Approximation with arbitrary order by certain linear positive operators

Abstract


This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence \((\lambda _{n})_{n\geq1}\) such that \(lim_{n\rightarrow\infty}\) \(\lambda_{n}=0\), how fast we want. Particular cases are presented.

Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Linear and positive operator · Korovkin theorem · Voronovskaya type theorems · Modulus of continuity · Order of approximation

Paper coordinates

O. Agratini, Approximation with arbitrary order by certain linear positive operators, Positivity, 22 (2018), pp. 1241-1254, https://doi.org/10.1007/s11117-018-0570-9

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Positivity

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Springer

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1385-1292
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1572-9281

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Approximation with arbitrary order by certain linear positive operators

Approximation with arbitrary order
by certain linear positive operators

Octavian Agratini and Sorin G. Gal Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Str. Kogălniceanu 1
400084 Cluj-Napoca, Romania
agratini@math.ubbcluj.ro University of Oradea
Department of Mathematics and Computer Science
Str. Universităţii 1
410087 Oradea, Romania
galso@uoradea.ro
Abstract.

This paper aims to highlight classes of linear positive operators of discrete and integral type for which the rates in approximation of continuous functions and in quantitative estimates in Voronovskaya type results are of an arbitrarily small order. The operators act on functions defined on unbounded intervals and we achieve the intended purpose by using a strictly decreasing positive sequence (λn)n1 such that limnλn=0, how fast we want. Particular cases are presented.

Keywords and phrases: Linear and positive operator, Korovkin theorem, Voronovskaya type theorems, modulus of continuity, order of approximation.

Mathematics Subject Classification: 41A36, 41A25.

1. Introduction

A special branch of Approximation Theory is the approximation of functions by using linear sequences of operators, say (Ln)n1, the essential feature being that of positivity. The cornerstone of this approach is Bohman-Korovkin criterion [5], [15]. This criterion says: if (Lnek)n1 converges to ek uniformly on [a,b], k{0,1,2}, for the test functions e0(x)=1, e1(x)=x, e2(x)=x2, then (Lnf)n1 converges to f uniformly on [a,b] for each fC([a,b]). Usually, for continuous functions defined on an unbounded interval only pointwise convergence occurs. It is one of the reasons why the study of such operators is more productive.

We recall that two types of positive approximation processes are used – the discrete, respectively continuous form. In what follows we consider as a starting point a linear positive process of discrete type which approximate functions defined on unbounded intervals. Since a linear substitution maps the interior of any unbounded interval J onto (0,) or , we have only two options. In our study we consider the benchmark interval J=[0,)=+ as it exhibits the problems caused by a finite endpoint and by the nonboundedness of the interval. Operators in questions are often designed as follows

(Lnf)(x)=k=0ak(n;x)f(xn,k),n,x+, (1.1)

where ak(n;):++ are continuous functions for each n, k{0}=0, and Δn=(xn,k)k0 is a net on +. Here f belongs to a function space for which the right-hand side of the relation (1.1) is well defined.

The most common case for classical operators is the equidistant network xn,k=k/n, k0. We mention that in (1.1) could be a finite sum. For example, if we take

ak(n;x)=(nk)(1+x)nxk for 0kn,ak(n;x)=0 for k>n,

and we choose the net Δn=(k/(nk+1))k=0,n¯, then Ln, n, become Bleiman-Butzer-Hahn operators [4].

The disadvantage of the positive linear approximation processes is definitely determined by the fact that they have a low rate of convergence. For some spaces of functions considered, the error of approximation |(Lnf)(x)f(x)| is assessed by using the modulus of continuity ω of f:I with argument δ defined by

ω(f;δ)=sup{|f(x)f(y)|;x,yI,|xy|δ},δ0. (1.2)

We are considering functions for which the supremum is finite, for example bounded functions, uniformly continuous functions, Lipschitz functions.

For a wide majority of linear positive operators, the error of approximation is described by using ω(f;1/n) [8, page 40], consequently the order of approximation is 𝒪(n1/2). Is it possible to construct classes of operators having a faster rate of convergence? Recently, such classes of operators acting on certain function spaces have been obtained.

Starting from (1.1) we consider a general class of discrete operators with the property that the order of approximation is improved, meaning that becomes arbitrarily small. Also, we prove that the same phenomenon holds for quantitative estimates in Voronovskaya type theorems. The next section consists in associating to this new process an integral generalization of Kantorovich type, obtaining for it the same kind of results. Particular sequences are identified in our construction, such that the result established can be applied to several cases.

2. A discrete class of operators

Let λ=(λn)n1 be a strictly decreasing sequence of positive numbers with the property

limnλn=0. (2.1)

We modify the operators defined by (1.1) replacing n with λn1. Also we consider the equidistant network having the nodes xn,k=kλn. The discrete class of operators is given as follows

(Lnλf)(x)=k=0ak(λn;x)f(kλn),n,x0, (2.2)

where ak(λn;) are non-negative functions belonging to C(+) and

f𝒟:={gC(+):the series in(2.2)is absolutely convergent}. (2.3)

We are working under assumption that the monomials ej, 0j2, belong to the above domain.

Clearly, Lnλ, n, are linear and positive operators. At this point we introduce the first and the second central moments of Lnλ operators, i.e.,

i(Lnλ;x)=(Lnλφxi)(x), where φx(t)=tx,(t,x)+×+,

and i{1,2}. Concerning the operators defined by (2.2), we consider that for each x+ and n they satisfy the following properties

k=0ak(λn;x)=1, (2.4)
1(Lnλ;x)=λnl1(x), (2.5)
2(Lnλ;x)=λnl2(x)+λn2l3(x), (2.6)

where ljC(+), j{1,2,3}.

Taking into account Bohman-Korovkin theorem, (Lnλ)n1 is an approximation process on the space C(K), endowed with the uniform norm, K+ compact, if

limnLnλej=ej, 0j2,

uniformly on K.

Based on the requirements (2.4), (2.5), (2.6), the above limits are achieved. On the other hand, our hypothesis package means much more. The real problem that arises here is whether there are classes of (Lnλ)n1 operators which meet all assumptions. We certify this fact presenting two examples. We also indicate successive steps taken to obtain them.

Example 2.1.

Ismail [13] introduced a generalization of Szász operators given by

(Tnf)(x)=enxH(1)A(1)k=0pk(nx)f(kn), (2.7)

where pk are the Sheffer polynomials defined by

A(t)exH(t)=k=0pk(x)tk,x0,|t|<R,

with

A(z)=k=0ckzk,H(z)=k=1hkzk, (2.8)

analytic functions in a disk |z|<R, R>1, such that A(1)0, H(1)=1, ck for k0, hk for k, c0h10 and supposing that pk(x)0 for all (k,x)0×+.

By using two sequences of non-negative real numbers (αn)n, (βn)n with the property that (βn/αn)n is strictly decreasing having null limit, Cetin and Ispir [6] introduced and studied a notable generalization of Szász operators attached to analytic functions f of exponential growth in a compact disk of the complex plane, |z|<R,

Sn(f;αn,βn)(z)=eαnz/βnk=01k!(αnzβn)kf(kβnαn).

For the particular case z+, αn=n, βn=1, these operators turn into genuine Szász operators [20].

In [10] Gal considered the Ismail’s kind generalization (2.7), this time the operators being applied to certain real-valued functions defined on +. More precisely, the following operators are studied

Tn(f;αn,βn)(x)=eαnxH(1)/βnA(1)k=0pk(αnxβn)f(kβnαn), (2.9)

x+, under the hypothesis on A,H and pk specified at (2.8).

Setting λn:=βn/αn, condition (2.1) is satisfied and we observe that these operators are of the form (2.2) with

ak(λn;x)=exH(1)/λnA(1)pk(x/λn),x0.

For λn=1/n we reobtain Tn operators which have the form indicated in (1.1). Following [10, Lemma 2.1], the author proved

Tn(e0;αn,βn)=1,
1(Tn;x)=βnαnA(1)A(1),
2(Tn;x)=xβnαn(H′′(1)+1)+βn2αn2A(1)+A′′(1)A(1).

Comparing these identities to those describing by relations (2.4), (2.5), (2.6), we deduce

l1(x)=A(1)A(1),l2(x)=(H′′(1)+1)x,l3(x)=A(1)+A′′(1)A(1),x0.
Example 2.2.

Set

E2(+)={fC(+):f(x)/(1+x2) is convergent as x}.

Firstly we present Mastroianni operators [17]. Let φn, n, be real functions on + which are infinitely differentiable and strictly monotone on +, satisfying the following properties:

(P1) φn(0)=1,n;

(P2) φn, n are completely monotone on +, i.e. (1)kφn(k)(x)0 for each x+ and k0;

(P3) for every (n,k)×0 there exists a positive integer p(n,k) and a real function αn,k:+ such that

φn(i+k)(x)=(1)kφp(n,k)(i)(x)αn,k(x),i0,x+, and
limnnp(n,k)=limnαn,k(0)nk=1.

To the sequence (φn)n1 Mastroianni associated the linear and positive operators Mn, n1, acting from E2(+) to C(+) as follows

(Mnf)(x)=k=0(1)kk!φn(k)(x)f(kn),x+. (2.10)

We consider φn(x)=(1+x)n, x0. These functions are infinitely differentiable and strictly decreasing on the domain. Moreover, φn(0)=1 and φn(k)(x)=(1)kn(n+1)(n+k1)(1+x)nk, k1, consequently conditions (P1) and (P2) are fulfilled. We choose

p(n,k)=n+k and αn,k(x)=n(n+1)(n+k1)(1+x)k

which implies φn(i+k)(x)=(1)kφn+k(i)(x)αn,k(x). Therefore, property (P3) also occurs. In this particular case, the operators defined by (2.10) turn into the celebrated Baskakov operators [3]

(Vnf)(x) =(1+x)nf(0)
+(1+x)nk=1n(n+1)(n+k1)k!(x1+x)kf(kn).

Recently, Gal and Opriş [11] introduced a modified Baskakov type operator. In the definition of φn(x)=(1+x)n, n, the authors replaced 1/n by λn, consequently λ=(λn)n1 satisfies relation (2.1). The new operators are expressed for x0 as follows

(Vnλf)(x)=(1+x)1/λnf(0)
+(1+x)1/λnk=11k!1λn(1+1λn)(k1+1λn)(x1+x)kf(kλn) (2.11)

Regarding these operators, the authors proved [11, Corollary 2] the following relations

(Vnλe0)(x)=1,1(Vnλ;x)=0,2(Vnλ;x)=λnx(1+x),x0.

Comparing these relations with the requirements (2.4), (2.5), (2.6), we conclude l1(x)=l3(x)=0 and l2(x)=x(1+x), x0.

The space of real valued uniformly continuous functions defined on + is denoted by UC(+).

Theorem 2.3.

Let the operators Lnλ, n, be defined by (2.2) such that the relations (2.4) and (2.6) hold. For fUC(+), x0, we have

|(Lnλf)(x)f(x)|(1+l2(x)+λnl3(x))ω(f;λn). (2.12)

Proof. First of all we mention that ω(f;) is well defined for any fUC(+), see, e.g., [8, Chapter 2, §6]. The quantitative result (2.12) given in terms of the modulus of continuity is a direct consequence of the following statement proved by Shisha and Mond [19]: if Λ is a linear positive operator defined on C(K), K compact, then one has

|(Λf)(x)f(x)| |f(x)||(Λe0)(x)1|
+((Λe0)(x)+1δ(Λe0)(x)(Λφx2)(x))ω(f;δ) (2.13)

for every xK and δ>0.

Even if this result was proved for continuous functions defined on compact intervals, the reasonings are the same for functions belonging to UC(+). In (2) we choose Λ=Lnλ. Taking into account that Lnλ operators reproduce the constants and relation (2.6) holds, we can write

|(Lnλf)(x)f(x)|(1+1δ2(Lnλ;x))ω(f;δ),x0.

Taking δ=λn, we get inequality (2.12) and the proof is ended.

Returning to the two examples, relation (2.12) becomes as follows:

(i) For Szász-type operators (2.9),

|Tn(f;λn)(x)f(x)|
(1+(H′′(1)+1)x+λnA(1)+A′′(1)A(1))ω(f;λn),

where λn=βn/αn.

The result coincides with that obtained in [10, Theorem 2.2].

(ii) For Baskakov type operators (2),

|Vnλf)(x)f(x)|(1+x(x+1))ω(f;λn).

The result is similar with that obtained in [11, Corollary 2.1(ii)].

In what follows, we will show that the operators of the form Lnλ, n, defined by (2.2), may give an arbitrary small order of approximation for quantitative estimates in Voronovskaya type theorems too. Let us recall that Voronovskaya type results for linear positive operators applied to functions defined on [0,+) were obtained, for example, in [12] for operators reproducing e0 and e1 and in [2] for operators reproducing only e0. For the simplicity of presentation, we will use below the results in [12] applied to the above particular cases.

For this purpose, firstly we need the following notations: for m with m2 and k{0}, the Pǎltǎnea [18] modulus of continuity is given by

ωφm(f;h)=sup{|f(x)f(y)|;x,y0,|xy|hφm(x+y2)},h0,

where φm(x)=x1+xm, for all x[0,+),

Ck[0,+)={f:[0,+);f(k) continuous on [0,+)},
Ck[0,+)
={fC[0,+);M>0, such that |f(x)|M(1+xk),x0},
Wm[0,+)
={f:[0,+);fe2,fevUC(+), for v=12m+1}.
Theorem 2.4.

Let the Szász-type operators Tn(f;λn)(x) be given by (2.9) and for the classical case, n, m2 and λn0 as fast we want.

If k=max{m+3,6,2m}, then Tn(f;λn)(x) is well defined on Ck[0,+) and for all fC2[0,+)Ck[0,+), with f′′Wm[0,+), all x(0,+) and n, we have

|Tn(f;λ)(x)f(x)λnxf′′(x)2|
(λnx+2An,m(x))ωφm(f;λn3/2λn2+25xλn+15x2). (2.14)

Here An,m(x)=Tn([1+(x+|tx|2)m]2;λ)(x), is bounded with respect to n for any fixed x(0,+) and m2, and can be explicitly calculated by a similar formula with that in [12], formula (3.2).

Proof. It is a consequence of Theorem 2.2 in [12], reasoning as in the proof of Theorem 3.1 there, but for the operator Tn(f;λn)(x) instead of the classical Szász-type operator.

Theorem 2.5.

Let the Baskakov-type operators Vnλf)(x) be given by (2), n, m2 and λn0 as fast we want.

If k=max{m+3,4}, then Vnλf)(x) is well defined on Ck[0,+) and for all fC2[0,+)Ck[0,+), with f′′Wm[0,+), all x(0,+) and n, we have

|Vnλf)(x)f(x)λnx(1+x)f′′(x)2|
λn2[x(1+x)+2x(1+x)Cn,2,m(x)]
ωφm(f;λn(1+6x+6x2)λn+3x(1+x)). (2.15)

Here Cn,2,m(x) is bounded with respect to n for any fixed x(0,+) and m2, and can be explicitly calculated by a similar formula with that in the statement of Theorem 2.3 in [12].

Proof. It is a consequence of Theorem 2.3 in [12], reasoning as in the proof of Theorem 3.5 there, but for the operator Vnλf)(x) instead of the classical Baskakov-type operator.

The above two theorems are new.

3. A Kantorovich type extension

Starting from the operators Lnλ, n, designed as in (2.2) which verify the assumptions (2.4), (2.5), (2.6), we associate them the following integral generalization of Kantorovich type

(Knλf)(x)=1λnk=0ak(λn;x)kλn(k+1)λnf(t)𝑑t,x0. (3.1)

The construction involves the mean values of the approximating function f on the subintervals [kλn,(k+1)λn], k0.

In the above f must be locally integrable function on + such that the antiderivative of f belongs to the domain 𝒟 defined at (2.3). Our aim is to show that this linear and positive integral process can approximate certain classes of functions with an arbitrary good order of approximation on each compact interval included in +.

Lemma 3.1.

Let the operators Knλ, n, be defined by (3.1). The following identities

(Knλe0)(x)=1,(Knλe1)(x)=(Lnλe1)(x)+12λn,
(Lnλe2)(x)=(Lnλe2)(x)+λn(Lnλe1)(x)+13λn2

hold for each x0.

Proof. The first identity is a consequence of relation (2.4). Further on, we can write

(Knλe1)(x) =1λnk=0ak(λn;x)(k+12)λn2=(Lnλe1)(x)+12λn

and

(Knλe2)(x) =1λnk=0ak(λn;x)(k2+k+13)λn3
=(Lnλe2)(x)+λn(Lnλe1)(x)+13λn2.

By using Lemma 3.1 we can express the first and the second central moments of Knλ operators with the help of corresponding moments of Lnλ. For each x0 we get

1(Knλ;x)=1(Lnλ;x)+12λn, (3.2)
2(Knλ;x)=2(Lnλ;x)+λn1(Lnλ;x)+13λn2. (3.3)

A question arises: are well defined the operators Lnλ and Knλ for functions of UC(+)? Since f belongs to this class, its growth is linear, i.e., the non-negative real constants c1 and c2 exist such that |f(x)|c1x+c2, x0. Consequently, the answer is positive.

Theorem 3.2.

Let the operators Knλ, n, be defined by (3.1) such that the relations (2.4), (2.5) and (2.6) hold. For every fUC(+)

|(Knλf)(x)f(x)|(1+l2(x)+(l1(x)+l3(x)+13)λn)ω(f;λn) (3.4)

takes place, where li, 1i3, are given at relations (2.4), (2.5), (2.6).

Proof. We apply the reasoning used at the proof of Theorem 2.3. Taking into account that Knλe0=e0 and relations (3.2), (3.3), inequality (2) becomes as follows

|(Knλf)(x)f(x)|(1+1δ2(Lnλ;x)+λn1(Lnλ;x)+13λn2)ω(f;δ).

Further, we use the identities (2.4), (2.5), (2.6) and choose δ:=λn. After a few calculations we arrive at the desired relation.

It is known that the function ω defined by (1.2) is continuous at δ=0, i.e.

limδ0+ω(f;δ)=ω(f;0)=0,

if and only if f is uniformly continuous on its domain, see [8, page 40]. Examining relations (3.4) and (2.1), the following pointwise convergence

limn(Knλf)(x)=f(x),x+,

takes place. These operators can also uniformly approximate any function fUC(+) on each compact, say K, included in +. Relative to the functions liC(+), i{1,2,3}, introduced by relations (2.4)-(2.6), we denote

maxxKl2(x)=α,maxxK(l1(x)+l3(x)+13)=β.

Relation (3.4) implies

|(Knλf)(x)f(x)|(1+α+βλn)ω(f;λn),xK,

therefore limnKnλf(x)=f(x), uniformly on the compact K.

We recall f:+ is Lipschitz continuous if there exists a real constant M0 such that |f(x)f(y)|M|xy| for any (x,y)+×+. We denote this set by LipM1. One hand, any Lipschitz continuous map is uniformly continuous; on the other hand, fLipM1 if and only if ωf(δ)Mδ. Taking in view these statements, we enunciate

Corollary 3.3.

Let the operators Knλ, n, be defined by (3.1) such that the relations (2.4), (2.5) and (2.6) hold. For every fLipM1, M0 and x0, it takes place

|(Knλf)(x)f(x)|M(1+l2(x)+(l1(x)+l3(x)+13)λn)λn.
Example 3.4.

Let us consider the Baskakov-Kantorovich operators, defined by the formula

BKnλ(f)(x)
=(1+x)1λnj=01j!1λn(1+1λn)(j1+1λn)xj(1+x)j1λnjλn(j+1)λnf(v)𝑑v.

Since BKnλ((tx)2)(x)=λn(x2+x+13λn), by Theorem 3.2 and Corollary 3.3 we get the estimates

|BKnλ(f)(x)f(x)|2ω1(f;λnx2+x+λn/3)

and

|BKnλ(f;λn)(x)f(x)|2Mλnx2+x+λn/3,

respectively, which are similar to Theorem 2.2 and Corollary 2.3 in Trifa [21].

Remark 3.5.

Analyzing the proofs of Theorems 2.3 and 3.2, we deduce that the upper bound error of approximation could be established if we relax requirements (2.5) and (2.6). More precisely, we can replace them with the followings

01(Lnλ;x)λnl1(x), (3.5)
2(Lnλ;x)λnl2(x)+λn2l3(x), (3.6)

where ljC(+), j{1,2,3}.

In what follows, we present a class of operators which has not been yet approached from the perspective of our work. Also we mention that this class of operators restarted to be studied in various recent papers, see, i.e., [1], [7], [9], published in 2016.

Example 3.6.

The starting point is the following Poisson-type distribution

ωβ(k,α)=αk!(α+kβ)k1e(α+kβ),k0,

where α>0 and |β|<1. Jain [14] introduced the sequence of linear operators by formula

(Pn[β]f)(x)=k=0ωβ(k,nx)f(kn),fC(+),

where 0β<1. It was been shown [14, Lemma 1]

k=0ωβ(k,α)=1. (3.7)

In order to become (Pn[β])n1 an approximation process, β should depend on n and its limit to be zero. We modify these operators following the model described by (2.2). Consequently, we choose β=λn, n1, a strictly decreasing sequence of positive numbers with the property (2.1). By following the notation used in this paper, we consider the modified operators

(Pnλf)(x)=k=0ωλn(k,xλn)f(kλn).

Based on (3.7) and [14, Eqs. (2.13), (2.14)], we get

(Pnλe0)(x)=1,
1(Pnλ;x)=λn1λnx,
2(Pnλ;x)=λn2(1λn)2x2+λn(1λ0)3x.

Since (λn)n1 is a positive strictly decreasing sequence, we can write

0 1(Pnλ;x)x1λ1λn,
2(Pnλ;x)x(1λ1)3λn+x2(1λ1)2λn2.

Considering formula (3.1), Kantorovich type extension will be read as follows

(K~nλf)(x)=1λnk=0ωλn(k,xλn)kλn(k+1)λnf(t)𝑑t,x0.

Set τ=(1λ1)1. For every fUC(+), in the light of relations (3.5) and (3.6), Theorem 2.3 implies

|(P~nλf)(x)f(x)|(1+τx(τ+λnx))ω(f;λn)

and Theorem 3.2 implies

|(K~nλf)(x)f(x)|(1+τ3x+(τ2x2+τx+13)λn)ω(f;λn).
Remark 3.7.

For the Kantorovich type operators in this section, we can use the results in [2] in order to obtain for them quantitative estimates of arbitrarily small rate.

Remark 3.8.

Usually, the linear positive operators provide an approximation error of order 𝒪(1/n). This rate of convergence can be modified by the mean of formulas (2.2), (3.1) such that its magnitude to be of order 𝒪(λn), where (λn)n1 can be chosen to converge arbitrarily fast to zero. Also, the results obtained have a strong unifying character, in the sense that for various choices of the nodes λn, one can recapture previous approximation results obtained for these operators by other authors, see, e.g., [6], [22], [23]. Finally, we mention that our results here target only uniformly continuous functions defined on unbounded intervals.

Remark 3.9.

The method in this paper could be applied in approximation by radial basis functions too, as follows. In the paper [16], it was formally introduced the sequences of the multivariate operators of the form

Ln(f)(x)=1ns(1r)jIn(Ωh)f(j/n)Φ(nxjnr1),

where xs, j is a multiindex and

In(Ωh)={js;[j/n,(j+1)/n]sΩh}.

Under the hypothesis in Theorem 2.2 in [16], the following quantitative estimate was obtained

Ln(f)fLp(Ω)cnγfWp,1(Ωh),

where Ω and Ωh are bounded domains.

Now, choosing λn0 as fast we want and defining

Ln(f;λn)(x)=λns(1r)jIλn(Ωh)f(jλn)Φ(x/λnj/λnr1),

reasoning exactly as in the proof of Theorem 2.2 in [16] and under the same hypothesis, one gets the much better estimate

Ln(f;λn)fLp(Ω)cλnγfWp,1(Ωh).

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1. Abel, U., Agratini, O.: Asymptotic behaviour of Jain operators. Numer. Algorithm 71(3), 553–565 (2016)
2. Aral, A., Gonska, H., Heilmann, M., Tachev, G.: Quantitative Voronovskaya-type results for polynomially bounded functions. Results Math. 70, 313–324 (2016)
3. Baskakov, V.A.: An example of a sequence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk SSSR 113, 249–251 (1957). (in Russian)
4. Bleimann, G., Butzer, P.L., Hahn, L.A.: Bernstein-type operators approximating continuous functions on the semiaxis. Indag. Math. 42, 255–262 (1980)
5. Bohman, H.: On approximation of continuous and of analytic functions. Ark. Mat. 2(1952–1054), 43–56
6. Cetin, N., Ispir, N.: Approximation by complex modified Szász-Mirakjan operators. Studia Sci. Math. Hungar. 50(3), 355–372 (2013)
7. Deniz, E.: Quantitative estimates for Jain-Kantorovich operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat. 65(2), 121–132 (2016)
8. DeVore, R.A., Lorentz, G.G.: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303. Springer, Berlin (1993)
9. Dogru, O., MohapatraR. N., Örkü, M.:Approximation properties of generalized Jain operators. Filomat 30(1016), 2359-2366
10. Gal, S.G.: Approximation with an arbitrary order by generalized Szász-Mirakjan operators. Studia Univ. Babes-Bolyai Math. 59(1), 77–81 (2014)
11. Gal, S.G., Opris, B.D.: Approximation with an arbitrary order by modified Baskakov type operators. Appl. Math. Comput. 265, 329–332 (2015)
12. Gupta, V., Tachev, G.: General form of Voronovskaja’s theorem in terms of weighted modulus of continuity. Results Math. 69, 419–430 (2016)
13. Ismail, M.E.H.: On a generalization of Szász operators. Mathematica (Cluj) 39, 259–267 (1974)
14. Jain, G.C.: Approximation of functions by a new class of linear operators. J. Aust. Math. Soc. 13(3), 271–276 (1972)
15. Korovkin, P.P.: On convergence of linear positive operators in space of continuous function. Dokl. Akad. Nauk SSSR (N.S.) 90, 961–964 (1953). (in Russian)
16. Mastroianni, G.: Su un operatore lineare e positivo. Rend. Acc. Sc. Fis. Mat. Napoli Serie IV 46, 161–176 (1979)
17. Paltanea, R.: Estimates of approximation in terms of a weighted modulus of continuity. Bull. Transilvania Univ. Brasov 4(53), 67–74 (2011)
18. Shisha, O., Mond, B.: The degree of convergence of linear positive operators. Proc. Nat. Acad. Sci. USA 60, 1196–1200 (1968)
19. Szász, O.: Generalization of S.Bernstein polynomials to the infinite interval. J. Res. Nat. Bur. Standards 45, 239–245 (1950)
20. Trifa, S.: Approximation with an arbitrary order by generalized Kantorovich-type and Durrmeyer-type operators on [0+∞). Studia Univ. Babes-Bolyai Math. (2017) 62(2) (in print)
21. Walczak, Z.: On approximation by modified Szász-Mirakjan operators. Glas. Mat. 37(57), 303–319 (2002)
22. Walczak, Z.: On modified Szász-Mirakjan operators. Novi Sad J. Math. 33(1), 93–107 (2003)

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