1. Introduction
In 1912, Bernstein [
1] constructed the following sequence of positive linear operators
to give a proof that every continuous function
can be uniformly approximated by a sequence of polynomial functions. Many papers were devoted to the study of the properties of these operators. In 1932, Voronovskaya [
2] obtained the speed of convergence of Bernstein operators to the identity operator. For every function
f which is twice differentiable on
and for every
, we have
In 1935, Popoviciu [
3] estimated the error of approximation of Bernstein operators
in terms of the usual modulus of continuity defined by
for every
and
. Later, other moduli of smoothness were constructed, in order to obtain characterizations of the functions for which we have a certain error of approximation.
For
and
, let us denote by
the operator defined for a real function
by the equality
where
for every nonnegative integer
k and real number
z. Here, we assume that the function
is such that the series on the right side of the equality (
1) converges. In this article, we will obtain results for the operator
similar to the results presented above for Bernstein operators.
For
,
, the operator
represents Ismail and May’s operator
(see [
4] relation (3.14)). For
, the operator
represents Szász–Mirakyan’s operator (see [
4] relation (3.6)). Jain [
5] considered the following positive linear operator
, where
,
and
is a sequence of real numbers from
converging to 0, and proved some approximation properties.
In
Section 2, we will present some basic approximation properties needed in the rest of the article. We include an evaluation of the error of the approximation in terms of the usual modulus of continuity.
In
Section 3, we study the uniform approximation properties of
, including an estimate of the rate of approximation in terms of the second-order modulus of Ditzian and Totik. We also obtain a characterization of the bounded functions which can be uniformly approximated by
. To our knowledge, these kinds of results have not been obtained for the particular operators of Ismail and May and the operators of Jain. We must mention that Totik [
6,
7] solved the problem of uniform approximation for the classical exponential-type operators. The operators
are of exponential type, but a direct application of Totik’s results is not possible in our case (see the first paragraph of
Section 3).
In the last section, we provide a pointwise complete asymptotic expansion for the newly introduced operators (
1). Similar results were obtained in [
8] for the Jain operators and functions with polynomial growth. We remark that our asymptotic results are valid for functions with exponential growth, as in the case of the operators of Ismail and May presented in [
9]. However, our study presents, in addition to the previous articles, two aspects: first, an explicit computation of the images of exponential functions using the Lambert function (see Theorem 3), and second, a direct and detailed proof of the asymptotic result, which, in the previous papers, was based on a result of Sikkema [
10] for functions with polynomial growth. As a corollary to our main result, we deduce a Voronovskaya-type theorem.
3. Uniform Approximation
In [
6], Totik has given general results for the uniform approximation of functions by positive linear operators which reproduce constant and linear functions and satisfy certain mild assumptions. He obtained a characterization of functions which can be uniformly approximated by such operators and estimated the rate of convergence in terms of the Ditzian–Totik modulus of smoothness of order 2. Let us consider the function
defined by
. The second-order modulus of Ditzian–Totik with respect to the weight function
(see also [
13]) is defined by
The results obtained by Totik are proven under some assumptions on the growth of the function
, which are not verified in our case (condition (3) from page 164 of [
6] is not valid at infinity, i.e.,
is not finite). This is why we have modified the method for obtaining such results.
We will present first an estimate of the rate of approximation in terms of the modulus . For this, we need the following auxiliary result.
Lemma 3. For every and we have Proof. Let us denote
. We have
Using (
2), we obtain
which proves (
7). □
Theorem 1. Let be a bounded function. Then,for some constant independent of α. Proof. Let us consider
D, the space of all absolutely continuous functions
g in every closed finite interval
with the property that
and the
K-functional
We know by ([
13] Theorem 2.1.1) that
for some constants
.
Let
. We have by a standard argument
It is easy to see that
Let us prove that
Consider first the case
.
Using Napier’s inequality
, for
, we obtain
Consider now the case
. Similarly, we have
We apply the operators
to the relation (
9) and we use inequality (
10) to estimate the error for the approximation of
g:
We decompose the function to be evaluated by
in
By simple computation, with
We obtain
Using relation (
7), we have
Using the equality (
11) and applying the operators
, we deduce
and
Passing to the infimum with
, we finally get
□
We have estimated the rate of convergence in terms of the second-order modulus of Ditzian and Totik. Let us denote by
the space of continuous functions on
that have a finite limit at infinity, endowed with the uniform norm, and let us prove that bounded functions which can be uniformly approximated are part of the space
. Let us consider the function
defined by
,
and the following modulus of continuity associated with the function
:
where
and
.
The modulus
is a particular case of a more general modulus studied in [
14,
15]. This modulus is suitable for the estimation of the uniform approximation rate for functions of the space
. Indeed, it can be expressed in terms of the usual modulus of continuity by
and the limit
is valid if and only if
is uniformly continuous on
. However, a function which is uniformly continuous on
has a finite limit at 0. In our case, the fact that
has a finite limit at 0 means that
f has a finite limit at infinity.
Theorem 2. Let be a bounded function. Then, converges uniformly to the function f if and only if f belongs to the space . Moreover, one has Proof. Let
. We prove that
. From the properties of the usual modulus of continuity, we have the following estimation
Applying the operators
, we get
For the evaluation of
, let us use the inequality
We apply the Cauchy–Schwarz inequality
In view of the liniarity and the preservation properties of the operators
presented in Lemmas 1 and 3, we obtain
Thus, we have proven that
and this implies the inequality (
13). This inequality and the properties of the modulus
prove that
converges uniformly to the function
f.
Suppose now that
converges uniformly to the bounded function
f. We will prove that
. It suffices to prove that
is uniformly continuous on
. Let us consider the function
defined by
,
, with the property that
, for
. Let us observe that
is uniformly continuous on
and
. Thus, it remains to prove that
is uniformly continuous on
.
Using the properties of the usual modulus of continuity, we can write
We will prove that each term of the right-hand side of the above inequality tends to 0, when
. Firstly,
Since
converges uniformly to
f, the left-hand side of the inequality from above tends to zero for all
.
Using the inequality from Remark 3, we obtain
We can choose an appropriate
such that the right-hand side tends to zero. This proves that
tends to 0 as
. Using the properties of the usual modulus of continuity, the function
is uniformly continuous on
. □
4. A Voronovskaya-Type Result
For a given , the solution of the equation defines the Lambert function (also called product logarithm), denoted . We need this function to express the image of exponential functions through the operators .
Theorem 3. Let . Then, there is such thatfor every , and every . Proof. With the notation
, we have
If
, then
and the series converges for every
. In this case, we can choose
.
Consider now the case . Since , there is such that , for every . This proves that is correctly defined for every , and every .
Since
, there is
v such that
. Using the Lambert function, we have
. The image of the exponential function can be written
□
Corollary 1. For every and every , Proof. Using the relation (
14) with the notation
, we have
Since
W is a differentiable function on
with
, we apply the l’Hospital rule for
. Consequently,
□
For
, let us denote by
the space of continuous functions on
satisfying the growth condition
as
. As we have seen from (
14), the operators
are correctly defined for every
. We can easily deduce from Corollary 1 that
where
is independent of
.
Based on the works of Abel and Agratini [
8] and Abel and Gupta [
9], we derive a complete asymptotic expansion for the operators
. We need
Lemma 4 (Lemma 3 of [
8]).
Let , and . Then,where are defined by Theorem 4. Let , and . For every function possessing a derivative of order at x, we havewhere the coefficients are given byThe coefficients have been defined in Lemma 4 and represent the Stirling numbers of the second kind. Proof. Let us remark that
We can apply Lemma 4 with
replaced by
and
replaced by
.
We can write the monomial function
(defined by
,
) using Stirling numbers of the second kind by
The
rth moment of the operators
will be expressed by
Exchanging the indexes of summation with
and
, we can write
The term corresponding to
in the above representation is
Using the binomial formula, the central moment of order
s of the operators
is represented by
The term independent of
is obtained by considering only the term corresponding to
in the representation of
, i.e.,
Replacing the other terms of
and changing the order of summation, we obtain
We know from ([
16] Lemma 2) that
This means that the index
k from the representation of
,
runs actually only from
to
.
Let us consider the Taylor expansion with the Peano remainder
where
is a function such that
. We apply the operators
and use the representation of the central moments we have derived. We interchange the sums with indexes
s and
k
and ignore the terms of higher order than
. It remains to prove that
Let . By the continuity of h, there is such that, for every with the property , we have .
For the other values of
t, we have
. Since
and all
,
belong to
, the function
belongs to
. Thus, for some constant
independent of
t, we have
We have proven that, for every
, we have
Applying Hölder inequality for positive linear operators,
Since
was chosen arbitrarily, letting
, the relation (
19) is proven. □
Corollary 2. Let and . For every function having a derivative of second order at x, it holds true Remark 4. The operators verify the conditions of ([17] Theorem 3.1), so the asymptotic relation (20) can be differentiated.