}(x)=x^{|\boldsymbol{\nu}|},x\geq 0$. Consequently, we can write

In the one-dimensional case, these operators are exactly the classical SzászDurrmeyer operators introduced and studied by Mazhar and Totik [13]

where $s_{n,k}(x)=e^{-nx}(nx)^{k}/k!,k\in\mathbb{N}_{0}$.
Finally, starting from $H$, we introduce

a key function that will be used to prove our results. Relying on (6), the properties of $H$ imply that $h$ is infinitely differentiable on a one-dimensional domain that includes a compact of the form $[0,R]$ with $R>1$.

At the beginning we determine the derivatives of the multiple Appell polynomials introduced in the previous section.

Lemma 1 For $j\in\mathbb{N}_{0}$ the following formula

holds.
Proof For $j=0$ the equation is evident. Differentiating both sides of relation (5) $j$ times $(j\in\mathbb{N})$ with respect to $x$, we obtain

Identifying the corresponding coefficients leads us to the desired formula. $\square$

Remark 2 From Lemma 1 it follows implicitly that $s_{v_{1},\ldots,v_{d}}^{(j)}(x)=0$ if $|\boldsymbol{v}|<j$.
Let $e_{r},r\in\mathbb{N}_{0}$, denote the monomials defined by $e_{r}(x)=x^{r}$. For technical reasons we consider in the following the slightly more general auxiliary operators $\bar{S}_{n,j}^{H},j\in\mathbb{N}_{0}$, given by

where $\Phi_{n,|\boldsymbol{v}|+j}$ is defined as in (9) and $f\in D_{j}\left(\mathbb{R}_{+}\right)$, where $D_{j}\left(\mathbb{R}_{+}\right)$is the space of functions $f$ for which $\bar{S}_{n,j}^{H}f$ is well defined for all $n\in\mathbb{N}$. As in the special case $j=0$, it can be seen that $\bar{S}_{n,j}^{H}f$ is well defined for all $f\in E\left(\mathbb{R}_{+}\right)$, provided that $n$ is sufficiently large.

Obviously, we have $\bar{S}_{n}^{H}\equiv\bar{S}_{n,0}^{H}$.

Lemma 2 For each $r$ and $j$ belonging to $\mathbb{N}_{0}$, the moments $\bar{S}_{n,j}^{H}e_{r}$ of the operators $\bar{S}_{n,j}^{H}$ are given by

Proof Examining relation (9), we notice that for any $p\in\mathbb{N}_{0}$ we can write

Hence, using relations (5) and (12), we get

Application of the Leibniz rule for differentiation yields the relation (13).

The next step is to express the central moments of the auxiliary operator $\bar{S}_{n,j}^{H}$. For each real number $x$, we define the function $\psi_{x}$ by $\psi_{x}=e_{1}-xe_{0}$.

Lemma 3 For each $s\in\mathbb{N}_{0}$, the central moment of order s of the operator $\bar{S}_{n,j}^{H}$ is given by

Proof Since $\bar{S}_{n,j}^{H}e_{0}=e_{0}$, by (13) we have

Using the binomial identity $\binom{s}{r}\binom{r}{k}=\binom{s}{k}\binom{s-k}{r-k}$ and index change replacing $r$ with $r+k$, we obtain

Further, we apply Leibniz rule to the product.
Also, involving the Kronecker delta symbol, we observe that

Assembling the results, we obtain relation (14).

We highlight some global and local approximation properties of $\bar{S}_{n}^{H},n\in\mathbb{N}$, operators. Using Lemma 2 and knowing that $h(1)=1$, see (11), through a direct calculation, we obtain the first three moments of the operators namely

Also, applying Lemma 3 and the above three identities, the central moments of order $s,s\in\{1,2\}$, are given as follows

Clearly, we get

However, we cannot apply the classical Korovkin theorem to obtain the uniform approximation on compacts of functions from the space ( $\mathbb{R}_{+}$) by the sequence of
operators $\left(\bar{S}_{n}^{H}\right)_{n}$, because the interval $\mathbb{R}_{+}$is not compact. It is only locally compact. For this we will appeal to the following Korovkin-type theorem of Altomare [5, Theorem 3.5]. Using the notations given there, this theorem reads as follows.

Theorem A Let ( $X,d$ ) be a locally compact metric space and consider a lattice subspace $E$ of $F(X)$ containing the constant function $\mathbf{1}$ and all the functions $d_{x}^{2}(x\in X)$. Let $\left(L_{n}\right)_{n\geq 1}$ be a sequence of positive linear operators from $E$ into $F(X)$ and assume that
(i) $\lim_{n\rightarrow\infty}L_{n}(\mathbf{1})=\mathbf{1}$, uniformly on compact subsets of $X$;
(ii) $\lim_{n\rightarrow\infty}L_{n}\left(d_{x}^{2}\right)(x)=0$, uniformly on compact subsets of $X$.

Then, for every $f\in E\cap C_{b}(X),\lim_{n\rightarrow\infty}L_{n}(f)=f$, uniformly on compact subsets of $X$.

In this theorem, $F(X)$ denotes the space of real functions defined on $X,\mathbf{1}$ denotes the constant function equal to $1,d_{x}$ denotes the function $d_{x}(y)=d^{2}(x,y),y\in X$ and $C_{b}(X)$ denotes the space of real continuous bounded functions defined on $X$.

We also need of the following limit, which follows immediately from (15):

We denote by $C_{B}\left(\mathbb{R}_{+}\right)$the Banach lattice of all real-valued continuous and bounded functions on $\mathbb{R}_{+}$, endowed with the sup-norm $\|\cdot\|_{\infty},\|f\|_{\infty}=\sup_{x\geq 0}|f(x)|$. If $K\subset\mathbb{R}_{+}$ is a compact interval, the same norm is valid in the space $C(K)$.

Remark3 For each $n\in\mathbb{N},\bar{S}_{n}^{H}$ maps continuously $C_{B}\left(\mathbb{R}_{+}\right)$into itself. This means that $C_{B}\left(\mathbb{R}_{+}\right)\subset D\left(\mathbb{R}_{+}\right)$. For $f\in C_{B}\left(\mathbb{R}_{+}\right)$, we have $\left|\Phi_{n,|\boldsymbol{v}|}(f)\right|\leq\|f\|_{\infty},|\boldsymbol{v}|\in\mathbb{N}_{0}$, and based on the value of $S_{n,|\boldsymbol{v}|}^{H}e_{0}$ we deduce $\left\|S_{n}^{H}f\right\|_{\infty}\leq\|f\|_{\infty}$.

Theorem 1 Let the operators $\bar{S}_{n}^{H},n\in\mathbb{N}$, be defined by (10). For any compact interval $K\subset\mathbb{R}_{+}$, the following relation

occurs, provided $f\in C_{B}\left(\mathbb{R}_{+}\right)$.
Proof We can apply the theorem of Altomare, given above, with the following choices: $X=\mathbb{R}_{+},d(x,y)=|x-y|,x,y\in X,E=D\left(\mathbb{R}_{+}\right),C_{b}(X)=C_{B}\left(\mathbb{R}_{+}\right),\mathbf{1}=e_{0}$, $d_{x}^{2}=\psi_{x}^{2}$ and $L_{n}=\bar{S}_{n}^{H}$.

We take into account that $C_{B}\left(\mathbb{R}_{+}\right)\cap D\left(\mathbb{R}_{+}\right)=C_{B}\left(\mathbb{R}_{+}\right)$.
The rate of convergence is established by using modulus of continuity defined for a function $f:\mathbb{R}_{+}\rightarrow\mathbb{R}$ by

if this supremum is finite.

Denote by $B_{\omega}\left(\mathbb{R}_{+}\right)$the space of locally integrable functions $f:\mathbb{R}_{+}\rightarrow\mathbb{R}$ for which $\omega(f;\delta)<\infty$, for all $\delta>0$. It is immediate that $C_{B}\left(\mathbb{R}_{+}\right)\subset B_{\omega}\left(\mathbb{R}_{+}\right)$and $UC\left(\mathbb{R}_{+}\right)\subset B_{\omega}\left(\mathbb{R}_{+}\right)$, where $UC\left(\mathbb{R}_{+}\right)$is the space of uniformly continuous functions on $\mathbb{R}_{+}$.

Remark 4 Operators $\bar{S}_{n}^{H},n\in\mathbb{N}$, are well defined on $B_{\omega}\left(\mathbb{R}_{+}\right)$. Indeed, if $f\in B_{\omega}\left(\mathbb{R}_{+}\right)$, then $|f(t)|\leq(1+t/\delta)\omega(f;\delta)+|f(0)|$, for $t\geq 0,\delta>0$. It follows that the series of the absolute values of the terms of series (10) is bounded by $|f(0)|+(1+\left.(1/\delta)\left(\bar{S}_{n}^{H}e_{1}\right)(x)\right)\omega(f;\delta)<\infty$.

Theorem 2 Let the operators $\bar{S}_{n}^{H},n\in\mathbb{N}$, be defined by (10). For any function $f\in B_{\omega}\left(\mathbb{R}_{+}\right)$, we have

where $c(h)=2+4h^{\prime}(1)+h^{\prime\prime}(1)$.
Proof We use the following property proved by Shisha and Mond [17, Theorem 1] that says: if $L$ is a linear positive operator, then one has

for every continuous function $f$ and $\delta>0$. We mention that in order to fulfill the condition $\omega(f;\delta)<\infty$, the original relation was given for continuous functions defined on a compact. The proof of the Shisha and Mond theorem can be extended without any difficulty to space $\boldsymbol{B}_{\omega}\left(\mathbb{R}_{+}\right)$

Considering $L:=\bar{S}_{n}^{H}$, using the following two identities, $\bar{S}_{n}^{H}e_{0}=e_{0}$ and (15) and finally choosing $\delta=1/\sqrt{n}$, we arrive at the stated relation.

Corollary 3 For any $f\in UC\left(\mathbb{R}_{+}\right)$we have

Proof We take into account that $UC\left(\mathbb{R}_{+}\right)\subset B_{\omega}\left(\mathbb{R}_{+}\right)$and hence Theorem 2 can be applied. Also, because $f\in UC\left(\mathbb{R}_{+}\right)$, we have

uniformly if $x\in K,K$ being a compact subset of $\mathbb{R}_{+}$.

Note that the result in this Corollary cannot be compared with the result in Theorem 1 , because the spaces $C_{B}\left(\mathbb{R}_{+}\right)$and $UC\left(\mathbb{R}_{+}\right)$are not comparable.

The following result concerns the asymptotic expansion of the operators. To achieve this we require certain conditions to be satisfied by the function $f\in D\left(\mathbb{R}_{+}\right)$, these being similar to those appearing in Sikkema’s paper [18, Definition]. For $x>0$ and $s\in\mathbb{N}$, let $K[s,x]$ denote the functions possessing the properties: $f$ is $s$ times differentiable at $x,f$ is bounded on every bounded interval of the domain and $f(t)=\mathcal{O}\left(t^{s}\right)$ as $t\rightarrow\infty$. Moreover, we assume that all $f\in K[s,x]$ are locally integrable on $\mathbb{R}_{+}$.

Theorem 4 Let the operators $\bar{S}_{n,j}^{H},(n,j)\in\mathbb{N}\times\mathbb{N}_{0}$, be defined by (12). Given $x>0$ and $f\in K[2q,x]$, the asymptotic relation

holds, where the coefficients $c_{k,j}(f,x)$ are given by

Proof In particular Lemma 3 guarantees

Following Lemma 3 and Sikkema’s result, [18, Theorem 3], we infer that

with