Convergence of Special Sequences of Semi-Exponential Operators

: 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.


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.
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 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 ⊆ R be an interval and C b (I) 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 (I).
Let Z x , Z x 1 , Z x 2 , . . .be I-valued random variables with probability distributions depending on a parameter x ∈ I. Suppose that for each f ∈ C b (I) the functions x → E f (Z x ) and x → E f (Z x n ), x ∈ I, n ≥ 1, are continuous on I.We will be concerned with positive linear operators For each s ∈ R, we consider the function t → e ist , t ∈ I, and define and similarly for L(e ist ; x).

Theorem 1 ([15]
). Suppose that for each s ∈ R and x ∈ I, and for each x ∈ I the function s → L(e ist ; x) is continuous on R. Then for all f ∈ C b (I) and x ∈ 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, ∞) or (−∞, +∞), depending on the structure of the involved operators.
Section 2 is devoted to a basic definition and some examples illustrating the properties (C 1 ) and (C 2 ) 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 (C 1 ) introduced in Definition 1. Similar results are presented in Section 4 for semi-exponential Post-Widder operators.Obviously, the properties (C 1 ) and (C 2 ) 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 (C 1 ), shared by the semi-exponential Gauss-Weierstrass operators and the Bernstein operators.

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 (L n ) n≥1 has the property (C 1 ), if for each m ≥ 1 there exists an operator R m such that We say that the sequence (L n ) n≥1 has the property (C 2 ), if for each m ≥ 1 there exists an operator Q m such that Remark 1.It is easy to verify (see, e.g., ([15] Proposition 12.1, Proposition 12.2)) that Denote by B n , V n and S n the classical Bernstein operators, Baskakov operators and Szász-Mirakjan operators, respectively Example 2. The sequence (S m ) has the property (C 1 ) with R m = S m and, moreover, it satisfies (6).We will show that it also has the property (C 2 ) with Q m = Id, where Id is the identity operator.Indeed, let us choose L m = S m and Q m = Id.Then Example 3. Consider the Post-Widder operators indexed by integers m ≥ 1 (see [19] (9.1.9)), The operators given by (8) are called in [1] Gamma operators and denoted by G m (see also [15] (10.1)).Thus We know (see [15] Theorem 12.2 and Example 12.3) that and so Of course, (9) can also be proved directly.It shows that the sequence (P m ) 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 (V(u)) u≥0 approximated by iterates of P n , namely For all the details, see, e.g., [7] and the references therein.From ( 9) and ( 10), we see that In fact (see [7]), is the solution to the problem For ν ≥ 0 set f ν (t) := f (νt), t ≥ 0. From ( 11)-( 13), we deduce . The Ismail-May exponential operators defined as (see [8,20]) satisfy (see [20] Lemma 1) It is easy to verify that for each m ≥ 1, s ∈ R and x ≥ 0, lim n→∞ T mn e isnt ; x n = e isx = Id(e ist ; x) and lim n→∞ T m e ist/n ; nx = e isx = Id(e ist ; x).
According to Theorem 1, it follows that and Consequently, the sequence (T n ) n≥1 has the properties (C 1 ) and (C 2 ).

Semi-Exponential Bernstein Operators
In [18], Abel et al. determined the semi-exponential Bernstein operators as follows where β > 0 is a given real number.By a straightforward calculation, we obtain the moments and the central moments up to order 2 for thus, (B From ( 16), we infer lim n→∞ B mn e isnt ; x n = e mx(e is/m −1) = S m (e ist ; s).
On the other hand, from ( 16), it follows that and a slight extension of Theorem 1 shows that ( 15) is valid.

Semi-Exponential Post-Widder Operators
The semi-exponential Post-Widder operators are determined as (see [16,18]) Let e r (t) = t r , r = 0, 1, 2, ... It immediately follows that and for n > Ax, (P 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 A few moments of the operators are given below.
Denote the m-th order central moment of and the first central moments are Then Proof.Using (19) with A = isn, we obtain It follows that lim n→∞ P β mn e isnt ; x n = e isx = Id(e ist ; x).
To prove (22), we use again (19), this time with A = is, s ∈ R. We get A slight extension of Theorem 1 concludes the proof.
They are related to the Jakimovski-Leviatan operators associated with the Appell polynomials having the generating function a(t) = e βt .These operators are given by (see [15] Example 12.1) The relation is the following one, According to ([15] Example 12.1)
The Appell polynomials (p k (x)) k≥0 of class A (2) are given by the following generating function where A(t) and B(t) are power series defined in the disk |z| < R, R > 1.
The associated sequence of operators is given by It is easy to see that the sequence (T n ) n≥1 satisfies (6) and, consequently, has the property (C 1 ).On the other hand and it follows that (T n ) n≥1 has the property (C 2 ) with Q m = Id (see also Example 2).

Compositions of Post-Widder operators and semi-exponential Szász-Mirakjan operators
As another example, we consider the operators As compositions of Post-Widder operators and the semi-exponential Szász-Mirakjan operators, L β n are Baskakov type operators.Indeed, we can write It is easy to see that Consequently, the sequence (L β n ) has the properties (C 1 ) and (C 2 ).
Proof.According to (25), we have x n = e mx(e is/m −1) = S m (e ist ; x) and an application of Theorem 1 concludes the proof of (27).

Composition of Post-Widder operators and Szász-Durrmeyer operators
In what follows, we denote by S n the Szász-Durrmeyer operators defined as Consequently, the sequence (M n ) has the properties (C 1 ) and (C 2 ).
Proof.Let us remark that for s ∈ R, and an application of Theorem 1 concludes the proof of (31).
On the other hand, by a direct calculation, we find that The sequence of the classical Baskakov operators has the properties (C 1 ) and (C 2 ) 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 (C 1 ) with R m = S m .