# On Sequences of J. P. King-Type Operators

Authors: Tuncer Acar; Mirella Cappelletti Montano; Pedro Garrancho; Vita Leonessa
Journal: Journal of Function Spaces
 (2019)

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## Abstract

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 andx2 on [0,1]. 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.


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## Body

## 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 (Vn,rn)n≥1 of positive linear Bernstein-type operators defined, for every f∈C[0,1], n≥1 and 0≤x≤1, by(1)Vn,rnf;x=∑k=1nnkrnxk1-rnxn-kfkn,rn:[0,1]→[0,1] being continuous functions for every n≥1. Such operators turn into the classical Bernstein operators Bn whenever, for any n≥1 and 0≤x≤1, rn(x)=x, but unlike the Bn’s, they are not in general polynomial-type operators. In fact, for every n≥1 and 0≤x≤1,(2)Vn,rn1=1,Vn,rne1=rn,Vn,rne2=rn2+rn1-rnn,where, for any t∈[0,1], 1(t)=1, and ei(t)=ti for i=1,2. By applying Korovkin theorem to Vn,rn, for every f∈C[0,1], and x∈[0,1], limn→∞Vn,rn(f;x)=f(x) if and only if limn→∞rn(x)=x. Among all possible choices, King focuses his attention on the operators Vn,rn∗ that fix e2, obtained by means of the generating functions(3)rn∗x=x2ifn=1,-12n-1+nx2n-1+14n-12ifn≥2.He shows that (Vn,rn∗)n≥1 is a positive approximation process in C[0,1]. Moreover, the operator Vn,rn∗ interpolates f at the end points 0 and 1, 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 Vn,rn∗(f;x) to f(x) is at least as good as the order of approximation of Bn(f;x) to f(x) for 0≤x<1/3.A systematic study of the operatorsVn,rn∗ 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 Vn,rn∗m as m→+∞.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 Sections3 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 ([4–7]). In [5] Cárdenas-Morales, Garrancho, and Muñoz-Delgado present a family of sequences of linear Bernstein-type operators Bn,α (n>1), depending on a real parameter α≥0, and fixing the polynomial function e2+αe1 (note that Bn,0=Vn,rn∗). Among other things, the authors prove that if f is convex and increasing on [0,1], then f(x)≤Bn,α(f;x)≤Bn(f;x) for every x∈[0,1]. Section 3.1 is indeed devoted to the operators Bn,α. 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 [1/2,1[.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 1 and e2 and obtain better error estimates on the whole interval [0,∞).A study in full generality is undertaken in [4]. In fact, in that article, Agratini indicates how to construct sequences (Ln∗)n≥1 of positive linear operators of discrete type that act on a suitable weighted subspace of C[0,∞) and preserve 1 and e2. 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 Aktug˘lu [10] in which Szász-Mirakyan-Beta type operators preserving e2 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 [1/2,∞) than the classical Szász-Mirakjan-Kantorovich operators.Post Widder and Stancu operators are instead object of a modification that preservese2 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 [0,1] or on [0,∞), and preserving both the constants and the polynomial e2+αe1 (α≥0). 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 functione2, have been introduced in [16–20] (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 reproducee1 and e2 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 {1,e1,e2}, (see, e.g., [23, 24]).Another general approach deserves to be mentioned. Coming back to the classical Bernstein operatorsBn, in [25] Gonska, Piţul, and Raşa construct a sequence of King-type operators Vnτ which preserve 1 and a strictly increasing function τ∈C[0,1], such that τ(0)=0 and τ(1)=1. Such operators are defined as Vnτ(f)=Bn(f)∘(Bnτ)-1∘τ, and they are a positive approximation process in C[0,1]. Moreover, they preserve some global smoothness properties. The authors also discuss the monotonicity of the sequence (Vnτf)n≥1 when f 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 operatorsVnτ has been deepened by Birou in [26], where he finds some conditions under which Vnτ’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 τ is a polynomial of degree at most 2, or τ(x)=(ebx-1)/(eb-1) (x∈[0,1],b<0).Very soon, the construction of the operatorsVnτ motivates other works.In [27] the operators Bnτ(f)=Bn(f∘τ-1)∘τ which fix the function τ are studied and, among other things, they are compared with Bn’s and Vnτ’s in the approximation of functions which are increasing and convex with respect to τ. The authors focus on the case for which Bnτ and Vnτ fix polynomials of degree m (see [28] for other generalizations of Bn’s reproducing 1 and a strictly increasing polynomial). For more details about Bnτ, see Section 3.2.Subsequently, the above idea has been applied to other positive linear operators (see [29–33]).In particular, in [32] the authors propose a generalization of the classical Szász-Mirakyan operators Sn by setting Snρ(f)=Sn(f∘ρ-1)∘ρ, where ρ is a continuously differentiable function on [0,∞) with ρ(0)=0 and infx≥0ρ′(x)≥1. 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 Snρ, 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 Snρ.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 preservingeλ0x and eλ1x, λ0,λ1∈R, λ0≠λ1, was already present in the literature (see [36, 37]).In [38] a modification of Szász-Mirakyan operators preserving constants and e2ax, a>0, is considered, while in [39] another modification of Szász-Mirakyan operators reproducing eax and e2ax (a>0) 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 [40–42] and [43–46], respectively.For modifications of linear operators preserving constants ande-x, constants and e-2x, or constants and eAx, A∈R cf. [47–51].We end this section underlying that King’s idea has been applied also to someq- or (p,q)- analogue operators (see, e.g., [52–56]) 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,C[0,1] is the space of all continuous real valued functions on [0,1], endowed with the sup norm ·∞ and the natural pointwise ordering. If k∈N, the symbol Ck[0,1] stands for the space of all continuously k-times differentiable functions on [0,1].We recall that the classical Bernstein operators are the positive linear operatorsBn:C[0,1]→C[0,1] defined by setting, for every n≥1, f∈C[0,1], and 0≤x≤1, (4)Bnf;x=∑k=0nnkxk1-xn-kfkn.It is very well known that the sequence(Bn)n≥1 is an approximation process in C[0,1]; i.e., for every f∈C[0,1], limn→∞Bn(f)=f uniformly on [0,1].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 anyC[0,1],(5)Bnf;x-fx≤1+x1-x/nδ2ω1f,δ,where ω1(f,δ) 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{u,v} be an extended complete Tchebychev system on [0,1].A functionf:(0,1)→R is said to be convex with respect to {u} (in symbols f∈C(u)), whenever(6)ux0ux1fx0fx1≥0,0<x0<x1<1.Moreover, a functionf:(0,1)→R is said to be convex with respect to {u,v}, in symbol f∈C(u,v), whenever(7)ux0ux1ux2vx0vx1vx2fx0fx1fx2≥0,0<x0<x1<x2<1.Iff∈C[0,1], then (6) and (7) hold for 0≤x0<x1<x2≤1.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 α≥0, are defined: (8)Bn,αf;x≔∑k=0nnkrn,αxk1-rn,αxn-kfkn (n≥1, f∈C[0,1], x∈[0,1]), where {rn,α:[0,1]→R}n>1 is the sequence of functions defined by (9)rn,αx≔-nα+12n-1+nα+124n-12+nαx+x2n-10≤x≤1.It is easy to check that Bn,αf=(Bnf)∘(rn,α). Note that, if =0Bn,α’s turn into the classical King operators (1), while if α goes to infinity they become the classical Bernstein operators.The operatorsBn,α are positive and map C[0,1] into itself, and they fix the functions 1 and e2+αe1. Moreover, Bn,α(e1)=rn,α and Bn,α(e2)=(1/n)rn,α+(n-1/n)rn,α2.Korovkin theorem can be applied in order to conclude that, forf∈C[0,1], limn→∞Bn,α(f;x)=f(x) for 0≤x≤1 since, for all α≥0, rn,α(x) converges to x.Considering the first and second modulus of smoothness, the following quantitative estimates can be achieved:(10)Bn,αf;x-fx≤1+2x2+αx-rn,αxα+2xδ2ω1f,δ,(11)Bn,αf;x-fx≤rn,αx-xδω1f,δ+1+2x2+αx-rn,αxα+2x2δ2ω2f,δ.By comparing estimates (10) and (5), we have then the approximation error for the operators Bn,α is at least as good as the one for Bn’s on the interval [0,Hα], where Hα=(1-2α+1+8α+4α2)/6. Indeed, we have that the inequality (12)2x2+αx-rn,αxα+2x≤x1-xn holds if and only if (13)0≤x≤1+n-2nα+1+2n+n2+8n2α+4n2α221+3n.Note that the right-end term in the above inequalities decreases toHα as n goes to infinity. We point out that for H0=1/3 we recover the result due to King, while for α→+∞ we get Hα→1/2; therefore King’s result is improved.The operatorsBn,α share some shape preserving properties. We begin to recall that they map continuous and increasing functions into (continuous) increasing functions. Moreover, if f is convex and increasing, then Bn,α(f) is convex. Finally, if f is convex with respect to {1,e2+αe1}, then Bn,α(f)≥f on [0,1].The operatorsBn,α verify the following asymptotic formula: (14)limn→∞2nBn,αf;x-fx=x1-xf′′x-22x+αf′x, for all functions f∈C[0,1], which are two times differentiable at x∈(0,1).We end this subsection observing that if we impose additional conditions onf, we can get tangible improvements in the approximation error. In fact, if f∈C[0,1] is increasing and if the divided difference f[x0,x1,x2] of f on the nodes 0≤x0<x1<x2≤1 satisfy f[x0,x1,x2]≥M, M being a real strictly positive constant, there exists α¯≥0 such that (15)0≤Bn,αf;x-fx<Bnf;x-fx,forα≥α¯and0<x<1. In particular, α¯≔minα≥0:(f1-f0)/(1+α)≤M. Note that, if f∈C2[0,1] is increasing and strictly convex and M is the lower bound of f′′, then α¯=2f′(1)/M.


### 3.2. Polynomial Operators Fixing Increasing Functions

The operators considered in the previous section fix1,e2+αe1, 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 Bnτf=Bn(f∘τ-1)∘τ are considered, where τ is any infinitely times continuously differentiable function on [0,1], such that τ(0)=0, τ(1)=1 and τ′(x)>0. More precisely, (16)Bnτf;x=∑k=0nnkτxk1-τxn-kf∘τ-1kn,f∈C0,1,x∈0,1.The Bernstein operators can be obtained as a particular case for τ=e1. On the other hand, if τ=(e2+αe1)/(1+α), Bnτ is a polynomial-type operator and Bnτ(τ)=τ. For a Durrmeyer variant of the operators Bnτ we refer the readers to [29] (and for a genuine Durrmeyer variant see [33]).We note thatBnττ2=τ/n+(n-1/n)τ2. From the positivity of Bnτ, together with the fact that {1,τ,τ2} is an extended complete Tchebychev system on [0,1], we easily get that limn→∞Bnτ(f)=f uniformly on [0,1].Moreover, the operatorsBnτ map continuous and increasing functions into (continuous) and increasing functions. Finally, Bnτ(f) is τ-convex of order k provided that f is so too (if k∈N, we say that a function f∈Ck[0,1] is τ-convex of order k whenever Dτmf=Dm(f∘τ-1)∘τ, Dk being the usual k-th differential operator).For any functionf∈C[0,1], two times differentiable at x∈(0,1), we have that (17)limn→∞2nBnτfx-fx=τx1-τx-τ′′xf′xτ′3+f′′xτ′2.We end this subsection by comparing the operatorsBnτ with Bn’s.First, if we take a positive constantK, whose existence is guaranteed by Freud [60], such that K(t-x)2≤τ′2forallt,x∈[0,1]; we have the following estimate: for f∈C[0,1], δ>0, and x∈[0,1], (18)Bnτf;x-fx≤ω1f,δ1+τ′xτx1-τxnKδ2.Moreover, the following statement holds.Theorem 1.
Letf∈C2[0,1]. Suppose that there exists n0∈N such that (19)fx≤Bnτf;x≤Bnf;x,∀n≥n0,x∈0,1. Then(20)f′′x≥τ′′xτ′xf′x≥1-x1-xτ′2τx1-τxf′′x,x∈0,1.In particular, f′′(x)≥0.
Conversely, if (20) holds with strict inequalities at a given point x0∈(0,1), then there exists n0∈N such that for n≥n0(21)fx0<Bnτf;x0<Bnf;x0.The proof is based on the comparison between the expressionx(1-x) and τ(x)(1-τ(x))-τ′′(x)f′(x)/τ′3+f′′(x)/τ′2 in the asymptotic formulae for Bn’s and Bnτ’s, respectively.


### 3.3. Fixing Increasing Exponential Functions

In this section we discuss the operators defined in [43]. From now on, set an(x)≔(eμx/n-1)/(eμ/n-1) and recall that expμ(x)≔eμx (μ>0). We define the sequence of positive linear operators Gn as (22)Gnf;x=expμxBnfexpμ;anx, or, equivalently, (23)Gnf;x=∑k=0nnkanxk1-anxn-kfkne-μk/neμx, for f:[0,1]→R, n≥1, and 0≤x≤1. The functions fixed by these operators are expμ and expμ2 (μ>0). Moreover, for any x∈[0,1] and n≥1, the following identities hold: (24)Gn1;x=eμx-1eμ/n+1-eμx/nn,Gnexpμ3;x=eμxeμx+1/n+eμx/n-eμ/nn,Gnexpμ4;x=eμxeμx+2/n+eμx+1/n+eμx/n-eμ/n-e2μ/nn.Since{1,expμ,expμ2} is an extended complete Tchebychev system, and the operators Gn are positive, they are an approximation process in C[0,1] (i.e., for each f∈C[0,1], limn→∞Gn(f;x)=f(x) uniformly w.r.t. x∈[0,1]).Other (shape preserving) properties that this sequence verifies are(i)
iff/expμ is increasing, then it is Gn(f)/expμ;(ii)
iff/expμ is increasing and convex, then Gnf/expμ is convex;(iii)
iff∈C(expμ), then Gn(f)∈C(expμ) (see (6)).Moreover,(25)Gnf;x-fx≤fxGn1;x-1+Gn1;x+e2μxGn1;x-1δ2ω1f∘logμ;δ,for f∈C[0,1], x∈(0,1), and δ>0. Here logμ denotes the inverse function of expμ. If μ≥1, then ω1f∘logμ;δ can be replaced by ω1f;δ.For the operatorsGn, the following Voronovskaya-type result holds: (26)limn→∞2nGnf;x-fx=x1-xf′′x-3μf′2fx, if f∈C[0,1] has second derivative at a point x∈(0,1).As in the previous subsection, by comparing the asymptotic formulae forBn and Gn, we are able to get an improvement in the approximation by means of operators Gn with respect to the operators Bn under certain conditions.Theorem 2.
Letf∈C2[0,1]. Suppose that there exists n0∈N such that (27)fx≤Gnf;x≤Bnf;x,∀n≥n0,x∈0,1. Then(28)f′′x≥3μf′2fx≥0,x∈0,1.In particular, f′′(x)≥0.
Conversely, if (28) holds with strict inequalities at a given point x∈(0,1), then there exists n0∈N such that for n≥n0(29)fx<Gnf;x<Bnf;x.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.
Iff∈C[0,1] is such that f∈C(expμ) and f∈C(expμ,expμ2), then for all n∈N and all x∈[0,1], one has that f(x)≤Gn(f;x)≤Bn(f;x).


## 3.1. Bernstein-Type Operators Fixing Polynomials

In [5], the following Bernstein-type operators, depending on a real parameter α≥0, are defined: (8)Bn,αf;x≔∑k=0nnkrn,αxk1-rn,αxn-kfkn (n≥1, f∈C[0,1], x∈[0,1]), where {rn,α:[0,1]→R}n>1 is the sequence of functions defined by (9)rn,αx≔-nα+12n-1+nα+124n-12+nαx+x2n-10≤x≤1.It is easy to check that Bn,αf=(Bnf)∘(rn,α). Note that, if =0Bn,α’s turn into the classical King operators (1), while if α goes to infinity they become the classical Bernstein operators.The operatorsBn,α are positive and map C[0,1] into itself, and they fix the functions 1 and e2+αe1. Moreover, Bn,α(e1)=rn,α and Bn,α(e2)=(1/n)rn,α+(n-1/n)rn,α2.Korovkin theorem can be applied in order to conclude that, forf∈C[0,1], limn→∞Bn,α(f;x)=f(x) for 0≤x≤1 since, for all α≥0, rn,α(x) converges to x.Considering the first and second modulus of smoothness, the following quantitative estimates can be achieved:(10)Bn,αf;x-fx≤1+2x2+αx-rn,αxα+2xδ2ω1f,δ,(11)Bn,αf;x-fx≤rn,αx-xδω1f,δ+1+2x2+αx-rn,αxα+2x2δ2ω2f,δ.By comparing estimates (10) and (5), we have then the approximation error for the operators Bn,α is at least as good as the one for Bn’s on the interval [0,Hα], where Hα=(1-2α+1+8α+4α2)/6. Indeed, we have that the inequality (12)2x2+αx-rn,αxα+2x≤x1-xn holds if and only if (13)0≤x≤1+n-2nα+1+2n+n2+8n2α+4n2α221+3n.Note that the right-end term in the above inequalities decreases toHα as n goes to infinity. We point out that for H0=1/3 we recover the result due to King, while for α→+∞ we get Hα→1/2; therefore King’s result is improved.The operatorsBn,α share some shape preserving properties. We begin to recall that they map continuous and increasing functions into (continuous) increasing functions. Moreover, if f is convex and increasing, then Bn,α(f) is convex. Finally, if f is convex with respect to {1,e2+αe1}, then Bn,α(f)≥f on [0,1].The operatorsBn,α verify the following asymptotic formula: (14)limn→∞2nBn,αf;x-fx=x1-xf′′x-22x+αf′x, for all functions f∈C[0,1], which are two times differentiable at x∈(0,1).We end this subsection observing that if we impose additional conditions onf, we can get tangible improvements in the approximation error. In fact, if f∈C[0,1] is increasing and if the divided difference f[x0,x1,x2] of f on the nodes 0≤x0<x1<x2≤1 satisfy f[x0,x1,x2]≥M, M being a real strictly positive constant, there exists α¯≥0 such that (15)0≤Bn,αf;x-fx<Bnf;x-fx,forα≥α¯and0<x<1. In particular, α¯≔minα≥0:(f1-f0)/(1+α)≤M. Note that, if f∈C2[0,1] is increasing and strictly convex and M is the lower bound of f′′, then α¯=2f′(1)/M.


## 3.2. Polynomial Operators Fixing Increasing Functions

The operators considered in the previous section fix1,e2+αe1, 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 Bnτf=Bn(f∘τ-1)∘τ are considered, where τ is any infinitely times continuously differentiable function on [0,1], such that τ(0)=0, τ(1)=1 and τ′(x)>0. More precisely, (16)Bnτf;x=∑k=0nnkτxk1-τxn-kf∘τ-1kn,f∈C0,1,x∈0,1.The Bernstein operators can be obtained as a particular case for τ=e1. On the other hand, if τ=(e2+αe1)/(1+α), Bnτ is a polynomial-type operator and Bnτ(τ)=τ. For a Durrmeyer variant of the operators Bnτ we refer the readers to [29] (and for a genuine Durrmeyer variant see [33]).We note thatBnττ2=τ/n+(n-1/n)τ2. From the positivity of Bnτ, together with the fact that {1,τ,τ2} is an extended complete Tchebychev system on [0,1], we easily get that limn→∞Bnτ(f)=f uniformly on [0,1].Moreover, the operatorsBnτ map continuous and increasing functions into (continuous) and increasing functions. Finally, Bnτ(f) is τ-convex of order k provided that f is so too (if k∈N, we say that a function f∈Ck[0,1] is τ-convex of order k whenever Dτmf=Dm(f∘τ-1)∘τ, Dk being the usual k-th differential operator).For any functionf∈C[0,1], two times differentiable at x∈(0,1), we have that (17)limn→∞2nBnτfx-fx=τx1-τx-τ′′xf′xτ′3+f′′xτ′2.We end this subsection by comparing the operatorsBnτ with Bn’s.First, if we take a positive constantK, whose existence is guaranteed by Freud [60], such that K(t-x)2≤τ′2forallt,x∈[0,1]; we have the following estimate: for f∈C[0,1], δ>0, and x∈[0,1], (18)Bnτf;x-fx≤ω1f,δ1+τ′xτx1-τxnKδ2.Moreover, the following statement holds.Theorem 1.
Letf∈C2[0,1]. Suppose that there exists n0∈N such that (19)fx≤Bnτf;x≤Bnf;x,∀n≥n0,x∈0,1. Then(20)f′′x≥τ′′xτ′xf′x≥1-x1-xτ′2τx1-τxf′′x,x∈0,1.In particular, f′′(x)≥0.
Conversely, if (20) holds with strict inequalities at a given point x0∈(0,1), then there exists n0∈N such that for n≥n0(21)fx0<Bnτf;x0<Bnf;x0.The proof is based on the comparison between the expressionx(1-x) and τ(x)(1-τ(x))-τ′′(x)f′(x)/τ′3+f′′(x)/τ′2 in the asymptotic formulae for Bn’s and Bnτ’s, respectively.


## 3.3. Fixing Increasing Exponential Functions

In this section we discuss the operators defined in [43]. From now on, set an(x)≔(eμx/n-1)/(eμ/n-1) and recall that expμ(x)≔eμx (μ>0). We define the sequence of positive linear operators Gn as (22)Gnf;x=expμxBnfexpμ;anx, or, equivalently, (23)Gnf;x=∑k=0nnkanxk1-anxn-kfkne-μk/neμx, for f:[0,1]→R, n≥1, and 0≤x≤1. The functions fixed by these operators are expμ and expμ2 (μ>0). Moreover, for any x∈[0,1] and n≥1, the following identities hold: (24)Gn1;x=eμx-1eμ/n+1-eμx/nn,Gnexpμ3;x=eμxeμx+1/n+eμx/n-eμ/nn,Gnexpμ4;x=eμxeμx+2/n+eμx+1/n+eμx/n-eμ/n-e2μ/nn.Since{1,expμ,expμ2} is an extended complete Tchebychev system, and the operators Gn are positive, they are an approximation process in C[0,1] (i.e., for each f∈C[0,1], limn→∞Gn(f;x)=f(x) uniformly w.r.t. x∈[0,1]).Other (shape preserving) properties that this sequence verifies are(i)
iff/expμ is increasing, then it is Gn(f)/expμ;(ii)
iff/expμ is increasing and convex, then Gnf/expμ is convex;(iii)
iff∈C(expμ), then Gn(f)∈C(expμ) (see (6)).Moreover,(25)Gnf;x-fx≤fxGn1;x-1+Gn1;x+e2μxGn1;x-1δ2ω1f∘logμ;δ,for f∈C[0,1], x∈(0,1), and δ>0. Here logμ denotes the inverse function of expμ. If μ≥1, then ω1f∘logμ;δ can be replaced by ω1f;δ.For the operatorsGn, the following Voronovskaya-type result holds: (26)limn→∞2nGnf;x-fx=x1-xf′′x-3μf′2fx, if f∈C[0,1] has second derivative at a point x∈(0,1).As in the previous subsection, by comparing the asymptotic formulae forBn and Gn, we are able to get an improvement in the approximation by means of operators Gn with respect to the operators Bn under certain conditions.Theorem 2.
Letf∈C2[0,1]. Suppose that there exists n0∈N such that (27)fx≤Gnf;x≤Bnf;x,∀n≥n0,x∈0,1. Then(28)f′′x≥3μf′2fx≥0,x∈0,1.In particular, f′′(x)≥0.
Conversely, if (28) holds with strict inequalities at a given point x∈(0,1), then there exists n0∈N such that for n≥n0(29)fx<Gnf;x<Bnf;x.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.
Iff∈C[0,1] is such that f∈C(expμ) and f∈C(expμ,expμ2), then for all n∈N and all x∈[0,1], one has that f(x)≤Gn(f;x)≤Bn(f;x).


## 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 byC[0,∞) the space of all continuous real valued functions on [0,∞). We also indicate by Cb[0,∞) the subspace of all continuous bounded functions on [0,∞). The space Cb[0,∞), 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 C∗[0,∞).In what follows, letφ be a weight function on [0,∞); we define(30)Bφ0,∞=f:0,∞→R∣thereexistsMf≥0suchthatfx≤Mfφx∀x≥0.Clearly,Bφ[0,∞) is a normed space when endowed with the weighed norm(31)fφ=supx≥0fxφxf∈Bφ0,∞.Moreover, we denote byCφ[0,∞) the space of all continuous functions in Bφ[0,∞), and by Cφ∗[0,∞) the space consisting of all functions in Cφ[0,∞) that converge at infinity. Finally, we say that f∈Uφ[0,∞) if f/φ 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 setting(32)Snf;x≔∑k=0∞e-nxnxkk!fknn≥1,x≥0,for all functions f:[0,∞)→R for which the series at the right-hand side is absolutely convergent. This space includes, in particular, all functions f:[0,∞)→R such that fx≤Mexp(αx)(x≥0), for some M≥0 and α∈R.In particularSn’s map Cb[0,∞) and C∗[0,∞) into themselves.It might be useful for the following subsections to recall that (see [64, Lemma 3]), Sn(1)=1, Sn(e1)=e1, and Sn(e2)=e2+(1/n)e1.Moreover, for everyx≥0,(33)Snψxt;x=0,Snψx2t;x=xn,where, for every y≥0, ψx(y)=y-x.It is well known that the sequence(Sn)n≥1 is an approximation process in C∗[0,∞); more precisely, for every f∈C∗[0,∞), limn→∞Sn(f;x)=f(x) uniformly w.r.t. x∈[0,∞).In particular, we recall that, taking (33) into account, for every f∈Cb[0,∞), x≥0 and n≥1 (see, for example, [65, Theorem 5.1.2]),(34)Snf;x-fx≤2ω1f,Snψx2t;x=2ω1f,xn,where ω1(f,δ) 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 ρ:[0,∞)→R be a function satisfying the following properties:(a)
ρ is continuously differentiable on [0,∞);(b)
ρ(0)=0 and infx≥0ρ′(x)≥1.From now on, we set(35)φx=1+ρ2xx≥0,and we consider the weighted spaces Bφ[0,∞), Cφ[0,∞), Cφ∗[0,∞), and Uφ[0,∞).Ifρ(x)=x for each x≥0 the space Cφ[0,∞) (resp., Cφ∗[0,∞)) becomes the classical weighed space (36)E2=f∈C0,∞:supx≥0fx1+x2∈R (resp., (37)E2∗=f∈C0,∞:limx→+∞fx1+x2∈R.The following result, proven in [66], shows that {1,ρ,ρ2} is a Korovkin set in Cφ∗[0,∞).Theorem 4.
Consider a sequence(Ln)n≥1 of positive linear operators from Cφ[0,∞) into Bφ[0,∞). If(38)limn→∞Lnρν-ρνφ=0forν=0,1,2,and, then, for every f∈Cφ∗[0,∞), (39)limn→∞Lnf-fφ=0.After these preliminaries, setN1≔{n∈N∣n≥n0}, for a suitable n0∈N. Given an interval I⊂[0,∞), consider two sequences (αn)n≥1, (βn)n≥1 of functions on I such that, for any n∈N1,(i)
αn,βn:I→R are positive functions on I;(ii)
βn(x)-αn(x)≥0 for every x∈I.In [34], the authors introduced and studied the sequence of the generalized Szász-Myrakjan operators, defined as(40)S~nρf;x=e-αnx∑k=0∞βnxkk!f∘ρ-1knfor every f∈C(I),n∈N1 and x∈I.Some conditions have to be imposed in order that the sequence(S~nρ)n≥n0 is an approximation process in Cφ∗[0,∞), and, in particular, in order to verify (38).More precisely, for anyn≥n0, there exist un,vn:I→R such that, for every x∈I,(41)unx≤un0,vnx≤vn0,limn→∞un0=limn→∞vn0=0,(42)S~nρ1;x=1+unx,(43)S~nρρ;x=ρx+vnx.Evaluating the operatorsS~nρ on 1 and ρ, it is easy to connect the sequences (αn)n≥n0 and (βn)n≥n0 with (un)n≥n0 and (vn)n≥n0, taking (40), (42), and (43) into account. More precisely, for every x∈I and n≥n0, (44)αnx=nρx+vnx1+unx-log1+unx,βnx=nρx+vnx1+unx.Accordingly, for anyn≥n0, f∈C(I) and x∈I,(45)S~nρf;x=e-nρx+vnx/1+unx1+unx∑k=0∞1k!nρx+vnx1+unxkf∘ρ-1kn.The operatorsS~nρ map Cφ[0,∞) into Bφ[0,∞). Moreover, since easy calculations show that, for every x∈I and n≥n0, (46)S~nρρ2;x=ρx+vnx21+unx+ρx+vnxn, by applying Theorem 4 to an extension of the operators S~nρ(f) to [0,∞), (47)limn→∞supx∈IS~nρf;x-fxφx=0.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 f∈Cφ[0,∞) and δ>0, (48)ωρf;δ=supx,t≥0ρt-ρx≤δ⁡ft-fxφt+φx. In particular, by using the results in [67], it can be proven that, for every f∈Cφ[0,∞) and n≥n0,(49)S~nρf-fφ3/2≤7+4un0+22vn0+vn02+2n+2vn0nωρf;δn, where (50)δn=16n+4n2+3un0+20vn0+22vn0n+4vn0n2+8vn02+6vn02n+vn03+21+un02n+un0+4vn0+2vn0n+vn02.Moreover, sincelimδ→0ωρ(f;δ)=0 if f∈Uφ[0,∞), from the latter formula and (41), we get that (51)limn→∞S~nρf-fφ3/2=0 for every f∈Uφ[0,∞).Further, under suitable assumptions, it is possible to determine a Voronovskaya-type result involvingS~nρ’s. More precisely, assume that (52)limn→∞nunx=l1,limn→∞nvnx=l2.Moreover, consider a functionf∈Cφ[0,∞) for which the function f∘ρ-1 is twice differentiable. If the second derivative of f∘ρ-1 is bounded on [0,∞), then, for every x∈I,(53)limn→∞nS~nρf;x-fx=fxl1+l2-ρxl1f∘ρ-1′ρx+12ρxf∘ρ-1′′ρx.The following examples show that, for suitable choices of the sequences(un(x))n≥n0, (vn(x))n≥n0 and of the function ρ, 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) IfI=[0,∞), un(x)=vn(x)=0, and ρ(x)=x for every x≥0, the operators Sn~ρ turn into the classical Szász-Myrakjan operators (32), which, as it is well known, preserve the function e0 and ρ=e1.
(2) IfI=[0,∞), ρ(x)=x, un(x)=0, and vn(x)=-1/2n+4n2x2+1/2n-x, then operators Sn~ρ turn into (54)Dn∗f;x=e1-4n2x2+1/2∑k=0∞4n2x2+1-1k2kk!fkn(f∈E2,n≥1,x≥0), which were object of investigation in [7] and, in the spirit of King’s work, preserve the function e0 and ρ2=e2.
In particular, when applied toDn∗, (53) gives the following result. If f∈E2 is a function which is twice differentiable and whose second derivative is bounded on [0,∞), then, for every x≥0,(55)limn→∞nDn∗f;x-fx=-12f′x+x2f′′x.
Formula (55) holds true uniformly w.r.t. x≥0, if f′,f′′∈E2∗. 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 f∈Cb[0,∞), (56)Dn∗f;x-fx≤2ω1f,Dn∗ψx2x.
We point out that, as shown in [7], for every x≥0(57)Dn∗ψx2t;x=2x2+xn-x4n2x2+1n.
Easy calculations prove thatDn∗(ψx2(t);x)≤Sn(ψx2(t);x) for every x≥0, so that, at least for f∈Cb[0,∞), the operators Dn∗ provide a better approximation error than the classical Szász-Myrakjan operators Sn (see (34)).
(3) IfI=1/(n0-1),∞, un(x)=1/(nx-1), vn(x)=0, and ρ(x)=x, then Sn~ρ’s are exactly the operators studied in [71], given by (58)Sn∗f;x=nxe1-nxnx-1∑k=0∞nx-1kk!fkn(f∈E2,n≥n0,x∈I). Those operators fix the functions ρ=e1 and ρ2=e2. In this case, the Voronovskaya-type formula becomes (59)limn→∞nSn∗f;x-fx=fxx-f′x+x2f′′x, for all x∈I and all f∈E2 which are twice differentiable and whose second derivative is bounded.
(4) ForI=[0,∞), un(x)=vn(x)=0 for every x≥0 and considering an arbitrary function ρ satisfying (a) and (b), the operators Sn~ρ reduce to (60)Snρf;x=e-nρx∑k=0∞nρxkk!f∘ρ-1kn(f∈Cφ[0,∞),x∈I,n≥1) that were introduced and studied in [32] and preserve the functions e0 and ρ.
In particular, for everyf∈Cφ[0,∞), (61)Snρf-fφ3/2≤7+4nωρf;20n+8n.
Moreover, iff∈Cφ[0,∞) is a function such that f∘ρ-1 is twice differentiable and the second derivative of f∘ρ-1 is bounded on [0,∞), then, for every x≥0, (62)limn→∞nSnρf;x-fx=12ρxf∘ρ-1′′ρx.


### 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 functionsf∈C0,∞, such that the right-hand side below is absolutely convergent, Szász-Mirakyan operators reproducing the functions 1 and e2ax, a>0, are introduced in [38] and defined by (63)Rn∗f;x≔e-nαnx∑k=0∞nαnxkk!fkn(x≥0,n∈N), in such a way that the conditions(64)Rn∗e2at;x=e2axare satisfied for all x and all n. To provide condition (64), equality(65)αnx=2axne2a/n-1must be held (for more details see [38]).To investigate the approximation properties of the operatorsRn∗, some preliminaries are needed. First, if a≥0, we get(66)Rn∗eat;x=enαnxea/n-1=e2ax/ea/n+1,Rn∗1;x=1,Rn∗e1;x=αnxRn∗e2;x=αn2x+αnxn. Then, letting ψxkt≔t-xk, k=0,1,2,…, we have(67)Rn∗ψx0t;x=1,Rn∗ψx1t;x=αnx-x,Rn∗ψx2t;x=αnx-x2+αnxn.Moreover, considering equality (65), one can find(68)limn→∞n2axne2a/n-1-x=-ax,limn→∞n2axne2a/n-1-x2+2axn2e2a/n-1=x.In 1970, Boyanov and Veselinov [72] showed that uniform convergence of any sequence of positive linear operators acting on C∗[0,∞) can be checked as follows.Theorem 5.
The sequenceAn:C∗[0,∞)→C∗[0,∞) of positive linear operators satisfies the conditions(69)limn→∞Ane-kt;x=e-kt,k=0,1,2,uniformly in [0,∞), if and only if (70)limn→∞Anf;x=fxuniformly in [0,∞), for all f∈C∗[0,∞).A quantitative form for Theorem5 can be given using the modulus of continuity on C∗[0,∞) introduced in [73, Corollary 3.2] and defined as (71)ω∗f,δ=supx,t≥0e-x-e-t≤δ⁡fx-ft(δ>0,f∈C∗[0,∞)).Theorem 6.
Forf∈C∗[0,∞), we have(72)Rn∗f-f∞≤2ω∗f;2βn+γn,where (73)βn=Rn∗e-t;x-e-x∞,γn=Rn∗e-2t;x-e-2x∞.
Moreover,βn and γn tend to zero as n goes to infinity so that Rn∗f converges uniformly to f.To investigate pointwise convergence of the operatorsRn∗ a quantitative Voronovskaya theorem is presented in [38] as well. Such a result allows establishing the rate of pointwise convergence and an upper bound for the error of approximation.Theorem 7.
Letf,f′′∈C∗[0,∞). Then the inequality(74)nRn∗f;x-fx+axf′x-x2f′′x≤pnxf′x+qnxf′′x+22qnx+x+rnxω∗f′′;1nholds for any x∈[0,∞), where(75)pnx≔nRn∗ψxt;x+ax,qnx≔12nRn∗ψx2t;x-x,rnx≔n2Rn∗e-x-e-t4;xRn∗ψx4t;x.As a uniform approximation result let us recall, as explained in [38], that the spaces (C∗[0,∞),·[0,∞)) and (C0,1,·0,1) are isometrically isomorphic. Define ψy≔e-y,y∈[0,∞), and let T:C0,1→C∗[0,∞) be given by(76)Tfy=f∗y=fψy,f∈C0,1,y∈0,∞.We remark that (77)limt→∞f∗t=limt→∞fψt=f0.Clearly, T is linear and bijective. Moreover, for all f∈C0,1 one has(78)Tf0,∞=supt∈0,∞fψt=f0,1.Hence T is an isometric isomorphism and (79)T-1f∗=f∗∘ψ-1,forf∗∈C∗0,∞.Corollary 8.
For allf∗∈C∗[0,∞) (f=f∗∘ψ-1) and n large enough we have(80)Rn∗f∗-f∗0,∞≤ω1f;12γn+2βn0,1+2ω2f;12γn+2βn0,1.To see some of the advantages of new constructions of Szász-Mirakyan operators the following comparisons results were also presented in [38].First, note that the definition of generalized convexity considered in[0,1] (cf. (7)) can be given also in [0,∞) (see [59, 74]). More precisely, in this subsection we consider functions f∈C[0,∞) convex with respect to {1,v}, in short {1,v}-convex, where (81)vx=e2ax,a>0.Observe that this is equivalent tof∘v-1 being convex in the classical sense. Moreover, if function f∈C2[0,∞) (the space of twice continuously differentiable functions), then f is 1,v-convex if and only if(82)f′′x≥2af′x,x>0(see [26]).Theorem 9.
Letf∈C2[0,∞) be increasing and 1,v-convex. Then (83)fx≤Rn∗f;x≤Snf;xforx≥0.The above-mentioned modified sequence of Szász-Mirakyan operators reproduces the functions1 and e2ax, a>0. Another modification of Szász-Mirakyan operators reproducing the functions eax and e2ax, a>0, was introduced in [39] as(84)Rnf;x=e-nαnx∑k=0∞nβnxkk!fkn,n∈N,x∈0,∞,where(85)βnx=axnea/nea/n-1,αnx=ax2-ea/nnea/n-1.This choice provides that(86)Rneat;x=eax,Rne2at;x=e2ax. For the operators Rn, it can be shown that(1)
Rn1;x=eax(ea/n-1/ea/n),(2)
Rne1;x=(ax/nea/nea/n-1)eax(ea/n-1/ea/n),(3)
Rne2;x=ax/nea/nea/n-12+ax/n2ea/nea/n-1eax(ea/n-1/ea/n),and if one considers the central moment operatorμnsx=RnΨxs;x of order s (s=0,1,2…), the following formulae hold:(1)
μn0x=eaxea/n-1/ea/n,(2)
μn1x=ax/nea/nea/n-1-xeax(ea/n-1/ea/n),(3)
μn2x=ax/nea/nea/n-1-x2+ax/n2ea/nea/n-1eax(ea/n-1/ea/n).Now set(87)φx=1+e2axx≥0and consider the space Bφ[0,∞) (resp., Cφ[0,∞), Cφ∗[0,∞)) defined by (30) and (31).The first result on uniform convergence of sequence of the operatorsRn was given in [39] by the following.Theorem 10.
For each functionf∈Cφ∗[0,∞)(88)limn→∞Rnf-fφ=0.In order to approximate unbounded functions, the exponential weighed spaceCa[0,∞) (with a fixed a>0), consisting of f∈C[0,∞) satisfying the condition fx≤Meax, where M is a positive constant, is considered and this space is a normed space with the norm (89)fa=supx∈0,∞fxeax.Also letCak[0,∞) be subspace of all functions f∈Ca[0,∞) such that limx→∞(fx/eax)=k, where k is a positive constant. A weighted modulus of continuity is defined by (90)ω~f;δ=supt-x≤δ,x≥0ft-fxeat+eax,for f∈Cak[0,∞). We note that if f∈Cak[0,∞), then limδ→0ω~f;δ=0 and ω~f;mδ≤2mω~f;δ for any m∈N (for more details we refer the readers to [39, Section 5]).Theorem 11.
Forf∈Cak[0,∞)(91)Rnf-f5a/2≤anefa+Cω~f;1n,where C is positive constant.In [39, Section 6] a Voronovskaja-type result is also presented.Theorem 12.
Letf∈Cφ[0,∞). If f is twice differentiable in x∈[0,∞) and f′′ is continuous in x, and then (92)limn→∞nRnf,x-fx=a2xfx-32axf′x+x2f′′x.Finally, the following saturation results for the sequence(Rn)n≥1 hold (see [39, Section 7]).Theorem 13.
Letf∈Cφ[0,∞) and consider a bounded open interval J⊂[0,∞). Then, for each x∈J(93)nRnfx-fx=o1if  and  only  iff∈eax,e2ax.Theorem 14.
Letf∈Cφ[0,∞), M≥0 and let J⊂[0,∞) be a bounded open interval. Then, for each x∈J, one has that (94)nRnfx-fx≤M+o1if and only if (95)a2xft-32atf′t+t2f′′t≤M,for  almost  everyt∈J.


## 4.1. Generalized Szász-Mirakyan Operators

In this subsection, we examine the Szász-Mirakyan type operators studied in [34]. Let ρ:[0,∞)→R be a function satisfying the following properties:(a)
ρ is continuously differentiable on [0,∞);(b)
ρ(0)=0 and infx≥0ρ′(x)≥1.From now on, we set(35)φx=1+ρ2xx≥0,and we consider the weighted spaces Bφ[0,∞), Cφ[0,∞), Cφ∗[0,∞), and Uφ[0,∞).Ifρ(x)=x for each x≥0 the space Cφ[0,∞) (resp., Cφ∗[0,∞)) becomes the classical weighed space (36)E2=f∈C0,∞:supx≥0fx1+x2∈R (resp., (37)E2∗=f∈C0,∞:limx→+∞fx1+x2∈R.The following result, proven in [66], shows that {1,ρ,ρ2} is a Korovkin set in Cφ∗[0,∞).Theorem 4.
Consider a sequence(Ln)n≥1 of positive linear operators from Cφ[0,∞) into Bφ[0,∞). If(38)limn→∞Lnρν-ρνφ=0forν=0,1,2,and, then, for every f∈Cφ∗[0,∞), (39)limn→∞Lnf-fφ=0.After these preliminaries, setN1≔{n∈N∣n≥n0}, for a suitable n0∈N. Given an interval I⊂[0,∞), consider two sequences (αn)n≥1, (βn)n≥1 of functions on I such that, for any n∈N1,(i)
αn,βn:I→R are positive functions on I;(ii)
βn(x)-αn(x)≥0 for every x∈I.In [34], the authors introduced and studied the sequence of the generalized Szász-Myrakjan operators, defined as(40)S~nρf;x=e-αnx∑k=0∞βnxkk!f∘ρ-1knfor every f∈C(I),n∈N1 and x∈I.Some conditions have to be imposed in order that the sequence(S~nρ)n≥n0 is an approximation process in Cφ∗[0,∞), and, in particular, in order to verify (38).More precisely, for anyn≥n0, there exist un,vn:I→R such that, for every x∈I,(41)unx≤un0,vnx≤vn0,limn→∞un0=limn→∞vn0=0,(42)S~nρ1;x=1+unx,(43)S~nρρ;x=ρx+vnx.Evaluating the operatorsS~nρ on 1 and ρ, it is easy to connect the sequences (αn)n≥n0 and (βn)n≥n0 with (un)n≥n0 and (vn)n≥n0, taking (40), (42), and (43) into account. More precisely, for every x∈I and n≥n0, (44)αnx=nρx+vnx1+unx-log1+unx,βnx=nρx+vnx1+unx.Accordingly, for anyn≥n0, f∈C(I) and x∈I,(45)S~nρf;x=e-nρx+vnx/1+unx1+unx∑k=0∞1k!nρx+vnx1+unxkf∘ρ-1kn.The operatorsS~nρ map Cφ[0,∞) into Bφ[0,∞). Moreover, since easy calculations show that, for every x∈I and n≥n0, (46)S~nρρ2;x=ρx+vnx21+unx+ρx+vnxn, by applying Theorem 4 to an extension of the operators S~nρ(f) to [0,∞), (47)limn→∞supx∈IS~nρf;x-fxφx=0.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 f∈Cφ[0,∞) and δ>0, (48)ωρf;δ=supx,t≥0ρt-ρx≤δ⁡ft-fxφt+φx. In particular, by using the results in [67], it can be proven that, for every f∈Cφ[0,∞) and n≥n0,(49)S~nρf-fφ3/2≤7+4un0+22vn0+vn02+2n+2vn0nωρf;δn, where (50)δn=16n+4n2+3un0+20vn0+22vn0n+4vn0n2+8vn02+6vn02n+vn03+21+un02n+un0+4vn0+2vn0n+vn02.Moreover, sincelimδ→0ωρ(f;δ)=0 if f∈Uφ[0,∞), from the latter formula and (41), we get that (51)limn→∞S~nρf-fφ3/2=0 for every f∈Uφ[0,∞).Further, under suitable assumptions, it is possible to determine a Voronovskaya-type result involvingS~nρ’s. More precisely, assume that (52)limn→∞nunx=l1,limn→∞nvnx=l2.Moreover, consider a functionf∈Cφ[0,∞) for which the function f∘ρ-1 is twice differentiable. If the second derivative of f∘ρ-1 is bounded on [0,∞), then, for every x∈I,(53)limn→∞nS~nρf;x-fx=fxl1+l2-ρxl1f∘ρ-1′ρx+12ρxf∘ρ-1′′ρx.The following examples show that, for suitable choices of the sequences(un(x))n≥n0, (vn(x))n≥n0 and of the function ρ, 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) IfI=[0,∞), un(x)=vn(x)=0, and ρ(x)=x for every x≥0, the operators Sn~ρ turn into the classical Szász-Myrakjan operators (32), which, as it is well known, preserve the function e0 and ρ=e1.
(2) IfI=[0,∞), ρ(x)=x, un(x)=0, and vn(x)=-1/2n+4n2x2+1/2n-x, then operators Sn~ρ turn into (54)Dn∗f;x=e1-4n2x2+1/2∑k=0∞4n2x2+1-1k2kk!fkn(f∈E2,n≥1,x≥0), which were object of investigation in [7] and, in the spirit of King’s work, preserve the function e0 and ρ2=e2.
In particular, when applied toDn∗, (53) gives the following result. If f∈E2 is a function which is twice differentiable and whose second derivative is bounded on [0,∞), then, for every x≥0,(55)limn→∞nDn∗f;x-fx=-12f′x+x2f′′x.
Formula (55) holds true uniformly w.r.t. x≥0, if f′,f′′∈E2∗. 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 f∈Cb[0,∞), (56)Dn∗f;x-fx≤2ω1f,Dn∗ψx2x.
We point out that, as shown in [7], for every x≥0(57)Dn∗ψx2t;x=2x2+xn-x4n2x2+1n.
Easy calculations prove thatDn∗(ψx2(t);x)≤Sn(ψx2(t);x) for every x≥0, so that, at least for f∈Cb[0,∞), the operators Dn∗ provide a better approximation error than the classical Szász-Myrakjan operators Sn (see (34)).
(3) IfI=1/(n0-1),∞, un(x)=1/(nx-1), vn(x)=0, and ρ(x)=x, then Sn~ρ’s are exactly the operators studied in [71], given by (58)Sn∗f;x=nxe1-nxnx-1∑k=0∞nx-1kk!fkn(f∈E2,n≥n0,x∈I). Those operators fix the functions ρ=e1 and ρ2=e2. In this case, the Voronovskaya-type formula becomes (59)limn→∞nSn∗f;x-fx=fxx-f′x+x2f′′x, for all x∈I and all f∈E2 which are twice differentiable and whose second derivative is bounded.
(4) ForI=[0,∞), un(x)=vn(x)=0 for every x≥0 and considering an arbitrary function ρ satisfying (a) and (b), the operators Sn~ρ reduce to (60)Snρf;x=e-nρx∑k=0∞nρxkk!f∘ρ-1kn(f∈Cφ[0,∞),x∈I,n≥1) that were introduced and studied in [32] and preserve the functions e0 and ρ.
In particular, for everyf∈Cφ[0,∞), (61)Snρf-fφ3/2≤7+4nωρf;20n+8n.
Moreover, iff∈Cφ[0,∞) is a function such that f∘ρ-1 is twice differentiable and the second derivative of f∘ρ-1 is bounded on [0,∞), then, for every x≥0, (62)limn→∞nSnρf;x-fx=12ρxf∘ρ-1′′ρx.


## 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 functionsf∈C0,∞, such that the right-hand side below is absolutely convergent, Szász-Mirakyan operators reproducing the functions 1 and e2ax, a>0, are introduced in [38] and defined by (63)Rn∗f;x≔e-nαnx∑k=0∞nαnxkk!fkn(x≥0,n∈N), in such a way that the conditions(64)Rn∗e2at;x=e2axare satisfied for all x and all n. To provide condition (64), equality(65)αnx=2axne2a/n-1must be held (for more details see [38]).To investigate the approximation properties of the operatorsRn∗, some preliminaries are needed. First, if a≥0, we get(66)Rn∗eat;x=enαnxea/n-1=e2ax/ea/n+1,Rn∗1;x=1,Rn∗e1;x=αnxRn∗e2;x=αn2x+αnxn. Then, letting ψxkt≔t-xk, k=0,1,2,…, we have(67)Rn∗ψx0t;x=1,Rn∗ψx1t;x=αnx-x,Rn∗ψx2t;x=αnx-x2+αnxn.Moreover, considering equality (65), one can find(68)limn→∞n2axne2a/n-1-x=-ax,limn→∞n2axne2a/n-1-x2+2axn2e2a/n-1=x.In 1970, Boyanov and Veselinov [72] showed that uniform convergence of any sequence of positive linear operators acting on C∗[0,∞) can be checked as follows.Theorem 5.
The sequenceAn:C∗[0,∞)→C∗[0,∞) of positive linear operators satisfies the conditions(69)limn→∞Ane-kt;x=e-kt,k=0,1,2,uniformly in [0,∞), if and only if (70)limn→∞Anf;x=fxuniformly in [0,∞), for all f∈C∗[0,∞).A quantitative form for Theorem5 can be given using the modulus of continuity on C∗[0,∞) introduced in [73, Corollary 3.2] and defined as (71)ω∗f,δ=supx,t≥0e-x-e-t≤δ⁡fx-ft(δ>0,f∈C∗[0,∞)).Theorem 6.
Forf∈C∗[0,∞), we have(72)Rn∗f-f∞≤2ω∗f;2βn+γn,where (73)βn=Rn∗e-t;x-e-x∞,γn=Rn∗e-2t;x-e-2x∞.
Moreover,βn and γn tend to zero as n goes to infinity so that Rn∗f converges uniformly to f.To investigate pointwise convergence of the operatorsRn∗ a quantitative Voronovskaya theorem is presented in [38] as well. Such a result allows establishing the rate of pointwise convergence and an upper bound for the error of approximation.Theorem 7.
Letf,f′′∈C∗[0,∞). Then the inequality(74)nRn∗f;x-fx+axf′x-x2f′′x≤pnxf′x+qnxf′′x+22qnx+x+rnxω∗f′′;1nholds for any x∈[0,∞), where(75)pnx≔nRn∗ψxt;x+ax,qnx≔12nRn∗ψx2t;x-x,rnx≔n2Rn∗e-x-e-t4;xRn∗ψx4t;x.As a uniform approximation result let us recall, as explained in [38], that the spaces (C∗[0,∞),·[0,∞)) and (C0,1,·0,1) are isometrically isomorphic. Define ψy≔e-y,y∈[0,∞), and let T:C0,1→C∗[0,∞) be given by(76)Tfy=f∗y=fψy,f∈C0,1,y∈0,∞.We remark that (77)limt→∞f∗t=limt→∞fψt=f0.Clearly, T is linear and bijective. Moreover, for all f∈C0,1 one has(78)Tf0,∞=supt∈0,∞fψt=f0,1.Hence T is an isometric isomorphism and (79)T-1f∗=f∗∘ψ-1,forf∗∈C∗0,∞.Corollary 8.
For allf∗∈C∗[0,∞) (f=f∗∘ψ-1) and n large enough we have(80)Rn∗f∗-f∗0,∞≤ω1f;12γn+2βn0,1+2ω2f;12γn+2βn0,1.To see some of the advantages of new constructions of Szász-Mirakyan operators the following comparisons results were also presented in [38].First, note that the definition of generalized convexity considered in[0,1] (cf. (7)) can be given also in [0,∞) (see [59, 74]). More precisely, in this subsection we consider functions f∈C[0,∞) convex with respect to {1,v}, in short {1,v}-convex, where (81)vx=e2ax,a>0.Observe that this is equivalent tof∘v-1 being convex in the classical sense. Moreover, if function f∈C2[0,∞) (the space of twice continuously differentiable functions), then f is 1,v-convex if and only if(82)f′′x≥2af′x,x>0(see [26]).Theorem 9.
Letf∈C2[0,∞) be increasing and 1,v-convex. Then (83)fx≤Rn∗f;x≤Snf;xforx≥0.The above-mentioned modified sequence of Szász-Mirakyan operators reproduces the functions1 and e2ax, a>0. Another modification of Szász-Mirakyan operators reproducing the functions eax and e2ax, a>0, was introduced in [39] as(84)Rnf;x=e-nαnx∑k=0∞nβnxkk!fkn,n∈N,x∈0,∞,where(85)βnx=axnea/nea/n-1,αnx=ax2-ea/nnea/n-1.This choice provides that(86)Rneat;x=eax,Rne2at;x=e2ax. For the operators Rn, it can be shown that(1)
Rn1;x=eax(ea/n-1/ea/n),(2)
Rne1;x=(ax/nea/nea/n-1)eax(ea/n-1/ea/n),(3)
Rne2;x=ax/nea/nea/n-12+ax/n2ea/nea/n-1eax(ea/n-1/ea/n),and if one considers the central moment operatorμnsx=RnΨxs;x of order s (s=0,1,2…), the following formulae hold:(1)
μn0x=eaxea/n-1/ea/n,(2)
μn1x=ax/nea/nea/n-1-xeax(ea/n-1/ea/n),(3)
μn2x=ax/nea/nea/n-1-x2+ax/n2ea/nea/n-1eax(ea/n-1/ea/n).Now set(87)φx=1+e2axx≥0and consider the space Bφ[0,∞) (resp., Cφ[0,∞), Cφ∗[0,∞)) defined by (30) and (31).The first result on uniform convergence of sequence of the operatorsRn was given in [39] by the following.Theorem 10.
For each functionf∈Cφ∗[0,∞)(88)limn→∞Rnf-fφ=0.In order to approximate unbounded functions, the exponential weighed spaceCa[0,∞) (with a fixed a>0), consisting of f∈C[0,∞) satisfying the condition fx≤Meax, where M is a positive constant, is considered and this space is a normed space with the norm (89)fa=supx∈0,∞fxeax.Also letCak[0,∞) be subspace of all functions f∈Ca[0,∞) such that limx→∞(fx/eax)=k, where k is a positive constant. A weighted modulus of continuity is defined by (90)ω~f;δ=supt-x≤δ,x≥0ft-fxeat+eax,for f∈Cak[0,∞). We note that if f∈Cak[0,∞), then limδ→0ω~f;δ=0 and ω~f;mδ≤2mω~f;δ for any m∈N (for more details we refer the readers to [39, Section 5]).Theorem 11.
Forf∈Cak[0,∞)(91)Rnf-f5a/2≤anefa+Cω~f;1n,where C is positive constant.In [39, Section 6] a Voronovskaja-type result is also presented.Theorem 12.
Letf∈Cφ[0,∞). If f is twice differentiable in x∈[0,∞) and f′′ is continuous in x, and then (92)limn→∞nRnf,x-fx=a2xfx-32axf′x+x2f′′x.Finally, the following saturation results for the sequence(Rn)n≥1 hold (see [39, Section 7]).Theorem 13.
Letf∈Cφ[0,∞) and consider a bounded open interval J⊂[0,∞). Then, for each x∈J(93)nRnfx-fx=o1if  and  only  iff∈eax,e2ax.Theorem 14.
Letf∈Cφ[0,∞), M≥0 and let J⊂[0,∞) be a bounded open interval. Then, for each x∈J, one has that (94)nRnfx-fx≤M+o1if and only if (95)a2xft-32atf′t+t2f′′t≤M,for  almost  everyt∈J.



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