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This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and

The aim of this paper is to provide a survey on a series of recent investigations which are centered around the problem of obtaining better properties by modifying properly some well known sequences of positive linear operators in the underlying Banach function spaces.

Such results are principally inspired by the pioneering work [

A systematic study of the operators

The A-statistical convergence of operators (

King’s idea inspires many other mathematicians to construct other modifications of well-known approximation processes fixing certain functions and to study their approximation and shape preserving properties.

In this review article we try to take stock of the situations and highlight the state of the art, hoping that this will be useful for all people that work in Approximation Theory and intend to apply King’s approach in some new contexts.

The paper is organized as follows: after a brief history on what has been done in this research area up to now, in Sections

From King’s work to nowadays, several investigations have been devoted to sequences of positive linear operators fixing certain (polynomial, exponential, or more general) functions. In this section we try to give some essential information about the construction of King-type operators. For all details we refer the readers to the references quoted in the text and we apologize in advance for any possible omission.

We begin to recall the contents of the first papers that generalize in some sense King’s idea ([

On the other hand, in [

The generalizations in [

In particular, in [

A study in full generality is undertaken in [

In [

Subsequently, other articles appear. First, we recall the paper due to Duman, Özarslan, and Aktu

Post Widder and Stancu operators are instead object of a modification that preserves

Another new general approach is considered by Agratini and Tarabie in [

Modifications which fix constants and linear functions, or the function

New King-type operators which reproduce

Another general approach deserves to be mentioned. Coming back to the classical Bernstein operators

Subsequently, the study of the operators

Very soon, the construction of the operators

In [

Subsequently, the above idea has been applied to other positive linear operators (see [

In particular, in [

We want to emphasize that the above constructions based on fixing suitable increasing functions do not recover the interesting case of linear operators fixing exponential functions, which has been a new and very popular direction in this research area in the last few years.

A sequence of Bernstein-type operators preserving

In [

Later, the idea of preserving exponential functions of different type has been applied to some other well-known linear positive operators, for which approximation and shape preserving properties, as well as quantitative estimates and Voronovskaya-type theorems, are proven.

For papers inspired by [

For modifications of linear operators preserving constants and

We end this section underlying that King’s idea has been applied also to some

In this section we review some results contained in [

Let us start with some preliminaries. Throughout this section,

We recall that the classical Bernstein operators are the positive linear operators

It is very well known that the sequence

In what follows, it will be useful to recall the following inequality which is an estimate of the rate of the above approximation presented by Shisha and Mond: for any

Besides the usual notion of convexity, other notions of convexity will be considered (see [

Let

A function

Moreover, a function

If

For the convenience of the reader we split up the discussion into three subsections.

In [

The operators

Korovkin theorem can be applied in order to conclude that, for

Considering the first and second modulus of smoothness, the following quantitative estimates can be achieved:

By comparing estimates (

Note that the right-end term in the above inequalities decreases to

The operators

The operators

We end this subsection observing that if we impose additional conditions on

The operators considered in the previous section fix

We note that

Moreover, the operators

For any function

We end this subsection by comparing the operators

First, if we take a positive constant

Moreover, the following statement holds.

Let

Conversely, if (

The proof is based on the comparison between the expression

In this section we discuss the operators defined in [

Since

Other (shape preserving) properties that this sequence verifies are

if

if

if

Moreover,

For the operators

As in the previous subsection, by comparing the asymptotic formulae for

Let

Conversely, if (

We end this section by observing that if the following conjecture is true, we might obtain an even better improvement in the approximation error.

If

In the present section we pass to discuss sequences of positive linear operators acting on spaces of continuous functions on unbounded intervals. To this end, we need to fix preliminarily some notations and recall definition and main results concerning the classical Szász-Mirakyan operators.

First of all, we denote by

In what follows, let

Clearly,

Moreover, we denote by

It is well known that Szász-Mirakyan operators were introduced independently in the 1940s by J. Favard ([

In particular

It might be useful for the following subsections to recall that (see [

Moreover, for every

It is well known that the sequence

In particular, we recall that, taking (

This last result might be useful to compare the Szász-Myrakyan operators with suitable generalizations that fix different functions.

In this subsection, we examine the Szász-Mirakyan type operators studied in [

From now on, we set

If

The following result, proven in [

Consider a sequence

After these preliminaries, set

In [

Some conditions have to be imposed in order that the sequence

More precisely, for any

Evaluating the operators

Accordingly, for any

The operators

Some estimates of the rate of convergence are available, by using a suitable modulus of continuity, introduced by Holhoş in [

Moreover, since

Further, under suitable assumptions, it is possible to determine a Voronovskaya-type result involving

Moreover, consider a function

The following examples show that, for suitable choices of the sequences

(1) If

(2) If

In particular, when applied to

Formula (

By means of [

We point out that, as shown in [

Easy calculations prove that

(3) If

(4) For

In particular, for every

Moreover, if

Another recent modification of the sequence of Szász-Mirakyan operators relies on the preservation of some exponential functions.

For functions

To investigate the approximation properties of the operators

In 1970, Boyanov and Veselinov [

The sequence

A quantitative form for Theorem

For

Moreover,

To investigate pointwise convergence of the operators

Let

As a uniform approximation result let us recall, as explained in [

For all

To see some of the advantages of new constructions of Szász-Mirakyan operators the following comparisons results were also presented in [

First, note that the definition of generalized convexity considered in

Observe that this is equivalent to

Let

The above-mentioned modified sequence of Szász-Mirakyan operators reproduces the functions

and if one considers the central moment operator

Now set

The first result on uniform convergence of sequence of the operators

For each function

In order to approximate unbounded functions, the exponential weighed space

Also let

For

In [

Let

Finally, the following saturation results for the sequence

Let

Let

The authors declare that they have no conflicts of interest.

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