Approximation of some classes of functions by Landau type operators

Abstract

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in \(L_p\) spaces and in weighted \(L_p\) spaces (1≤p<∞). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness.

Authors

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Ali Aral
Kirikkale University, Turkey

Keywords

Landau operator; weighted space; Korovkin theorem, modulus of smoothness

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O. Agratini, A. Aral, Approximation of some classes of functions by Landau type operators, Results in Mathematics, 76 (2021) art. no. 12, https://doi.org/10.1007/s00025-020-01319-9

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Approximation of some classes of functions by Landau type operators

Approximation of some classes of functions by Landau type operators

Octavian Agratini Babeş-Bolyai University,
Faculty of Mathematics and Computer Science,
Str. Kogălniceanu, 1, 400084 Cluj-Napoca, Romania
and
Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy,
Str. Fântânele, 57, 400320 Cluj-Napoca, Romania
agratini@math.ubbcluj.ro
 and  Ali Aral Kirikkale University, Department of Mathematics,
71450 Yahşihan, Kirrikale, Turkey
aliaral73@yahoo.com
Abstract.

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in Lp spaces and in weighted Lp spaces (1p<). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness.

Keywords and phrases: Landau operator, weighted space, Korovkin theorem, modulus of smoothness.

Mathematics Subject Classification: 41A36, 41A25.

1. Introduction

A strongly rooted part of Approximation Theory is the approximation of various signals by linear and positive operators. Over time, many mathematicians have begun to generalize and modify classic processes of this type, thus providing a deeper study of them. This note follows this path, investigating a generalization of Landau operators. In [11, Eq. (2)], Edmund Landau proved the following relation

limn01f(ξ)(1(ξx)2)n𝑑ξ201(1u2)n𝑑u=f(x),

the convergence being uniform over an interval [a,b], where 0<a<b<1. However, as is stated in the paper, the function f can be extended by continuity on a larger interval. Further we point out some results obtained about Landau’s operators without claiming to include all important achievements related to these operators.

In [10], Jackson obtained their order of approximation. Sikkema [17, p. 44] proved a qualitative Voronovskaja result. A discrete version of this class of operators was studied by Gao [8]. Pendina [14] investigated iterations of Landau’s operators. Generalizations of operators in different forms have been introduced. Approximation by Landau operators relative to the Hausdorff metric appeared in [18]. Mamedov [12] presented the following generalization

(Mnf)(x)=kn1/(2k)Γ(1/(2k))01f(t)(1(tx)2k)n𝑑t,x[0,1],

where n is fixed and Γ stands for Gamma function. The deeper investigation of Mn, n, operators was continued by the same author in [13]. Two other generalizations were constructed by Chen and Shih [4]. Gal and Iancu [7] replaced in the classical operators, the usual integral by the nonlinear Choquet integral.

Another generalization of Landau operators was proposed in [1]. These new operators reproduce the affine functions, a less common feature of integral type operators. The approximation properties were studied in two spaces, namely in C(), the space of real valued continuous functions on , and in the weighted space Bρm(), where

Bρm()={f:|f(x)|Mfρm(x),x},

with the polynomial weight ρm(x)=1+x2m, m fixed.

In this paper we pursue two aims.

The first purpose is to continue the investigation of the operators introduced in [1] by their study in Lp(), 1p<, normed spaces which play a central role in many questions in analysis. The special importance of them derive from the fact that they offer a partial but useful generalization of the fundamental L2() space of square integrable functions. The second purpose is to extend the operators for vector functions, this assuming new challenges for error evaluation.

The architecture of this article still includes three paragraphs, in the first we present the class of operators and in the other two we investigate the aspects depicted above.

2. Preliminaries

Trying to realize a self-contained exposure, we present the operators introduced in [1], specifing the meaning of the notations as well as some formulas.

Let α1 be fixed and λ=1/(2α)(0,1/2]. For each n we set

an=01(1y2α)n𝑑yandτp,n=B(n+1,pλ)B(n+1,λ),p, (2.1)

where B stands for Beta function. Using elementary calculations we get

an=λB(n+1,λ),
01yi(1y2α)n𝑑y=λB(n+1,(1+i)λ).

The sequences (an)n1 and (τp,n)n1 for p2, are strictly decreasing. Moreover, for p2,

limnτp,n=0. (2.2)

We consider the linear and positive operators defined as follows

(Lnf)(x) =1an01f(xy+τ2,n)(1y2α)n𝑑y (2.3)
=1anx1xf(t+τ2,n)(1(xt)2α)n𝑑t,x,

where f is Lebesgue measurable on the domain.

The behavior of the operators on the Korovkin type test functions ej, ej(x)=xj for j{0,1,2}, is described by the following relations

Lne0=e0,Lne1=e1,Lne2=e2+τ3,nτ2,n2. (2.4)

Further, we focus on the study of approximation properties in unweighted and weighted Lp spaces, 1p<.

3. Approximation in Lp spaces

As usual, we denote the norm of the Lp space, 1p<, by Lp.

Lemma 3.1.

Let Ln, n, be defined by (2.3). If fLp(), where 1p<, then we get

LnfLpfLp.

Proof. We can write successively

LnfLp =(|(Lnf)(x)|p𝑑x)1/p
=1an(|01f(xy+τ2,n)(1y2α)n𝑑y|p𝑑x)1/p
1an01(|f(xy+τ2,n|pdx)1/p(1y2α)ndy
fLp1an01(1y2α)n𝑑y
=fLp,

see (2.1).

The above property says that our operators are non-expansive on the spaces Lp().

A subtle measurement of the error of approximation is provided by modulus of smoothness. Let Th, h, denote the translation operator, Th(f;x)=f(x+h), and Δh:=ThI the difference operator, where I is the identity operator. The first modulus of smoothness of fLp() is defined by

ω(f;δ)p=sup0hδΔh(f;)Lp,δ0.

Among its properties, see, e.g., the monograph [5, Chapter 2, §7], we recall: it is a continuous and increasing function of δ with ω(f;0)p=0. It also takes place

ω(f;λδ)p(λ+1)ω(f;δ)p,λ>0. (3.1)

Actually, ω(;δ)p meets the conditions that ensure it is a seminorm on Lp(), p1.

Theorem 3.1.

Let Ln, n, be defined by (2.3). If fLp() for some 1p<, then the following relation

LnffLp2ω(f;τ3,nτ2,n2)p

holds, where τ2,n and τ3,n are defined at (2.1).

Proof. Since the operators reproduce the constants, see (2.4), we have

(Lnf)(x)f(x)=1an01(f(xy+τ2,n)f(x))(1y2α)n𝑑y. (3.2)

We deduce

LnffLp =1an(|01(f(xy+τ2,n)f(x))(1y2α)n𝑑y|p𝑑x)1/p
1an01(|f(xy+τ2,n)f(x)|p𝑑x)1/p(1y2α)n𝑑y
1an01ω(f;|τ2,ny|)p(1y2α)n𝑑y.

The first increase represents Minkowski’s integral inequality for two measure spaces, see [9, Problem 202]. The inversions of the order of integration are justified by Fubini’s Theorem.

Based on (3.1), for any δ>0 we can write

LnffLp 1anω(f;δ)p01(1+|τ2,ny|δ)(1y2α)n𝑑y
ω(f;δ)p(1+1δ(1an01(τ2,ny)2(1y2α)n𝑑y)1/2)
ω(f;δ)p(1+1δ(τ3,nτ2,n2)1/2).

We used both Cauchy-Schwarz inequality for integrals and the identity

1an01(τ2,ny)2(1y2α)n𝑑y=(Lne2)(0),

see (2.4).

Taking in view the relations (2.2), the choice of δ=(τ3,nτ2,n2)1/2 is appropriate and the conclusion of the theorem is completed.

Next, we explore the approximation properties of the operators in some weighted spaces Lp. For a given p[1,), we consider the weight

ν:(0,1],ν(x)=(1+x2m)p,

where m>1 is a fixed integer. We denote by Lp,ν() the linear space of p-absolutely integrable functions on with respect to the weight ν, i.e., for 1p<,

Lp,ν()={f::fν1pLp()}

endowed with the norm p,ν defined as follows

fp,ν=fν1pLp=(|f(t)|pν(t)𝑑t)1/p.
Lemma 3.2.

Each operator Ln defined by (2.3) maps Lp,ν() into Lp,ν().

Proof. Let n be fixed. It is enough to show that

Lnfp,νMfp,ν,fLp,ν, (3.3)

where M is a positive constant.

Using again Minkowski’s inequality, we can write

Lnfp,ν =(|(Lnf)(x)1+x2m|p𝑑x)1/p
=1an(1(1+x2m)p|01f(xy+τ2,n)(1y2α)n𝑑y|p𝑑x)1/p
1an01(|f(xy+τ2,n)1+x2m|p𝑑x)1/p(1y2α)n𝑑y.

Appealing to the inequality

1+(xy+τ2,n)2m22m1(1+x2m)(1+(τ2,ny)2m),

true for any m, we can continue to write

Lnfp,ν fp,ν22m1an01(1+(τ2,ny)2m)(1y2α)n𝑑y
fp,ν22m1an01(1+j=02m(2mj)τ2,nj)(1y2α)n𝑑y.

Considering the relations (2.1), we deduce τ2,n<1/2 and choosing in (3.3) the constant M=(22m+32m)/2, the proof of our lemma is finished. 

Theorem 3.2.

Let Ln, n, be defined by (2.3). For every fLp,ν()

limnLnffp,ν=0 (3.4)

holds.

Proof. The demonstration is based on a Korovkin type theorem for a general weighted space Lp,ω() established in [6, Theorem 1]. The proved theorem says: Let (Λn)n be a uniformly bounded sequence of positive linear operators from Lp,ω() into Lp,ω(), 1p<. We assume that

limnΛneieip,ω=0,i=0,1,2, (3.5)

where the weight ω is a positive continuous function on satisfying the condition

t2pω(t)𝑑t<. (3.6)

Then, for every fLp,ω(), we have

limnΛnffp,ω=0.

We consider a particular case of the weight ω, namely ω:=ν. Since p1 and m>1, this weight function satisfies condition (3.6). Also, Lemma 3.2 ensures that our operators LnΛn, n, are uniformly bounded. Using (2.4), the conditions of (3.5) are obviously fulfilled for i=0 and i=1. Knowing the expression of the Lne2 function, we get

Lne2e2p,ν=|τ3,nτ2,n2|e0p,ν.

Taking in view the relation (2.2), condition (3.5) also takes place for i=3. The proof of the identity (3.4) is ended.

Moduli of smoothness have become standard instrument to estimate errors of approximation. Yuksel and Ispir [20] proposed a weighted modulus of the following form

(f,δ)supx0hδ|f(x+h)f(x)|1+(x+h)2m,δ0,

for any function f satisfying a polynomial growth, more precisely

|f(x)|Mf(1+x2m),x,

Mf being a positive constant depending only on f. Starting from this construction, we define a weighted modulus of smoothness ωp,m as follows

ωp,m(f;δ)=sup0hδ(|f(x+h)f(x)1+(|x|+h)2m|p𝑑x)1/p,δ0, (3.7)

for any function fLp,ν(). We show that this modulus satisfies some classical properties of Lp-modulus. Clearly, ωp,m(f;0)=0 and it is a monotone increasing function with respect to δ.

Also, for any δ0 and any positive integer m,

ωp,m(f;δ)fp,ν(1+22m1(1+δ2m)),

takes place. The proof is simple, based on Minkowski’s inequality and on the following increase

1+x2m1+(|x|+h)2m22m1(1+δ2m) for x and 0hδ. (3.8)
Lemma 3.3.

Let ωp,m(f;) be defined by (3.7). For any non-negative real numbers δ and c the following relation

ωp,m(f;cδ)(c+1)ωp,m(f;δ),fLp,ν(), (3.9)

holds.

Proof. For δ=0 or c=0 the statement is evident. Further, we consider δ>0 and c>0.

At first step, for any positive integer n we can write

ωp,m(f;nδ) =sup0hnδ(|f(x+h)f(x)1+(|x|+h)2m|p𝑑x)1/p
=sup0hδk=1nf(+kh)f(+(k1)h)1+(|e1|+nh)2mp
ωp,m(f;δ)k=1n1+(|x|+h+(k1)h)2m1+(|x|+nh)2m
nωp,m(f;δ). (3.10)

In the above we replaced h with nh. Further, for an arbitrary c>0, using both the monotonicity of ωp,m(f;) and inequality established at (3), we get

ωp,m(f;cδ)ωp,m(f;([c]+1)δ)([c]+1)ωp,m(f;δ)(c+1)ωp,m(f;δ),

where [c] indicates the integer part of c. We arrived at (3.9).

Lemma 3.4.

Let ωp,m(f;) be defined by (3.7). It takes place

limδ0+ωp,m(f;δ)=0,fLp,μ(). (3.11)

Proof. For a positive real number a, let χ1a, χ2a, χa be the characteristic functions of the intervals [a,), (,a], [a,a], respectively.

Since fLp,ν(), for each ε>0 there exists a>0 large enough such that

(a|f(x)1+x2m|p𝑑x)1/p+(a|f(x)1+x2m|p)1/p<ε24m+3.

This inequality can be written fχ2ap,ν+fχ1ap,ν<2(4m+3)ε.

For δ>0, this relation implies

fχ2(a+δ)p,ν+fχ1a+δp,ν<ε24m+3.

Considering 0hδ, the above inequality involves

f(+h)χ2(a+δ)p,ν+f(+h)χ1a+δp,ν<22m1(1+δ2m)ε24m+3.

We used again inequalities shown in (3.8).

Taking δ1 we can write

f(+h)χ2(a+δ)p,ν+f(+h)χ1a+δp,ν<ε22m+3.

Thus, we have

ωp,m(f;δ) 22m1(1+δ2m)2sup0hδ(f(+h)f)χa+δ1+h2mp,ν
+22m1(1+δ2m)2ε22m+2
22m+1sup0hδ(f(+h)f)χa+δ1+h2mp,ν+ε2, 0<δ1. (3.12)

By the Weierstrass theorem, there exists a sequence (φn)n1, where φnC([a2δ,a+2δ]), such that

limn(fφn)χa+2δp,ν=0.

That is, for a given ε>0, there exists a rank n0 such that

(fφn)χa+2δp,ν<ε24m+4, (3.13)

whenever nn0 and δ>0. Thus, we get

(f(+h)φn(+h))χa+δp,ν22m(fφn)χa+2δp,ν<ε22m+4 (3.14)

for nn0.

Applying the Minkowski’s inequality yields

(f(+h)f)χa+δ1+h2mp,ν (f(+h)φn(+h))χa+δp,ν
+(φn(+h)φn)χa+δp,ν+(φnf)χa+δp,ν.

From (3.13) and (3.14) it follows that

sup0hδ(f(+h)f)χa+δ1+h2mp,ν<ε22m+3+sup0hδ(φn(+h)φn)χa+δp,ν (3.15)

for nn0 and δ>0.

Since φn, n, are continuous functions on a compact interval, based on Cantor’s theorem, they are uniformly continuous. Consequently, for 0hδ and nn0 we can write

|φn(x+h)φn(x)|ε22m+3χa+δp,ν,

where x[aδ,a+δ]. Rhus, we obtain

sup0hδ(φn(+h)φn)χa+δp,νε22m+3

and, from (3.15) we get

sup0hδ(f(+h)f)χa+δ1+h2mp,νε22m+2.

Returning at (3), the following inequality ωp,m(f;δ)<ε is true for 0<δ<1, which leads us to the statement (3.11).

Theorem 3.3.

Let Ln, n, be defined by (2.3). For every fLp,ν()

Lnffp,ν2ωp,m(f;τ3,nτ2,n2) (3.16)

holds, where ωp,m is given by (3.7) and τ2,n, τ3,n are defined at (2.1).

Proof. Starting from relation (3.2), we have

Lnffp,ν 1an(|01f(xy+τ2,n)f(x)1+x2m(1y2α)n𝑑y|p𝑑x)1/p
1an01(|f(xy+τ2,n)f(x)1+x2m|p𝑑x)1/p(1y2α)n𝑑y
1an01ωp,m(f;|τ2,ny|)(1y2α)n𝑑y.

From this point the proof runs similarly to that of Theorem 3.1.

By using (3.9) and taking c=|τ2,ny|/δ, δ>0, we get

Lnffp,ν 1anωp,m(f;δ)01(1+|τ2,ny|δ)(1y2α)n𝑑y
ωp,m(f;δ)(1+1δ(1an01(τ2,ny)2(1y2α)n𝑑y)1/2)
ωp,m(f;δ)(1+1δτ3,nτ2,n2)

and the relation (3.16) follows.

In the final part of this section we make a stop over global smoothness preservation property.

Theorem 3.4.

Let Ln, n, be defined by (2.3) and let ωp,m be given by (3.7). For every fLp,ν() and δ>0

ωp,m(Lnf,δ)22m1(1+τ3,nτ2,n2)ωp,m(f;δ). (3.17)

Proof. By identity (3.2) we get

Jh :=(|Ln(f;x+h)Ln(f;x)1+(|x|+h)2m|p𝑑x)1/p
1an(|01f(xy+τ2,n+h)f(xy+τ2,n)1+(|x|+h)2m(1y2α)n𝑑y|p𝑑x)1/p.

Using the inequality

1+(|xy+τ2,n|+h)2m22m1(1+(|x|+h)2m)(1+(τ2,ny)2m),

we have

Jh 22m1an01(|f(xy+τ2,n+h)f(xy+τ2,n)1+(|xy+τ2,n|+h)2m|p𝑑x)1/p
×(1+|τ2,ny|)(1y2α)ndy
22m1anωp,m(f;δ)01(1+|τ2,ny|)(1y2α)n𝑑y
22m1ωp,m(f;δ)(1+(Lnφx2)(x)).

In the above Lnφx2 represents the second order central moment of our operators. Based on (2.4), we get (Lnφx2)(x)=τ3,nτ2,n2. Taking sup0hδJh, we arrive at relation (3.17) and the proof is completed.

4. Multidimensional Landau type operators

For a positive integer p2 we consider the Euclidean space p. During this section we use the following notations: 𝒙=(xi)1ipp, 𝒏=(ni)1ipp, D=[0,1]p. According to the model offered by the relation (2.1) we consider the real numbers τk,ni=B(ni+1,kλ)/B(ni+1,λ), k, and the vector 𝝉k,𝒏=(τk,ni)1ip. Set

An=i=1pani,

where ani is defined as in formula (2.1). A variant of the Landau p-dimensional operators which generalize those introduced by (2.3), can be described as follows

(L~𝒏f)(𝒙)=1AnDf(𝒙𝒚+𝝉2,𝒏)j=1p(1yj2α)njdy1dyp, (4.1)

where f is a real-valued function defined on p such that the right hand side exists and is finite. We turn our attention to establishing approximation properties of these operators for the class of continuous functions. To achieve this, we introduce the following p+2 reference functions. Let Xp. The constant function on X of constant value 1 is denoted by 𝟏. For each j=1,,p we shall denote by prj:X the j-th canonical projection which is defined by

prj(𝒙)=xj, for every 𝒙=(xi)1ipX.

Set πp=j=1pprj2, πp(𝒙) representing the Euclidean inner product of 𝒙 with itself.

C(X) stands for the space of all real-valued continuous functions on X endowed with the norm of the uniform convergence ,

f=sup𝒙X|f(𝒙)|. (4.2)
Theorem 4.1.

Let the operators L~𝐧 be defined by (4.1). If

L~𝒏𝟏10, (4.3)
L~𝒏prjprj0,j=1,,p, (4.4)
L~𝒏πpπp0, (4.5)

then L~𝐧ff0 for any fC(X), where X is a convex compact included in p and the norm is defined by (4.2).

Proof. Clearly, the operators are linear and positive. We use the multivariate Korovkin theorem. In this line we recall that

𝒦={𝟏,pr1,,prp,πp} (4.6)

is a Korovkin system of test functions in C(X), see, e.g., [2, Eq. (4.4.22)]. For two dimensional case this result was first established by Volkov [19]. The p-dimensional analogue of Korovkin’s theorem was proved by Shashkin [15]. Based on the definition of our operators, we have

(L~𝒏𝟏)(𝒙)=1,

consequently the identity (4.3) takes place. Further, using (4.1) along with the formulas (2.3) and (2.4), for each j=1,,p, we obtain

(L~𝒏prj)(𝒙) =1An01(xjyj+τn,j)(k=1kjp01(1yk2α)nk𝑑yk)(1yj2α)nj𝑑yj
=1anj01(xjyj+τn,j)(1yj2α)nj𝑑yj=(Lne1)(xj)=xj,

which ensures the fulfillment of the conditions required at (4.4).

Following a similar route to the one presented above, we can write

(L~𝒏πp)(𝒙)=j=1p(L~𝒏prj2)(𝒙)
=j=1p1An01(xjyj+τn,j)2(k=1kjp01(1yk2α)nk𝑑yk)(1yj2α)nj𝑑yj
=j=1p1anj01(xjyj+τn,j)2(1yj2α)nj𝑑yj=j=1p(Lne2)(xj)
=πp(𝒙)+j=1p(τ3,njτ2,nj2).

We deduce L~𝒏πpπpj=1p|τ3,njτ2,nj2|. The result established at (2.2) guarantees that relation (4.5) is achieved. The proof is completed. 

We present an estimate of the error of approximation in terms of the corresponding quantities for the test functions referred to in (4.6). For linear positive operators univariate case, a general result was establishing by Shisha and Mond [16]. For p-dimensional case, the general result was obtained by Censor [3]. Such estimates involve moduli of smoothness. For real multivariate functions we consider the following type of modulus [3, Eq. (1)]

ω(f;δ)=max𝒕,𝒙Xd(𝒕,𝒙)δ|f(𝒕)f(𝒙)|,δ0, (4.7)

where Xp is a convex compact, d(,) stands for the Euclidean distance in p and f is a real valued function continuous on X.

Theorem 4.2.

Let the operators L~𝐧 be defined by (4.1). The following relation

L~𝒏ff2ω(f;d(𝝉3,𝒏,𝝉2,𝒏2)) (4.8)

holds, where ω is given at (4.7) and for s=2, s=3,

𝝉s,𝒏=(τs,ni)1ipp.

Proof. Using the notions and the notations specified above, the proof of formula (4.8) is based on [3, Theorem 1] which says: if Λn is positive linear operator on C(X) such that Λn𝟏=1, then

Λnff2ω(f;μn)

holds, where μn2=Λn(k=1p(ξkxk)2;x1,,xp). Here Λn operates on a function of ξ1,,ξp and the resulting function is evaluated at the point (x1,,xp).

In our case we have

L~𝒏(k=1p(prkxk)2;𝒙) =k=1pL~𝒏(prk22xkprk+xk2𝟏;𝒙)
=k=1p(τ3,nkτ2,nk2)2=d(τ3,𝒏,τ2,𝒏2),

see the computations accomplished in the proof of Theorem 4.1. The components of the vectors 𝝉s,𝒏, s{2,3}, are explicitly defined at (2.1). Formula (4.8) follows.

References

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  • [2] Altomare, F., Campiti, M., Korovkin-type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, Vol. 17, Walter de Gruyter & Co., Berlin, New York, 1994.
  • [3] Censor, E., Quantitative results for positive linear approximation operators, J. Approx. Theory, 4(1971), 442-450.
  • [4] Chen, Zhanben, Shih, Tsimin, A new class of generalized Landau linear positive operator sequence and its properties of approximation, Chinese Quart. J. Math., 13(1998), no. 1, 29-43.
  • [5] De Vore, R.A., Lorentz, G.G., Constructive Approximation, A Series of Comprehensive Studies in Mathematics, Vol. 303, Springer Verlag, Berlin, Heidelberg, 1993.
  • [6] Gadjiev, A.D., Aral, A., Weighted Lp-approximation with positive linear operators on unbounded sets, Appl. Math. Lett., 20(2007), Issue 10, 1046-1051.
  • [7] Gal, S.G., Iancu, I., Quantitative approximation by nonlinear convolution operators of Landau-Choquet type, Carpathian J. Math., 37(2021), Issue 1 (to appear).
  • [8] Gao, J.B., Approximation properties of a kind of generalized discrete Landau operator, (Chinese), J. Huazhong Univ. Sci. Tech., 12(1984), no. 5, 1-4.
  • [9] Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities, Cambridge Mathematical Library, Cambridge University Press, 1988.
  • [10] Jackson, D., A proof of Weierstrass theorem, Amer. Math. Monthly, 41(1934), no. 5, 309-312.
  • [11] Landau, E., Über die Approximation einer stetingen Funktion durch eine ganze rationale Funktion, Rend. Circ. Mat. Palermo, 25(1908), 337-345.
  • [12] Mamedov, R.G., Approximation of functions by generalized linear Landau operators, (Russian), Dokl. Akad. Nauk SSSR, 139(1961), no. 1, 28-30.
  • [13] Mamedov, R.G., On the order and on the asymptotic value of the approximation of functions by generalized linear Landau operators, (Russian), Akad. Nauk Azerbadzan SSR, Trudy Inst. Mat. Meh., 2(10)(1963), 49-65.
  • [14] Pendina, T.P., Iterations of positive linear operators of exponential type and of Landau polynomials, (Russian), In: Geometric Problems of the Theory of Functions and Sets (Russian), Kalinin. Gos. Univ., Kalinin, 1987, 105-111.
  • [15] Shashkin, Yu.A., Korovkin systems in spaces of continuous functions, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 26(1962), Issue 4, 495-512; translated in Amer. Math. Soc. Transl., Series 2, 54(1996), 125-144.
  • [16] Shisha, O., Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60(1968), 1196-1200.
  • [17] Sikkema, P.C., Approximation formulae of Voronovskaya type for certain convolution operators, J. Approx. Theory, 26(1979), no. 1, 26-45.
  • [18] Veselinov, V., Certain estimates to the approximation of functions by de la Vallée-Poussin and Landau operators, (Russian), Ann. Univ. Sofia Fac. Math., 66(1974), 153-158.
  • [19] Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables, (Russian), Dokl. Akad. Nauk SSSR (N.S.), 115(1957), 17-19.
  • [20] Yuksel, I., Ispir, N., Weighted Approximation by a Certain Family of Summation Integral-Type Operators, Comput. Math. Appl., 52(2006), 1463-1470.

[1] Agratini, O., Gal, S.G., On Landau type appproximation operators. Mediterr. J. Math. (sent to publication)
[2]Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, vol. 17. Walter de Gruyter & Co., Berlin, New York (1994), Book Google Scholar
[3] Censor, E., Quantitative results for positive linear approximation operators. J. Approx. Theory 4, 442–450 (1971), MathSciNet Article Google Scholar
[4] Chen, Zhanben, Shih, Tsimin, A new class of generalized Landau linear positive operator sequence and its properties of approximation. Chin. Q. J. Math. 13(1), 29–43 (1998), Google Scholar
[5] De Vore, R.A., Lorentz, G.G., Constructive Approximation, A Series of Comprehensive Studies in Mathematics, vol. 303. Springer, Berlin, Heidelberg (1993) Google Scholar
[6] Gadjiev, A.D., Aral, A., Weighted Lp-approximation with positive linear operators on unbounded sets. Appl. Math. Lett. 20(10), 1046–1051 (2007), MathSciNet Article Google Scholar
[7] Gal, S.G., Iancu, I.: Quantitative approximation by nonlinear convolution operators of Landau-Choquet type. Carpathian J. Math. 37, Issue 1 (to appear) (2021)
[8] Gao, J.B., Approximation properties of a kind of generalized discrete Landau operator, (Chinese). J. Huazhong Univ. Sci. Tech. 12(5), 1–4 (1984), MathSciNet Google Scholar
[9] Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities. Cambridge University Press, Cambridge Mathematical Library (1988), MATH Google Scholar
[10] Jackson, D., A proof of Weierstrass theorem. Am. Math. Mon. 41(5), 309–312 (1934), MathSciNet Article Google Scholar
[11] Landau, E., Über die Approximation einer stetingen Funktion durch eine ganze rationale Funktion. Rend. Circ. Mat. Palermo 25, 337–345 (1908), Article Google Scholar
[12] Mamedov, R.G., Approximation of functions by generalized linear Landau operators, (Russian). Dokl. Akad. Nauk SSSR 139(1), 28–30 (1961), MathSciNet Google Scholar
[13] Mamedov, R.G., On the order and on the asymptotic value of the approximation of functions by generalized linear Landau operators, (Russian), Akad. Nauk Azerbadzan SSR, Trudy Inst. Mat. Meh., 2(10), 49–65 (1963)
[14] Pendina, T.P., Iterations of positive linear operators of exponential type and of Landau polynomials, (Russian). In: Geometric Problems of the Theory of Functions and Sets (Russian), Kalinin. Gos. Univ., Kalinin, 105–111, (1987)
[15] Shashkin, Y.A.: Korovkin systems in spaces of continuous functions, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 26, Issue 4, 495–512 (1962), translated in Am. Math. Soc. Transl., Series 2, 54, 125–144 (1996)
[16] Shisha, O., Mond, B., The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 60, 1196–1200 (1968), MathSciNet Article Google Scholar
[17] Sikkema, P.C., Approximation formulae of Voronovskaya type for certain convolution operators. J. Approx. Theory 26(1), 26–45 (1979), MathSciNet Article Google Scholar
[18] Veselinov, V., Certain estimates to the approximation of functions by de la Vallée–Poussin and Landau operators, (Russian). Ann. Univ. Sofia Fac. Math. 66, 153–158 (1974)Google Scholar
[19] Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR (N.S.), 115, 17–19 (1957) (Russian)
[20] Yuksel, I., Ispir, N., Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52, 1463–1470 (2006), MathSciNet Article Google Scholar

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