1. Introduction
In approximation theory, constructing Kantorovich-type operators to reduce errors is a well known method. The Kantorovich-type operators and their approximation properties have attracted a lot of attention since the 1990s. Since then, different types of Kantorovich operators were constructed [
1,
2]. Taking the classical Szász operator as an example, it is usually used to approximate continuous functions, while Szász–Kantorovich operator can be utilized to approximate a broad class of functions, such as integrable ones. Later, in order to improve the approximation rate, various new operators were constructed, which preserve the test functions that emerged in this field and a lot of advancements have been made regarding this subject to acquire better approximation. In [
3], King introduced a sequence of positive linear operators, which approximate each continuous function on [
] while preserving the function
, and gave quantitative estimates. Different King-type operators were constructed and had been achieved [
3,
4,
5,
6,
7]. For some recent studies on linear positive operators preserving exponential functions, we refer the readers to [
8,
9,
10,
11,
12,
13,
14]. Following the idea of King, and the further developed in references [
3,
4,
6,
7], we introduced a kind of King-type of Szász–Kantorovich operators [
5],
where
In [
5], we presented the direct and converse estimate of
. In [
15], we introduced other kinds of Szász operators
, which preserve constants and
. The operators
are given by the following: for
,
where
The rate of convergence and the Voronovskaja asymptotic relationship of Szász operators (
1) in the sense of uniform convergence on the
interval were obtained [
15]. A natural question is: what happens if we combine the idea of function extension with that of integral averaging? In this paper, we construct new Kantorovich operators corresponding to
as follows: for
,
where the definition of
can be found in (
1).
In this study, we will give some fundamental properties, which play a significant role in the uniform approximation, we obtain the uniform approximation results of
on
space. At the same time, operator plays important role in computer-aided geometric design, which deals with computational aspects of geometric objects in mathematical developments of curves and surfaces. Inspired by the ideas of Zhang Chungou and others [
16,
17,
18,
19,
20,
21,
22,
23,
24], some shape-preserving properties of these operators are obtained.
Remark 2. Throughout this paper, represents the space of continuous functions on the interval; represents the space of continuous bounded functions on the interval; exists and is limited }, .
The complete structure of the manuscript constitutes six sections. The remaining part of this paper is organized as follows. In
Section 2, we give some basic properties of the operators, such as the moments for proving the convergence theorems. In
Section 3, we establish the approximation theorems of the positive theorem and the Voronovskaja-type weak inverse theorem for continuous functions. In
Section 4, we present some new shape preserving properties of these operators (
2). In
Section 5, we will demonstrate some numerical experiments which verify the validity of the theoretical results and the potential superiority of these new operators. Finally, in
Section 6, some conclusions are provided.
2. Definitions and Lemmas
Definition 1 ([
1])
. For , the rth forward differences are given byor equivalently by Definition 2 ([
1])
. For , the continuous modulus is defined as: Definition 3 ([
16,
17])
. Let f be continuous on , the average function of f is defined as follows: for all , Definition 4 ([
16])
. If function is continuous and is increasing (or decreasing) on , then said to be starshaped with respect to the origin. Alternatively, it can be defined by the following: for each α, , (or ). Definition 5 ([
16])
. , if , then is called semi-additivity; if , then is called super-additivity. For the Szász–Mirakjan operators, the following lemmas are known (see, for instance, [
16]).
Lemma 1 ([
15] Lemma 2.1)
. Let , we have Lemma 2 ([
15] Lemma 2.2)
. Let , it holds that.
Lemma 3 ([
15] Lemma 2.3)
. Let , then one has
Lemma 4. Let , then we have
Proof. Using the definition of the operators
(
2), combining Lemma 1 (1)–(3), by directly calculating, the result is satisfied. We only take (3) as an example to prove, as (1) and (2) are similar.
□
Remark 3. Remark 4. We need also a well-known result (see Ref. [
25]).
Lemma 5 ([
25])
. Let be a sequence of linear positive operators from to , satisfying , and the above convergence is uniform if and only if uniformly in for all . Lemma 6. Let , then we have
Proof. Indeed, utilizing the definition of
(
2), noting Lemma 2 (1)–(4), after some simple calculations, the assertion is true. We only take (4) as an example to prove, as (1)–(3) are similar.
□
Remark 5. Remark 6. With the help of Lemma 6, the limit values for the central moments can be obtained.
Lemma 7. Let , then we have
Proof. By the definition of
(2), utilizing the results of Lemmas 3 and 6, (1) and (2) are satisfied. For the proof of
, it follows from definition of
, Lemma 3
and Lemma 6 (1)–(3) that
and here, we use the fact that
□
Lemma 8 ([
2])
. For any continuous (not identical to 0), there exists a concave continuous modulus such that for , one has , where the constant 2 can not be any smaller and is defined as follows: if is continuous, non-decreasing, semi-additive, and , then is said to be a modules of continuity. Lemma 9 ([
23])
. The function is convex (or concave) on equivalent to for any , is convex (or concave) on , where Holhos [
26] proposed the concept of modulus
: for any
and
,
The relationship between the above modulus and the classical modulus is [
26]:
where
Lemma 10 ([
26])
. The : are positive linear operators, and letIf all the vanish at infinity, then for any , we have the following conclusion: 4. Shape Preservation
For
, the operators can also be expressed in the form
where
Theorem 4 (Monotonicity). Let be monotonically increasing (or decreasing) on , for so are all the operators
Proof. If
is monotonically increasing on
, then
is also monotonically increasing, i.e.,
is convex. If
have convexity preserving property [
24], then
are convex,
which implies
are monotonically increasing.
Similarly, we see that if is monotonically decreasing on , so are the operators . □
Theorem 5 (Convexity). Let be convex (or concave) on , so are all the operators .
Proof. We shall use the representation
If is convex, from Lemma 9 it is known that is also convex, and we have , i.e., So, are also convex.
Similarly, we see that if is concave on , so are the operators . □
Theorem 6 (Starshapeness). Let be non-negative on , , be decreasing on ; then, for , so are all the operators . But in general, if is increasing on , are no longer increasing.
Proof. For the first part of the Theorem, if is non-negative on , , is decreasing on , and combining with Definition 4, we have the following:
When , choosing , we obtain ;
When
, choosing
, we obtain
, by (
7), we know
therefore,
is decreasing on
.
Second, setting
, from Lemma 6, it is known that
It follows easily that, for , are increasing. For , are decreasing. So, if are increasing on , are no longer increasing. □
Theorem 7 (Semi-additivity). Let be non-negative, , semi-additive and increasing on , then for , so are all the operators . But in general, if is super-additive and decreasing on , are no longer super-additive.
Proof. First for all
,
let
, then
, it can be written that
If
is semi-additive on
, then,
In addition, if
is increasing on
, for
,
, then for any
,
and therefore
On the other hand, setting
, then
. In fact, by Lemma 6 and direct calculation gives that for the case
, one has
that means
are no longer super-additive for
□
Theorem 8 (Average convexity). Let be non-negative on , , if is convex (or concave), so are all the operators .
Proof. Now, let us take the second order derivative of
. It follows from Formula (
6) that
If are convex, for , , thus , i.e., (x) are convex. Similarly, we see that if are concave on , so are the operators . □
Theorem 9 (Average starshapeness). Let be non-negative on , , be decreasing on , then for , so are all the operators . But if is increasing on , are no longer increasing.
Proof. First, for
is decreasing on
,
noting Relation (
6), we write
So, is decreasing.
Second, choosing
, from Lemma 6, it is known that
It is known that, for , are increasing; for , are decreasing. are no longer increasing. □
Theorem 10. Let be non-negative, and increasing on , , for , one has .
Proof. Since
for
, from Formulas (
8) and (
9), we write
□
6. Conclusions
In this paper, we present a type of Szász–Kantorovich operators using the following ideas: (1) integral averaging leads to Kantorovich-type operators; (2) a function extension improves the approximation abilities; (3) the introduction of a parameter can be fine tune the approximation ability of the operators.
All these features combined provide a better approximation procedure. We further investigate the convergence of these operators, as well as attain the quantitative estimates, some shape preserving properties, while some important approximation tools, such as the forward differences, the modulus of continuity and the concave continuous modulus, are utilized. Numerical examples are used to verify the validity of our Szász–Kantorovich operators.
However, in this paper, we only considered the direct theorems of the Szász–Kantorovich operators, the functions are univariate. The converse results and higher dimensional case will be investigated in our future work.