Convergence of Special Sequences of Semi-Exponential Operators

Abstract

Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results of this type have been obtained with analytic methods. Semi-exponential operators have also been introduced, extending the theory of exponential operators. We combine these two approaches, applying the semi-exponential operators in a new context and enlarging the list of operators representable as limits of other operators.

1. Introduction

The theory of approximation by positive linear operators is a multifaceted theory. Some classical topics include convergence toward the identity operator, rate of convergence, Voronovskaya-type results, saturation, complete asymptotic expansions, shape-preserving properties, and iterates. We recall here two special topics.

(A)

Operators representable as limits of sequences of other operators. Making intensive use of probabilistic methods, it is possible to show that certain positive linear operators can be represented as limits of suitably modified classical operators (see, e.g., [1] and several subsequent papers by J. de la Cal and co-authors [2,3,4,5,6] and the references therein).

(B)

Exponential operators. Some important operators, called exponential operators, were intensively investigated in several papers (see, e.g., [7,8,9,10,11,12,13,14]).

The research related to (A) and (B) is still active. In [15], using the mentioned probabilistic methods, the authors obtained a general result, expressed in purely analytic terms, for studying the representation of certain operators as limits of other operators. This led to a large list of new results and new examples.

In [16,17,18], the theory of exponential operators was extended by considering so-called semi-exponential operators.

In this paper, we combine these two new approaches. We apply the general technique from [15] to sequences of semi-exponential operators and to sequences of compositions of operators. On the one hand, this casts the theory of semi-exponential operators in a new light. On the other hand, it enlarges the list of operators representable as limits of other operators, in the sense described in (A).

The main result of [15], which will be instrumental in our paper, can be presented as follows:

Let $I\subseteq \mathbb{R}$ be an interval and $C_b\left(I\right)$ be the space of all real-valued continuous and bounded functions on I. The functions considered in this paper are assumed to be in $C_b\left(I\right)$.

Let $Z^x,Z_1^x,Z_2^x,\cdots$ be I-valued random variables with probability distributions depending on a parameter $x\in I$. Suppose that for each $f\in C_b\left(I\right)$ the functions $x\to Ef\left(Z^x\right)$ and $x\to Ef\left(Z_n^x\right),x\in I,n\ge 1$, are continuous on I.

We will be concerned with positive linear operators

$$L_n:C_b\left(I\right)\to C_b\left(I\right),n\ge 1,\mathrm{and}L:C_b\left(I\right)\to C_b\left(I\right),$$

defined for $f\in C_b\left(I\right)$ and $x\in I$ by

$$L_n(f\left(t\right);x):=Ef\left(Z_n^x\right),L(f\left(t\right);x):=Ef\left(Z^x\right).$$

(1)

For each $s\in \mathbb{R}$, we consider the function $t\to e^{ist},t\in I$, and define

$$L_n(e^{ist};x)=L_n(cos\left(st\right);x)+i{L}_n(sin\left(st\right);x)$$

and similarly for $L(e^{ist};x)$.

Theorem 1 

([15]). Suppose that for each $s\in \mathbb{R}$ and $x\in I$,

$${\displaystyle \underset{n\to \infty }{lim}{L}_n(e^{ist};x)=L(e^{ist};x),}$$

(2)

and for each $x\in I$ the function $s\to L(e^{ist};x)$ is continuous on $\mathbb{R}$. Then

$${\displaystyle \underset{n\to \infty }{lim}{L}_n(f\left(t\right);x)=L(f\left(t\right);x)}$$

(3)

for all $f\in C_b\left(I\right)$ and $x\in I$.

In the next sections, we will be concerned with classical positive linear operators $L_n$ and L for which it is easy to identify the corresponding random variables. The interval I will be $[0,1]$ or $[0,\infty )$ or $(-\infty ,+\infty )$, depending on the structure of the involved operators.

Section 2 is devoted to a basic definition and some examples illustrating the properties $\left(C_1\right)$ and $\left(C_2\right)$ introduced by this definition. In Section 3, we consider the semi-exponential Bernstein operators and apply Theorem 1 to prove that they have the property $\left(C_1\right)$ introduced in Definition 1. Similar results are presented in Section 4 for semi-exponential Post–Widder operators. Obviously, the properties $\left(C_1\right)$ and $\left(C_2\right)$ are related to compositions of operators. This fact is illustrated in Section 5. Here we discuss the case of Jakimovski–Leviatan-type operators, the composition of Post-Widder and semi-exponential Szász–Mirakjan operators, the composition of Post–Widder and Szász–Durrmeyer operators and the case of Balázs–Szabados operators. In Section 6, we present a property, similar to $\left(C_1\right)$, shared by the semi-exponential Gauss–Weierstrass operators and the Bernstein operators.

2. Definitions and Examples

An appropriate modification of a sequence of operators generates a new sequence with a prescribed limit. We describe two such modifications. Other modifications are described in [15] or will be described in the next sections.

Definition 1. 

We say that the sequence ${\left(L_n\right)}_{n\ge 1}$ has the property $\left(C_1\right)$, if for each $m\ge 1$ there exists an operator $R_m$ such that

$${\displaystyle \underset{n\to \infty }{lim}{L}_{mn}\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)=R_m(f\left(t\right);x),f\in C_b\left(I\right),x\in I.}$$

(4)

We say that the sequence ${\left(L_n\right)}_{n\ge 1}$ has the property $\left(C_2\right)$, if for each $m\ge 1$ there exists an operator $Q_m$ such that

$${\displaystyle \underset{n\to \infty }{lim}{L}_m\left(f\left(\frac{t}{n}\right);nx\right)=Q_m(f\left(t\right);x),f\in C_b\left(I\right),x\in I.}$$

(5)

Remark 1. 

It is easy to verify (see, e.g., ([15] Proposition 12.1, Proposition 12.2)) that

$$R_{m\nu }\left(f\left(\nu t\right);\frac{x}{\nu }\right)=R_m\left(f\left(t\right);x\right),m\ge 1,\nu \ge 1,f\in C_b\left(I\right),x\in I.$$

(6)

$$Q_m\left(f\left(\frac{t}{\nu }\right);\nu x\right)=Q_m\left(f\left(t\right),x\right),m\ge 1,\nu \ge 1,f\in C_b\left(I\right),x\in I.$$

(7)

Denote by $B_n$, $V_n$ and $S_n$ the classical Bernstein operators, Baskakov operators and Szász–Mirakjan operators, respectively

$$\begin{array}{cc}\hfill & B_n(f;x)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)x^j{(1-x)}^{n-j}f\left(\frac{j}{n}\right),\hfill \\ \hfill & V_n(f\left(t\right);x)=\sum _{k=0}^{\infty }\left(\genfrac{}{}{0pt}{}{n+k-1}{k}\right)\frac{x^k}{{(1+x)}^{n+k}}f\left(\frac{k}{n}\right),\hfill \\ \hfill & S_n(f;x)=\sum _{j=0}^{\infty }{e}^{-nx}\frac{{\left(nx\right)}^j}{j!}f\left(\frac{j}{n}\right).\hfill \end{array}$$

Example 1 (see ([1] Theorem 2)). (a) The sequence of the classical Baskakov operators has the properties $\left(C_1\right)$ and $\left(C_2\right)$ with $R_m=S_m$ and $Q_m=G_m$ (for the definition of $G_m$ see Example 3 below).

(b) The sequence of Bernstein operators has the property $\left(C_1\right)$ with $R_m=S_m$.

Example 2. 

The sequence $\left(S_m\right)$ has the property $\left(C_1\right)$ with $R_m=S_m$ and, moreover, it satisfies (6). We will show that it also has the property $\left(C_2\right)$ with $Q_m=Id$, where $Id$ is the identity operator.

Indeed, let us choose $L_m=S_m$ and $Q_m=Id$. Then

$$\underset{n\to \infty }{lim}{S}_m\left(f\left(\frac{t}{n}\right);nx\right)=\underset{n\to \infty }{lim}{S}_{mn}(f\left(t\right);x)=f\left(x\right)=Id(f\left(t\right);x),$$

so that (5) is verified.

Example 3. 

Consider the Post–Widder operators indexed by integers $m\ge 1$ (see [19] (9.1.9)),

$$P_m\left(f\left(t\right);x\right)={\displaystyle \frac{1}{(m-1)!}}{\left({\displaystyle \frac{m}{x}}\right)}^m{\int }_0^{\infty }{e}^{-mu/x}{u}^{m-1}f\left(u\right)du,$$

where $f\in C_b[0,\infty )$, $x\in [0,\infty )$. (If $x=0$, $P_m(f\left(t\right);0)=f\left(0\right))$.

With $u={\displaystyle \frac{t}{m}}$ we get

$${\displaystyle P_m(f\left(t\right);x)={\displaystyle \frac{x^{-m}}{(m-1)!}}{\int }_0^{\infty }{t}^{m-1}{e}^{-t/x}f\left({\displaystyle \frac{t}{m}}\right)dt.}$$

(8)

The operators given by (8) are called in [1] Gamma operators and denoted by $G_m$ (see also [15] (10.1)). Thus

$$P_m(f\left(t\right);x)=G_m(f\left(t\right);x).$$

We know (see [15] Theorem 12.2 and Example 12.3) that

$$G_m\left(f\left(\frac{t}{\nu }\right);\nu x\right)=G_m(f\left(t\right);x),m\ge 1,\nu >0,t\ge 0,$$

and so

$$P_m\left(f\left(\frac{t}{\nu }\right);\nu x\right)=P_m(f\left(t\right);x),m\ge 1,\nu >0,t\ge 0.$$

(9)

Of course, (9) can also be proved directly. It shows that the sequence $\left(P_m\right)$ satisfies (7) and consequently has the property ($C_2$).

Many other examples can be found in [15].

Remark 2. 

Consider now the semigroup of operators ${\left(V\left(u\right)\right)}_{u\ge 0}$ approximated by iterates of $P_n$, namely

$$V\left(u\right)(f\left(t\right);x)=\underset{n\to \infty }{lim}{P}_n^{\left[nu\right]}(f\left(t\right);x),u\ge 0,x\ge 0.$$

(10)

For all the details, see, e.g., [7] and the references therein. From (9) and (10), we see that

$$V\left(u\right)\left(f\left(\frac{t}{\nu }\right);\nu x\right)=V\left(u\right)(f\left(t\right);x),u\ge 0,x\ge 0,\nu >0.$$

(11)

In fact (see [7]),

$$V\left(u\right)(f\left(t\right);x)={\displaystyle \frac{1}{\sqrt{2\pi u}}}{\int }_{\mathbb{R}}f\left(x{e}^{y-u/2}\right)e^{-y^2/2u}dy.$$

Moreover, the function

$$v_f(x,u):=V\left(u\right)(f\left(t\right);x)$$

(12)

is the solution to the problem

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial v}{\partial u}}(x,u)={\displaystyle \frac{x^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}(x,u),x\ge 0,u\ge 0,\hfill \\ \hfill \\ v(x,0)=f\left(x\right),x\ge 0.\hfill \end{array}\right.$$

(13)

For $\nu \ge 0$ set $f_{\nu }\left(t\right):=f\left(\nu t\right),t\ge 0$. From (11)–(13), we deduce

$$v_{f_{\nu }}(x,u)=v_f(\nu x,u),x\ge 0,u\ge 0.$$

Example 4. 

Let $s_k\left(x\right)=e^{-x}{\displaystyle \frac{x^k}{k!}}$. The Ismail–May exponential operators defined as (see [8,20])

$$T_n(f\left(t\right);x)={\displaystyle \frac{n}{\sqrt{x}}}\sum _{k=1}^{\infty }{s}_k\left(n\sqrt{x}\right){\int }_0^{\infty }{s}_{k-1}\left({\displaystyle \frac{nt}{\sqrt{x}}}\right)f\left(t\right)dt+e^{-n\sqrt{x}}f\left(0\right),x>0,$$

and

$$T_n(f\left(t\right);0)=f\left(0\right),$$

satisfy (see [20] Lemma 1)

$$T_n(e^{At};x)=exp{\displaystyle \frac{nAx}{n-A\sqrt{x}}},n>\left|A\right|\sqrt{x}.$$

It is easy to verify that for each $m\ge 1$, $s\in \mathbb{R}$ and $x\ge 0$,

$${\displaystyle \underset{n\to \infty }{lim}{T}_{mn}\left(e^{isnt};\frac{x}{n}\right)=e^{isx}=Id(e^{ist};x)}$$

and

$${\displaystyle \underset{n\to \infty }{lim}{T}_m\left(e^{ist/n};nx\right)=e^{isx}=Id(e^{ist};x).}$$

According to Theorem 1, it follows that

$${\displaystyle \underset{n\to \infty }{lim}{T}_{mn}\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)=f\left(x\right),f\in C_b[0,\infty ),x\ge 0,}$$

and

$${\displaystyle \underset{n\to \infty }{lim}{T}_m\left(f\left(\frac{t}{n}\right);nx\right)=f\left(x\right),f\in C_b[0,\infty ),x\ge 0.}$$

Consequently, the sequence ${\left(T_n\right)}_{n\ge 1}$ has the properties $\left(C_1\right)$ and $\left(C_2\right)$.

3. Semi-Exponential Bernstein Operators

In [18], Abel et al. determined the semi-exponential Bernstein operators as follows

$${\displaystyle B_n^{\beta }(f\left(t\right);x)=e^{-\beta x}\sum _{j=0}^{\infty }{\displaystyle \frac{{\beta }^j}{j!}}{x}^j{(1-x)}^{n-j}\sum _{k=0}^{\infty }\left(\genfrac{}{}{0pt}{}{n-j}{k}\right){\left({\displaystyle \frac{x}{1-x}}\right)}^{k}f\left({\displaystyle \frac{j+k}{n}}\right),0\le x<\frac{1}{2},}$$

where $\beta >0$ is a given real number.

By a straightforward calculation, we obtain the moments and the central moments up to order 2 for $B_n^{\beta }$.

$$\begin{array}{cc}\hfill & B_n^{\beta }(1;x)=1,\hfill \\ \hfill & B_n^{\beta }(t;x)=x+{\displaystyle \frac{\beta x(1-x)}{n}},\hfill \\ \hfill & B_n^{\beta }(t^2;x)=x^2+{\displaystyle \frac{x(1-x)}{n}}+{\displaystyle \frac{\beta x(1-x)(-x^2\beta +\beta x+2nx-2x+1)}{n^2}},\hfill \\ \hfill & B_n^{\beta }(t-x;x)={\displaystyle \frac{\beta x(1-x)}{n}},\hfill \\ \hfill & B_n^{\beta }({(t-x)}^2;x)={\displaystyle \frac{x(1-x)(-{\beta }^2{x}^2+({\beta }^2-2\beta )x+n+\beta )}{n^2}}.\hfill \end{array}$$

Theorem 2. 

Let $f\in C_b[0,\infty )$ and $x\in [0,\infty )$. Then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}{B}_{mn}^{\beta }\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)=S_m(f\left(t\right);x),\beta >0,}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\displaystyle \underset{\beta \to 0}{lim}{B}_n^{\beta }(f\left(t\right);x)=B_n(f\left(t\right);x),}\hfill \end{array}$$

(15)

thus, $(B_n^{\beta })$ has the property $\left(C_1\right)$ with $R_m=S_m$.

Proof. 

Let us remark that for $s\in \mathbb{R}$

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^{\infty }\left(\genfrac{}{}{0pt}{}{n-j}{k}\right){\left({\displaystyle \frac{x}{1-x}}\right)}^k{e}^{isj/n}{\left(e^{is/n}\right)}^k}& {\displaystyle =e^{isj/n}\sum _{k=0}^{n-j}\left(\genfrac{}{}{0pt}{}{n-j}{k}\right){\left({\displaystyle \frac{x{e}^{is/n}}{1-x}}\right)}^k}\hfill \\ & =e^{isj/n}{\left({\displaystyle \frac{1-x+x{e}^{is/n}}{1-x}}\right)}^{n-j}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill B_n^{\beta }\left(e^{ist};x\right)& {\displaystyle =e^{-\beta x}\sum _{j=0}^{\infty }{\displaystyle \frac{{\beta }^j}{j!}}{x}^j{(1-x)}^{n-j}{\left(e^{is/n}\right)}^j{\left({\displaystyle \frac{1-x+x{e}^{is/n}}{1-x}}\right)}^{n-j}}\hfill \\ \hfill & {\displaystyle =e^{-\beta x}{(1-x+x{e}^{is/n})}^n\sum _{j=0}^{\infty }{\displaystyle \frac{1}{j!}}{\left({\displaystyle \frac{\beta x{e}^{is/n}}{1-x+x{e}^{is/n}}}\right)}^j.}\hfill \end{array}$$

This yields

$$B_n^{\beta }(e^{ist};x)={(1-x+x{e}^{is/n})}^{n}exp\left({\displaystyle \frac{\beta x(1-x)(e^{is/n}-1)}{1-x+x{e}^{is/n}}}\right).$$

(16)

From (16), we infer

$${\displaystyle \underset{n\to \infty }{lim}{B}_{mn}\left(e^{isnt};{\displaystyle \frac{x}{n}}\right)=e^{mx(e^{is/m}-1)}=S_m(e^{ist};s).}$$

Applying Theorem 1 we get (14).

On the other hand, from (16), it follows that

$${\displaystyle \underset{\beta \to 0}{lim}{B}_n^{\beta }\left(e^{ist};x\right)={(1-x+x{e}^{is/n})}^n=B_n(e^{ist};x),}$$

and a slight extension of Theorem 1 shows that (15) is valid. □

4. Semi-Exponential Post–Widder Operators

The semi-exponential Post–Widder operators are determined as (see [16,18])

$${\displaystyle P_n^{\beta }(f\left(t\right);x)={\displaystyle \frac{n^n}{e^{\beta x}{x}^n}}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(n\beta \right)}^k}{k!}}{\displaystyle \frac{1}{\mathsf{\Gamma }(n+k)}}{\int }_0^{\infty }{t}^{n+k-1}{e}^{-nt/x}f\left(t\right)dt.}$$

Let $e_r\left(t\right)=t^r,r=0,1,2,...$ It immediately follows that

$$\begin{array}{ccc}\hfill P_n^{\beta }(e_r\left(t\right);x)& =& e^{-\beta x}\sum _{k=0}^{\infty }\frac{{\left(x\beta \right)}^k}{k!\mathsf{\Gamma }(n+k)}\mathsf{\Gamma }(n+r+k){\left(\frac{x}{n}\right)}^r,\hfill \end{array}$$

(17)

$$P_n^{\beta }(e_{r+1}\left(t\right);x)={\displaystyle \frac{x^2\beta {(n+1)}^r}{n^{r+1}}}{P}_{n+1}^{\beta }(e_r\left(t\right);x)+{\displaystyle \frac{x(n+r)}{n}}{P}_n^{\beta }(e_r\left(t\right);x),$$

(18)

and for $n>Ax$,

$$\begin{array}{ccc}\hfill (P_n^{\beta }{e}^{At})\left(x\right)& =& \frac{n^n}{{(n-Ax)}^n}exp\left(\frac{A{x}^2\beta }{n-Ax}\right).\hfill \end{array}$$

(19)

From each of the Formulas (17)–(19), it is possible to compute explicitly the moments of the operators. To apply (19), we use the equation

$$\begin{array}{ccc}\hfill (P_n^{\beta }{e}_m)\left(x\right)& =& {\left[\frac{{\partial }^m}{\partial A^m}\frac{n^n}{{(n-Ax)}^n}exp\left(\frac{A{x}^2\beta }{n-Ax}\right)\right]}_{A=0}.\hfill \end{array}$$

A few moments of the operators are given below.

$$\begin{array}{cc}\hfill P_n^{\beta }(e_0\left(t\right);x)& =1,\hfill \\ \hfill P_n^{\beta }(e_1\left(t\right);x)& =x+\frac{\beta x^2}{n},\hfill \\ \hfill P_n^{\beta }(e_2\left(t\right);x)& =x^2+\frac{x^2+2\beta x^3}{n}+\frac{2\beta x^3+{\beta }^2{x}^4}{n^2},\hfill \\ \hfill P_n^{\beta }(e_3\left(t\right);x)& =x^3+\frac{3x^3+3\beta x^4}{n}+\frac{2x^3+9\beta x^4+3{\beta }^2{x}^5}{n^2}+\frac{6\beta x^4+6{\beta }^2{x}^5+{\beta }^3{x}^6}{n^3},\hfill \\ \hfill P_n^{\beta }(e_4\left(t\right);x)& =x^4+\frac{6x^4+4\beta x^5}{n}+\frac{11x^4+24\beta x^5+6{\beta }^2{x}^6}{n^2}\hfill \\ & +\frac{6x^4+44\beta x^5+30{\beta }^2{x}^6+4{\beta }^3{x}^7}{n^3}+\frac{24\beta x^5+36{\beta }^2{x}^6+12{\beta }^3{x}^7+{\beta }^4{x}^8}{n^4}.\hfill \end{array}$$

Denote the m-th order central moment of $P_n^{\beta }$ by ${\mu }_{n,m}^{\beta }\left(x\right)=P_n^{\beta }({(t-x)}^m;x)$. Then we have

$${\mu }_{n,m}^{\beta }\left(x\right)={\left[\frac{{\partial }^m}{\partial A^m}\frac{n^{\lambda }}{{(n-Ax)}^n}exp\left(\frac{A{x}^2\beta }{n-Ax}-Ax\right)\right]}_{A=0}$$

and the first central moments are

$$\begin{array}{cc}\hfill {\mu }_{n,0}^{\beta }\left(x\right)& =1,\hfill \\ \hfill {\mu }_{n,1}^{\beta }\left(x\right)& =\frac{\beta x^2}{n},\hfill \\ \hfill {\mu }_{n,2}^{\beta }\left(x\right)& =\frac{x^2}{n}+\frac{2\beta x^3+{\beta }^2{x}^4}{n^2},\hfill \\ \hfill {\mu }_{n,3}^{\beta }\left(x\right)& =\frac{2x^3+3\beta x^4}{n^2}+\frac{6\beta x^4+6{\beta }^2{x}^5+{\beta }^3{x}^6}{n^3},\hfill \\ \hfill {\mu }_{n,4}^{\beta }\left(x\right)& =\frac{3x^4}{n^2}+\frac{6x^4+20\beta x^5+6{\beta }^2{x}^6}{n^3}+\frac{24\beta x^5+36{\beta }^2{x}^6+12{\beta }^3{x}^7+{\beta }^4{x}^8}{n^4}.\hfill \end{array}$$

Theorem 3. 

Let $f\in C_b[0,\infty )$, $m\in \mathbb{N}$, $\beta >0$ and $x\in [0,\infty )$. Then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}{P}_{mn}^{\beta }\left(f\left(nt\right);\frac{x}{n}\right)=f\left(x\right),}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}{P}_n^{\beta }\left(f\left({\displaystyle \frac{t}{n}}\right);nx\right)=f(x+\beta x^2),}\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & {\displaystyle \underset{\beta \to 0}{lim}{P}_n^{\beta }\left(f\left(t\right);x\right)=P_n\left(f\left(t\right);x\right),n>0.}\hfill \end{array}$$

(22)

Proof. 

Using (19) with $A=isn$, we obtain

$$P_{mn}^{\beta }\left(e^{isnt};\frac{x}{n}\right)={\left({\displaystyle \frac{mn}{mn-isx}}\right)}^{mn}exp{\displaystyle \frac{is{x}^2\beta }{m{n}^2-isnx}}.$$

It follows that

$${\displaystyle \underset{n\to \infty }{lim}{P}_{mn}^{\beta }\left(e^{isnt};\frac{x}{n}\right)=e^{isx}=Id(e^{ist};x).}$$

(23)

Combining (23) and Theorem 1 leads to (20).

From (19) with $A=is/n,s\in \mathbb{R}$, we get

$$P_n^{\beta }(e^{ist/n};nx)={\left({\displaystyle \frac{n}{n-isx}}\right)}^{n}exp{\displaystyle \frac{is{x}^2\beta n}{n-isx}}.$$

Consequently,

$${\displaystyle \underset{n\to \infty }{lim}{P}_n^{\beta }(e^{ist/n};nx)=e^{is(x+\beta x^2)}.}$$

Consider the operator $L(f\left(t\right);z):=f(z+\beta z^2),z\ge 0.$

Then

$${\displaystyle \underset{n\to \infty }{lim}{P}_n^{\beta }(e^{ist/n};nx)=L(e^{ist};x).}$$

(24)

Now, (21) is a consequence of (24) combined with Theorem 1.

To prove (22), we use again (19), this time with $A=is$, $s\in \mathbb{R}$. We get

$$\begin{array}{c}\hfill {\displaystyle \underset{\beta \to 0}{lim}{P}_n^{\beta }(e^{ist};x)=\underset{\beta \to 0}{lim}{\left({\displaystyle \frac{n}{n-isx}}\right)}^{n}exp{\displaystyle \frac{is{x}^2\beta }{n-isx}}={\left(1-{\displaystyle \frac{isx}{n}}\right)}^{-n}=P_n(e^{ist};x).}\end{array}$$

A slight extension of Theorem 1 concludes the proof. □

Remark 3. 

Formula (20) shows that the sequence ${(P_n^{\beta })}_{n\ge 1}$ has the property ($C_1$) with $R_m=Id$, $m\ge 1$. Formula (21) should be compared with property $\left(C_2\right)$.

5. Composition of Operators

5.1. $S_m^{\beta }$ as a Limit of Baskakov Type Operators

The semi-exponential Szász-Mirakjan operators were described in [17] as follows

$${\displaystyle S_n^{\beta }(f\left(t\right);x)=\sum _{k=0}^{\infty }{e}^{-(n+\beta )x}{\displaystyle \frac{{\left((n+\beta )x\right)}^k}{k!}}f\left({\displaystyle \frac{k}{n}}\right).}$$

Under the name of Szász–Mirakjan–Schurer operators, $S_n^{\beta }$ are investigated in ([21] p. 338, [13,22]).

They are related to the Jakimovski–Leviatan operators associated with the Appell polynomials having the generating function $a\left(t\right)=e^{\beta t}$. These operators are given by (see [15] Example 12.1)

$${\displaystyle {\mathsf{\Psi }}_{n,\beta }(f\left(t\right);x)=e^{-nx-\beta }\sum _{k=0}^{\infty }{\displaystyle \frac{{(nx+\beta )}^k}{k!}}f\left({\displaystyle \frac{k}{n}}\right).}$$

The relation is the following one,

$${\mathsf{\Psi }}_{n,\beta x}(f\left(t\right);x)=e^{-(n+\beta )x}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left((n+\beta )x\right)}^k}{k!}}f\left({\displaystyle \frac{k}{n}}\right)=S_n^{\beta }(f\left(t\right);x).$$

According to ([15] Example 12.1)

$${\displaystyle \underset{n\to \infty }{lim}{V}_{mn}\left(f\left(nt\right);{\displaystyle \frac{(\beta +m)x}{mn}}\right)=S_m^{\beta }(f\left(t\right);x).}$$

5.2. Jakimovski-Leviatan Type Operators

Here, we consider an extension of the Jakimovski–Leviatan operators, based on Appell polynomials of class $A^{\left(2\right)}$ (see [23,24]).

The Appell polynomials ${\left(p_k\left(x\right)\right)}_{k\ge 0}$ of class $A^{\left(2\right)}$ are given by the following generating function

$${\displaystyle A\left(t\right)e^{xt}+B\left(t\right)e^{-xt}=\sum _{k=0}^{\infty }{p}_k\left(x\right)t^k,}$$

where $A\left(t\right)$ and $B\left(t\right)$ are power series defined in the disk $\left|z\right|<R,R>1$.

The associated sequence of operators is given by

$$T_n(f\left(t\right);x)={\displaystyle \frac{1}{A\left(1\right)e^{nx}+B\left(1\right)e^{-nx}}}\sum _{k=0}^{\infty }{p}_k\left(nx\right)f\left(\frac{k}{n}\right),x\ge 0,f\in C_b[0,\infty ).$$

It is easy to see that the sequence ${\left(T_n\right)}_{n\ge 1}$ satisfies (6) and, consequently, has the property $\left(C_1\right)$. On the other hand

$${\displaystyle \underset{n\to \infty }{lim}{T}_m\left(f\left({\displaystyle \frac{t}{n}}\right);nx\right)=\underset{n\to \infty }{lim}{T}_{mn}(f\left(t\right);x)=f\left(x\right),}$$

and it follows that ${\left(T_n\right)}_{n\ge 1}$ has the property $\left(C_2\right)$ with $Q_m=Id$ (see also Example 2).

5.3. Compositions of Post-Widder Operators and Semi-Exponential Szász-Mirakjan Operators

As another example, we consider the operators

$$L_n^{\beta }:=P_n\circ S_n^{\beta }.$$

As compositions of Post–Widder operators and the semi-exponential Szász–Mirakjan operators, $L_n^{\beta }$ are Baskakov type operators. Indeed, we can write

$$\begin{array}{ccc}\hfill L_n^{\beta }(f\left(t\right);x)& =& \sum _{k=0}^{\infty }\frac{n^n}{x^n}\frac{1}{\mathsf{\Gamma }\left(n\right)}{\int }_0^{\infty }{e}^{-nt/x}{t}^{n-1}{e}^{-(\beta +n)t}\frac{{\left((\beta +n)t\right)}^k}{k!}f\left(\frac{k}{n}\right)dt\hfill \\ & =& \frac{n^n}{x^n}\frac{1}{\mathsf{\Gamma }\left(n\right)}\sum _{k=0}^{\infty }\frac{{(\beta +n)}^k}{k!}f\left(\frac{k}{n}\right){\int }_0^{\infty }{e}^{-(\frac{n}{x}+\beta +n)t}{t}^{n+k-1}dt\hfill \\ & =& \frac{n^n}{x^n}\frac{1}{\mathsf{\Gamma }\left(n\right)}\sum _{k=0}^{\infty }\frac{{(\beta +n)}^k}{k!}f\left(\frac{k}{n}\right)\frac{x^{n+k}}{{(n+(\beta +n)x)}^{n+k}}{\int }_0^{\infty }{e}^{-u}{u}^{n+k-1}du.\hfill \end{array}$$

Then

$$L_n^{\beta }(f\left(t\right);x)=\frac{n^n}{{(\beta +n)}^n}\sum _{k=0}^{\infty }\left(\genfrac{}{}{0pt}{}{n+k-1}{k}\right)\frac{x^k}{{\left(\frac{n}{\beta +n}+x\right)}^{n+k}}f\left(\frac{k}{n}\right).$$

(25)

It is easy to see that

$$L_n^{\beta }(f\left(t\right);x)=V_n\left(f\left(t\right);\frac{(n+\beta )x}{n}\right).$$

Theorem 4. 

Let $f\in C_b[0,\infty )$ and $x\in [0,\infty )$. Then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{n\to \infty }{lim}{L}_m^{\beta }\left(f\left({\displaystyle \frac{t}{n}}\right);nx\right)=P_m\left(f\left(t\right);{\displaystyle \frac{m+\beta }{m}}x\right),}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{lim}{L}_{mn}^{\beta }\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)=S_m(f\left(t\right);x).\hfill \end{array}$$

(27)

Consequently, the sequence $(L_n^{\beta })$ has the properties $\left(C_1\right)$ and $\left(C_2\right)$.

Proof. 

According to (25), we have

$$L_m^{\beta }(e^{ist};x)={\left(1+{\displaystyle \frac{(\beta +m)x(e^{is/m}-1)}{m-(\beta +m)x(e^{is/m}-1)}}\right)}^m.$$

(28)

This yields

$${\displaystyle \underset{n\to \infty }{lim}{L}_m^{\beta }\left(e^{ist/n};nx\right)={\left({\displaystyle \frac{m}{m-\frac{\beta +m}{m}xis}}\right)}^m=P_m\left(e^{ist};{\displaystyle \frac{m+\beta }{m}}x\right),}$$

which, combined with Theorem 1, proves (26). Similarly, from (28), we derive

$${\displaystyle \underset{n\to \infty }{lim}{L}_{nm}^{\beta }\left(e^{ismt};{\displaystyle \frac{x}{n}}\right)=e^{mx(e^{is/m}-1)}=S_m(e^{ist};x)}$$

and an application of Theorem 1 concludes the proof of (27). □

5.4. Composition of Post-Widder Operators and Szász–Durrmeyer Operators

In what follows, we denote by ${\overline{S}}_n$ the Szász–Durrmeyer operators defined as

$${\overline{S}}_n(f\left(t\right);x)=n\sum _{k=0}^{\infty }{s}_{n,k}\left(x\right){\int }_0^{\infty }{s}_{n,k}\left(u\right)f\left(u\right)du,x\ge 0,$$

(29)

where $s_{n,k}\left(x\right)=e^{-nx}\frac{{\left(nx\right)}^k}{k!}$.

Consider the composition of Post–Widder operators and Szász–Durrmeyer operators

$${\overline{M}}_n=P_n\circ {\overline{S}}_n.$$

Then ${\overline{M}}_n$ are the Baskakov–Szász operators given explicitly by

$${\overline{M}}_n(f\left(t\right);x)=n\sum _{k=0}^{\infty }{v}_{n,k}\left(x\right){\int }_0^{\infty }{s}_{n,k}\left(u\right)f\left(u\right)du,x\ge 0,$$

(30)

where ${\displaystyle v_{n,k}\left(x\right)=\sum _{k=0}^{\infty }\left(\genfrac{}{}{0pt}{}{n+k-1}{k}\right)\frac{x^k}{{(1+x)}^{n+k}}.}$

Theorem 5. 

Let $f\in C_b[0,\infty )$ and $x\ge 0$. Then

$${\displaystyle \underset{n\to \infty }{lim}{\overline{M}}_m\left(f\left({\displaystyle \frac{t}{n}}\right);nx\right)=P_m(f\left(t\right);x),}$$

(31)

$${\displaystyle \underset{n\to \infty }{lim}{\overline{M}}_{nm}\left(f\left(nt\right);\frac{x}{n}\right)={\overline{S}}_m(f\left(t\right);x).}$$

(32)

Consequently, the sequence $\left({\overline{M}}_n\right)$ has the properties $\left(C_1\right)$ and $\left(C_2\right)$.

Proof. 

Let us remark that for $s\in \mathbb{R}$,

$${\displaystyle {\int }_0^{\infty }{s}_{m,k}\left(u\right)e^{isu}du={\displaystyle \frac{m^k}{{(m-is)}^{k+1}}},k\in {\mathbb{N}}_0.}$$

Therefore,

$${\overline{S}}_m(e^{ist};x)={\displaystyle \frac{m}{m-is}}exp{\displaystyle \frac{mxsi}{m-is}},$$

and

$${\overline{M}}_m(e^{ist};x)=P_m\left({\displaystyle \frac{m}{m-is}}{e}^{At};x\right),$$

where $A={\displaystyle \frac{msi}{m-is}}$.

Using (19) with $\beta =0$, we get

$${\overline{M}}_m(e^{ist};x)={\displaystyle \frac{m}{m-is}}{\left({\displaystyle \frac{m-is}{m-is-isx}}\right)}^m.$$

It follows that

$${\displaystyle \underset{n\to \infty }{lim}{\overline{M}}_m\left(e^{ist/n};nx\right)={\left({\displaystyle \frac{m}{m-isx}}\right)}^m=P_m(e^{ist};x)}$$

and an application of Theorem 1 concludes the proof of (31).

On the other hand, by a direct calculation, we find that

$${\overline{S}}_{nm}\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)={\overline{S}}_m(f\left(t\right);x),m,n\in \mathbb{N}.$$

(33)

Consequently,

$$\begin{array}{cc}\hfill {\displaystyle \underset{n\to \infty }{lim}{\overline{M}}_{nm}\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)}& {\displaystyle =\underset{n\to \infty }{lim}\left(P_{nm}\circ {\overline{S}}_{nm}\right)\left(f\left(nt\right);{\displaystyle \frac{x}{n}}\right)}\hfill \\ \hfill & {\displaystyle =\underset{n\to \infty }{lim}{P}_{nm}\circ {\overline{S}}_m\left(f\left(t\right);x\right)={\overline{S}}_m(f\left(t\right);x),}\hfill \end{array}$$

and this proves (32). □

Remark 4. 

Formula (33) shows that the sequence ${\left({\overline{S}}_n\right)}_{n\ge 1}$ satisfies (6), and, in particular, has the property $\left(C_1\right)$. It also has the property $\left(C_2\right)$. Indeed, we have

$${\overline{S}}_m\left(f\left(\frac{t}{n}\right);nx\right)={\overline{S}}_{mn}(f\left(t\right);x),$$

and therefore

$${\displaystyle \underset{n\to \infty }{lim}{\overline{S}}_m\left(f\left(\frac{t}{n}\right);nx\right)=f\left(x\right).}$$

5.5. Balázs-Szabados Operators

Balázs and Szabados in [25] considered the positive linear rational operators of Bernstein-type given by

$${\displaystyle R_n^{\left[\beta \right]}(f\left(t\right);x)={\displaystyle \frac{1}{{(1+n^{\beta -1}x)}^n}}\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right){\left(n^{\beta -1}x\right)}^{k}f\left({\displaystyle \frac{k}{n^{\beta }}}\right),}$$

where $x\ge 0$, $0<\beta <1$, $f\in C[0,\infty )$.

Theorem 6. 

Let $f\in C_b[0,\infty )$, $x\ge 0$ and $0<\alpha <\beta <1$. Then

$${\displaystyle \underset{n\to \infty }{lim}{R}_{nm}^{\left[\beta \right]}\left(f\left(n^{\alpha }t\right);\frac{x}{n^{\alpha }}\right)=f\left(x\right).}$$

(34)

Proof. 

We have

$$R_{mn}^{\left[\beta \right]}\left(e^{is{n}^{\alpha }t};\frac{x}{n^{\alpha }}\right)={\left(1+\frac{{\left(mn\right)}^{\beta -1}x\left(exp\left(\frac{is{n}^{\alpha }}{{\left(mn\right)}^{\beta }}\right)-1\right)}{n^{\alpha }+{\left(mn\right)}^{\beta -1}x}\right)}^{mn}.$$

Therefore,

$${\displaystyle \underset{n\to \infty }{lim}{R}_{mn}^{\left[\beta \right]}\left(e^{is{n}^{\alpha }t};\frac{x}{n^{\alpha }}\right)=e^{isx}=Id(e^{ist};x).}$$

Now, (34) is a consequence of Theorem 1. □

Remark 5. 

According to (34), the sequence ${(R_n^{\left[\beta \right]})}_{n\ge 1}$ has a property similar to $\left(C_1\right)$.

6. The Semi-Exponential Gauss–Weierstrass Operators

The exponential Gauss–Weierstrass operators are defined as (see, e.g., [18])

$$W_n(f\left(t\right);x)=\sqrt{{\displaystyle \frac{n}{2\pi }}}{\int }_{R}exp\left(-{\displaystyle \frac{n{(u-x)}^2}{2}}\right)f\left(u\right)du.$$

It is easy to verify that

$$W_{nm}\left(f\left(t\sqrt{n}\right);\frac{x}{\sqrt{n}}\right)=W_m(f\left(t\right);x).$$

(35)

The property (35) is similar to that expressed in (6).

The semi-exponential Gauss–Weierstrass operators were described in [17] and are given by (see also [18])

$$W_n^{\beta }(f\left(t\right);x)=\sqrt{\frac{n}{2\pi }}{\int }_{\mathbb{R}}exp\left(-\frac{n}{2}{\left(u-x-\frac{\beta }{n}\right)}^2\right)f\left(u\right)du.$$

By direct calculation, one finds that

$${\displaystyle \underset{n\to \infty }{lim}{W}_{nm}^{\beta }\left(f\left(t\sqrt{n}\right);\frac{x}{\sqrt{n}}\right)=W_m(f\left(t\right);x),m\ge 1.}$$

(36)

This relation is similar to property $\left(C_1\right)$.

In this context, a property similar to $\left(C_1\right)$ is shared by the Bernstein operators. Indeed, let $m\ge 1$ be fixed and

$$L_n(f\left(t\right);x)=B_{mn}\left(f\left(t\sqrt{n}\right);\frac{x}{\sqrt{n}}\right).$$

Then,

$${\displaystyle \underset{n\to \infty }{lim}{L}_n\left(e^{ist};x\right)=e^{isx}=Id(e^{ist};x).}$$

Now, Theorem 1 shows that

$${\displaystyle \underset{n\to \infty }{lim}{B}_{mn}\left(f\left(t\sqrt{n}\right);\frac{x}{\sqrt{n}}\right)=f\left(x\right).}$$

(37)

Another proof of (37) can be given by using Korovkin-type results.

7. Conclusions and Further Work

The following situation is often encountered in approximation theory. One starts with a family of linear positive operators, extends it, and investigates the properties of the new operators. In particular, the convergence of sequences of such operators is a subject of study. Here are two examples.

(A) In [1,2,3,4,5,6,15], some classical positive linear operators (Bernstein, Baskakov, Bleimann–Butzer–Hahn, …) are modified using a procedure inspired by the Poisson approximation to the binomial distribution. The limit of the new sequence is another classical operator, e.g., a Szász–Mirakjan operator. Details can be found in [15].

(B) The exponential operators are constructed using a differential equation. Modifying this equation, the so-called semi-exponential operators are introduced (see [8,16,17,18,20]).

In this paper, we apply procedures of type (A) to the semi-exponential operators and determine the limits of the new sequences. This sheds a new light on the semi-exponential operators and enlarges the family of the operators representable as limits of other operators. As further work, we intend to introduce other procedures of type (A) and to apply them to new families of positive linear operators.