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 on . Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King’s approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators.

1. Introduction

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 [1]. In that paper the author, J. P. King, introduces a new sequence of positive linear Bernstein-type operators defined, for every , and , by being continuous functions for every . Such operators turn into the classical Bernstein operators whenever, for any and , , but unlike the ’s, they are not in general polynomial-type operators. In fact, for every and ,where, for any , , and for . By applying Korovkin theorem to , for every , and , if and only if . Among all possible choices, King focuses his attention on the operators that fix , obtained by means of the generating functionsHe shows that is a positive approximation process in . Moreover, the operator interpolates at the end points and , and it is not a polynomial operator, because of (2) and (3). Through a quantitative estimate in terms of the classical first-order modulus of continuity, King also proves that the order of approximation of to is at least as good as the order of approximation of to for .

A systematic study of the operators is due to Gonska and Piţul [2], who determine new estimates for the rate of convergence in terms of the first and second moduli of continuity and, among the others, the behavior of the iterates as .

The A-statistical convergence of operators (1) is considered in [3].

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 3 and 4 we illustrate certain King-type modifications of the well-known Bernstein and Szász-Mirakyan operators.

2. A Brief History

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 ([47]). In [5] Cárdenas-Morales, Garrancho, and Muñoz-Delgado present a family of sequences of linear Bernstein-type operators (), depending on a real parameter , and fixing the polynomial function (note that ). Among other things, the authors prove that if is convex and increasing on , then for every . Section 3.1 is indeed devoted to the operators . More general results can be found in [8].

On the other hand, in [6] Duman and Özarslan apply the King’s original idea to Meyer-König and Zeller operators, and they obtain a better estimation error on the interval .

The generalizations in [4, 7] contain a different challenge: the authors propose King-type approximation processes in spaces of continuous functions on unbounded intervals.

In particular, in [7] (see also Examples 1) Duman and Özarslan consider the modified Szász-Mirakyan operators reproducing and and obtain better error estimates on the whole interval .

A study in full generality is undertaken in [4]. In fact, in that article, Agratini indicates how to construct sequences of positive linear operators of discrete type that act on a suitable weighted subspace of and preserve and . Besides the variant of Szász-Mirakjan operators, introduced independently in [7], he also constructs a variant of Baskakov and Bernstein-Chlodovsky operators.

In [9] Agratini investigates convergence and quantitative estimates for the bivariate version of the general operators previously considered in [4]. It is worthwhile noticing that the above results seem to be the only obtained in a multidimensional setting.

Subsequently, other articles appear. First, we recall the paper due to Duman, Özarslan, and Aktulu [10] in which Szász-Mirakyan-Beta type operators preserving are considered. Moreover, Duman and Özarslan, jointly with Della Vecchia ([11]), study a Kantorovich modification of Szász-Mirakjan type operators preserving linear functions, and they show their operators enable better error estimation on the interval than the classical Szász-Mirakjan-Kantorovich operators.

Post Widder and Stancu operators are instead object of a modification that preserves in polynomial weighted spaces, proposed by Rempulska and Skorupka in [12]. Also in this case better approximation properties than the original operators are achieved.

Another new general approach is considered by Agratini and Tarabie in [13] (see also [14]). The authors construct classes of discrete linear positive operators, acting on or on , and preserving both the constants and the polynomial (). Those classes of operators include the ones considered in [5] and a new modification of Szász-Mirakyan operators (see also [15]).

Modifications which fix constants and linear functions, or the function , have been introduced in [1620] (see also [21, Chapter 5]). In particular, such studies are concerned with modified Bernstein-Durrmeyer operators, Phillips operators, integrated Szász-Mirakjan operators, Beta operators of the second kind, and a Durrmeyer-Stancu type variant of Jain operators.

New King-type operators which reproduce and are studied in [22] by Braica, Pop and Indrea. Subsequently, Pop’s school deals with modifications of Kantorovich type operators, Durrmeyer type operators, Schurer operators, Bernstein-type operators, and Baskakov operators, fixing exactly two test functions from the set , (see, e.g., [23, 24]).

Another general approach deserves to be mentioned. Coming back to the classical Bernstein operators , in [25] Gonska, Piţul, and Raşa construct a sequence of King-type operators which preserve and a strictly increasing function , such that and . Such operators are defined as , and they are a positive approximation process in . Moreover, they preserve some global smoothness properties. The authors also discuss the monotonicity of the sequence when is a convex and decreasing function. They establish a Voronovskaja-type theorem, and finally they prove a recursion formula generalizing a corresponding result valid for the classical Bernstein operators. Note that the class of operators presented in [25] recovers the cases previously studied in [1, 5].

Subsequently, the study of the operators has been deepened by Birou in [26], where he finds some conditions under which ’s provide a lower approximation error than the classical Bernstein operators for the class of decreasing and generalized convex functions (see, also [27]). Moreover, he analyzes some shape preserving properties in the case is a polynomial of degree at most , or ().

Very soon, the construction of the operators motivates other works.

In [27] the operators which fix the function are studied and, among other things, they are compared with ’s and ’s in the approximation of functions which are increasing and convex with respect to . The authors focus on the case for which and fix polynomials of degree (see [28] for other generalizations of ’s reproducing and a strictly increasing polynomial). For more details about , see Section 3.2.

Subsequently, the above idea has been applied to other positive linear operators (see [2933]).

In particular, in [32] the authors propose a generalization of the classical Szász-Mirakyan operators by setting , where is a continuously differentiable function on with and . We want to point out that this class of operators does not include the ones studied in [7]. However, very recently (see [34]; cf. Section 4.1), Aral, Ulusoy, and Deniz generalize the operators , extending in this way the results contained in [7, 32]. See [35] for a modification of Baskakov-type operators in the spirit of what has been done for .

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 and , , , was already present in the literature (see [36, 37]).

In [38] a modification of Szász-Mirakyan operators preserving constants and , , is considered, while in [39] another modification of Szász-Mirakyan operators reproducing and () is studied. For more details about these two different variants, see Section 4.2.

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 [38, 39] we refer the readers to [4042] and [4346], respectively.

For modifications of linear operators preserving constants and , constants and , or constants and , cf. [4751].

We end this section underlying that King’s idea has been applied also to some or analogue operators (see, e.g., [5256]) and to some sequences of operators involving orthogonal polynomials (see, e.g., [57]).

3. On Bernstein-Type Operators

In this section we review some results contained in [5, 27, 43], where the authors deal with different modifications of the Bernstein operators based on King’s idea.

Let us start with some preliminaries. Throughout this section, is the space of all continuous real valued functions on , endowed with the sup norm and the natural pointwise ordering. If , the symbol stands for the space of all continuously -times differentiable functions on .

We recall that the classical Bernstein operators are the positive linear operators defined by setting, for every , , and ,

It is very well known that the sequence is an approximation process in ; i.e., for every , uniformly on .

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 ,where is the first-order modulus of continuity.

Besides the usual notion of convexity, other notions of convexity will be considered (see [58]; see also [59]).

Let be an extended complete Tchebychev system on .

A function is said to be convex with respect to (in symbols ), whenever

Moreover, a function is said to be convex with respect to , in symbol , whenever

If , then (6) and (7) hold for .

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

3.1. Bernstein-Type Operators Fixing Polynomials

In [5], the following Bernstein-type operators, depending on a real parameter , are defined: (, , ), where is the sequence of functions defined by It is easy to check that . Note that, if ’s turn into the classical King operators (1), while if goes to infinity they become the classical Bernstein operators.

The operators are positive and map into itself, and they fix the functions and . Moreover, and .

Korovkin theorem can be applied in order to conclude that, for , for since, for all , converges to .

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

By comparing estimates (10) and (5), we have then the approximation error for the operators is at least as good as the one for ’s on the interval , where . Indeed, we have that the inequality holds if and only if

Note that the right-end term in the above inequalities decreases to as goes to infinity. We point out that for we recover the result due to King, while for we get ; therefore King’s result is improved.

The operators share some shape preserving properties. We begin to recall that they map continuous and increasing functions into (continuous) increasing functions. Moreover, if is convex and increasing, then is convex. Finally, if is convex with respect to , then on .

The operators verify the following asymptotic formula: for all functions , which are two times differentiable at .

We end this subsection observing that if we impose additional conditions on , we can get tangible improvements in the approximation error. In fact, if is increasing and if the divided difference of on the nodes satisfy , being a real strictly positive constant, there exists such that In particular, . Note that, if is increasing and strictly convex and is the lower bound of , then .

3.2. Polynomial Operators Fixing Increasing Functions

The operators considered in the previous section fix , but they are not polynomial-type operators. The construction of polynomial-type operators fixing the above functions is presented in [27]. In that paper operators of the form are considered, where is any infinitely times continuously differentiable function on , such that , and . More precisely, The Bernstein operators can be obtained as a particular case for . On the other hand, if , is a polynomial-type operator and . For a Durrmeyer variant of the operators we refer the readers to [29] (and for a genuine Durrmeyer variant see [33]).

We note that . From the positivity of , together with the fact that is an extended complete Tchebychev system on , we easily get that uniformly on .

Moreover, the operators map continuous and increasing functions into (continuous) and increasing functions. Finally, is -convex of order provided that is so too (if , we say that a function is -convex of order whenever , being the usual -th differential operator).

For any function , two times differentiable at , we have that

We end this subsection by comparing the operators with ’s.

First, if we take a positive constant , whose existence is guaranteed by Freud [60], such that ; we have the following estimate: for , , and ,

Moreover, the following statement holds.

Theorem 1. Let . Suppose that there exists such that ThenIn particular, .
Conversely, if (20) holds with strict inequalities at a given point , then there exists such that for

The proof is based on the comparison between the expression and in the asymptotic formulae for ’s and ’s, respectively.

3.3. Fixing Increasing Exponential Functions

In this section we discuss the operators defined in [43]. From now on, set and recall that (). We define the sequence of positive linear operators as or, equivalently, for , , and . The functions fixed by these operators are and (). Moreover, for any and , the following identities hold:

Since is an extended complete Tchebychev system, and the operators are positive, they are an approximation process in (i.e., for each , uniformly w.r.t. ).

Other (shape preserving) properties that this sequence verifies are(i)if is increasing, then it is ;(ii)if is increasing and convex, then is convex;(iii)if , then (see (6)).

Moreover, for , , and . Here denotes the inverse function of . If , then can be replaced by .

For the operators , the following Voronovskaya-type result holds: if has second derivative at a point .

As in the previous subsection, by comparing the asymptotic formulae for and , we are able to get an improvement in the approximation by means of operators with respect to the operators under certain conditions.

Theorem 2. Let . Suppose that there exists such that ThenIn particular, .
Conversely, if (28) holds with strict inequalities at a given point , then there exists such that for

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

Conjecture 3. If is such that and , then for all and all , one has that .

4. On Szász-Mirakyan Type Operators

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 the space of all continuous real valued functions on . We also indicate by the subspace of all continuous bounded functions on . The space , endowed with the sup-norm and the natural pointwise ordering, is a Banach lattice. Moreover, the space of all continuous functions that converge at infinity will be denoted by .

In what follows, let be a weight function on ; we define

Clearly, is a normed space when endowed with the weighed norm

Moreover, we denote by the space of all continuous functions in , and by the space consisting of all functions in that converge at infinity. Finally, we say that if is uniformly continuous.

It is well known that Szász-Mirakyan operators were introduced independently in the 1940s by J. Favard ([61]), G. M. Mirakjan ([62]), and O. Szász ([63]), and they are defined by settingfor all functions for which the series at the right-hand side is absolutely convergent. This space includes, in particular, all functions such that , for some and .

In particular ’s map and into themselves.

It might be useful for the following subsections to recall that (see [64, Lemma 3]), , , and .

Moreover, for every ,where, for every , .

It is well known that the sequence is an approximation process in ; more precisely, for every , uniformly w.r.t. .

In particular, we recall that, taking (33) into account, for every , and (see, for example, [65, Theorem 5.1.2]),where denotes the classical first modulus of continuity.

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

4.1. Generalized Szász-Mirakyan Operators

In this subsection, we examine the Szász-Mirakyan type operators studied in [34]. Let be a function satisfying the following properties:(a) is continuously differentiable on ;(b) and .

From now on, we set and we consider the weighted spaces , , , and .

If for each the space (resp., ) becomes the classical weighed space (resp.,

The following result, proven in [66], shows that is a Korovkin set in .

Theorem 4. Consider a sequence of positive linear operators from into . Ifand, then, for every ,

After these preliminaries, set , for a suitable . Given an interval , consider two sequences , of functions on such that, for any ,(i) are positive functions on ;(ii) for every .

In [34], the authors introduced and studied the sequence of the generalized Szász-Myrakjan operators, defined asfor every and .

Some conditions have to be imposed in order that the sequence is an approximation process in , and, in particular, in order to verify (38).

More precisely, for any , there exist such that, for every ,

Evaluating the operators on and , it is easy to connect the sequences and with and , taking (40), (42), and (43) into account. More precisely, for every and ,

Accordingly, for any , and ,

The operators map into . Moreover, since easy calculations show that, for every and , by applying Theorem 4 to an extension of the operators to ,

Some estimates of the rate of convergence are available, by using a suitable modulus of continuity, introduced by Holhoş in [67]. More precisely, it is defined by setting, for every and , In particular, by using the results in [67], it can be proven that, for every and , where

Moreover, since if , from the latter formula and (41), we get that for every .

Further, under suitable assumptions, it is possible to determine a Voronovskaya-type result involving ’s. More precisely, assume that

Moreover, consider a function for which the function is twice differentiable. If the second derivative of is bounded on , then, for every ,

The following examples show that, for suitable choices of the sequences , and of the function , operators (45) turn into well known Szász-Myrakjan type operators that fix certain functions and the results in [34] can be applied to those operators. For quantitative Voronovskaya theorems and the study of a Durrmeyer-type variant of the operators (40) see [68] and [69], respectively.

Examples 1. (1) If , , and for every , the operators turn into the classical Szász-Myrakjan operators (32), which, as it is well known, preserve the function and .
(2) If , , , and , then operators turn into , which were object of investigation in [7] and, in the spirit of King’s work, preserve the function and .
In particular, when applied to , (53) gives the following result. If is a function which is twice differentiable and whose second derivative is bounded on , then, for every ,Formula (55) holds true uniformly w.r.t. , if . An estimate of convergence in (55) can be found in [70, Corollary 4].
By means of [65, Theorem 5.1.2], we have that, for every , We point out that, as shown in [7], for every Easy calculations prove that for every , so that, at least for , the operators provide a better approximation error than the classical Szász-Myrakjan operators (see (34)).
(3) If , , , and , then ’s are exactly the operators studied in [71], given by . Those operators fix the functions and . In this case, the Voronovskaya-type formula becomes for all and all which are twice differentiable and whose second derivative is bounded.
(4) For , for every and considering an arbitrary function satisfying (a) and (b), the operators reduce to that were introduced and studied in [32] and preserve the functions and .
In particular, for every , Moreover, if is a function such that is twice differentiable and the second derivative of is bounded on , then, for every ,

4.2. Fixing Increasing Exponential Functions

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

For functions , such that the right-hand side below is absolutely convergent, Szász-Mirakyan operators reproducing the functions and , , are introduced in [38] and defined by (), in such a way that the conditionsare satisfied for all and all . To provide condition (64), equalitymust be held (for more details see [38]).

To investigate the approximation properties of the operators , some preliminaries are needed. First, if , we get Then, letting , , we haveMoreover, considering equality (65), one can find