Approximation properties of a family of integral type operators

Abstract

In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer–Jain operators, Păltănea–Szász–Mirakjan operators and operators using Baskakov–Szász type bases. We focus on highlighting some approximation targeting different classes of functions. The main working tools are Bohman–Korovkin theorem, weighted K-functionals and moduli of smoothness.

Authors

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

Keywords

Positive linear operator; Weighted K-functional; Modulus of smoothness; Durrmeyer–Jain operator

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O. Agratini, Approximation properties of a family of integral type operators, Positivity, 25 (2021), 97–108, https://doi.org/10.1007/s11117-020-00752-y

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Approximation properties of a family of integral type operators

Approximation properties
of a family of integral 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
Abstract.

In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer-Jain operators, Păltănea-Szász-Mirakjan operators and operators using Baskakov-Szász type bases. We focus on highlighting some approximation targeting different classes of functions. The main working tools are Bohman-Korovkin theorem, weighted K-functionals and moduli of smoothness.

Keywords and phrases: Positive linear operator, weighted K-functional, modulus of smoothness, Durrmeyer-Jain operator.

Mathematics Subject Classification: 41A36, 41A25.

1. Introduction

On the last five decades the interest of the study of positive approximation processes have emerged with growing evidence. In what follows we consider linear positive operators which approximate real valued functions defined on the unbounded interval +=[0,). Among them, a special attention has been paid to operators which reproduce affine functions. Due to the linearity of the operators, this property is implied by the preservation of the test functions e0 and e1. Set ej, j0={0}, the monomial of degree j. The starting point is given by discrete operators of the following form

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

verifying the conditions

k=0λn,k=e0, (1.2)
k=0xn,kλn,k=e1, (1.3)

where λn,k is a continuous function defined on + with non-negative real values for each k0, (xn,k)k0 represents a net on + (0=xn,0<xn,1<) and function f is chosen such that the series from relation (1.1) is convergent. Because such operators are not suitable for approximating discontinuous functions, they are generalized to integral type operators. One of the usual techniques is known as Durrmeyer method [12] which leads us to an approximation process, say (Ln)n1, in spaces of integrable functions.

This type of integral generalization, studied in depth for the first time in 1981 by Derriennic [7], has been constant in the attention of many researches, among the most recent papers is that of Abel and Karsli [1].

For each n, let (μn,k)k1 be a sequence of continuous and positive functions defined on + such that the following relation is fulfilled

0μn,k(t)𝑑t=1,k. (1.4)

Setting In,k(f)=+μn,k(t)f(t)𝑑t, k, we introduce the operators

(Lnf)(x)=λn,0(x)f(0)+k=1λn,k(x)In,k(f), (1.5)

where f(+), this space consisting of all real valued functions f defined on + with the property λn,kf belongs to the Lebesgue space L1(+) for each k and the series from the right hand side of relation (1.5) is convergent. Denoting by CB(+) the space of all continuous and bounded functions defined on +, it is clear that CB(+)(+).

Remark 1.1.

We mention that identity (1.4) is not an essential requirement. It is only important that the integral exists and is finite. If

0μn,k(t)𝑑t=cn,k,

where cn,k+, k, then we can define

In,k(f)=cn,k10μn,k(t)f(t)𝑑t,

and, this way, we reach the same operators Ln given by (1.5).

It is obvious that Ln, n, operators are linear and positive. Moreover, relations (1.4) and (1.2) imply

Lne0=e0,n. (1.6)

This also gives a formula for the norm of the operators, Ln=1.

To motivate the consistency of our study, our first concern is to highlight families of such operators that have already been studied. Next, some approximation properties of our general operators are revealed.

We point out that, alternatively, by using a two-dimensional kernel, the operators can be rewritten as follows

(Lnf)(x)=0Kn(x,t)f(t)𝑑t,

where

Kn(x,t)=λn,0(x)δx(t)+k=1λn,k(x)μn,k(t),

δx(t) being Dirac generalized function.

Finally we introduce the j-th central moment of Ln operators (j0), i.e.,

j(Ln;x)=(Lnφxj)(x), where φx(t)=tx,(t,x)+×+. (1.7)

This can be accomplished by assuming that ej(+).

2. Examples of particular cases of the operators Ln

The section includes three examples of known operators defined following the structure given by the relation (1.5).

1. Operators using Baskakov-Szász type bases. Choosing in (1.5)

λn,k(x)=(n+k1k)xk(1+x)nk,x[0,),k0,

and

μn,k(t)=nk!ent(nt)k,t[0,),k,

the resulting operators (denoted by Mn) have been introduced in 2001 by Purshottam Agrawal and Ali Mohammad [4, Eq. (1.1)] in order to approximate a class of continuous functions of exponential growth. Their study was deepened by both the authors mentioned above [5] and by Gupta V., Gupta M.K. [13].

In this case, the conditions (1.2) and (1.4) are verified. Also, considering the network xn,k=k/n, k0, the relation (1.3) is fulfilled.

2. Szász-Mirakjan-Păltănea operators. The integral version we referred appeared in [18].

It depends on two parameters α>0, ρ>0. In this case we have

λα,k(x) :=sα,k(x)=eαx(αx)kk!,x[0,),k0,
μα,k(x) :=θα,kρ(t)=αρkρΓ(kρ)eαρt(αt)kρ1,t[0,),k,

and the class of operators is designed as follows

(Lαρf)(x)=eαxf(0)+k=1sα,k(x)0θα,kρ(t)f(t)𝑑t.

Both relations (1.2), (1.4) and relation (1.3) with xn,k=k/n, k0, are satisfied. Păltănea continued the study of these operators in [19]. Among the most recent papers that deepen the study of (Lαρ)α>0 we mention [17].

3. Durrmeyer-Jain operators. Using a Poisson-type distribution with two parameters given by

wβ(k;α)=αk!(α+kβ)k1e(α+kβ),k0,

for α>0 and β[0,1), Jain [14] introduced the following class of positive linear operators

(Pn[β]f)(x)=k=0wβ(k;nx)f(kn),x0,

where fC(+) whenever the above series is convergent. In [2] an extension in Durrmeyer sense of Pn[β] has been introduced, which was named Λn[β]. In this case, in (1.5) we choose as follows

λn,k(x)=wβ(k;nx),x[0,),k0, (2.1)
μn,k(t)=n1βent/(1β)(nt1β)k1,t[0,),k. (2.2)

Relations (1.2) and (1.4) take place but (1.3) fails. We get

Pn[β]e1=(1β)1e1.

For the special case β=0, LnΛn[0], n, turn into Phillips operators [20]. Among the first papers in which the approximation properties of these operators were studied, we mention [16].

Remark 2.1.

The integral operators presented in Examples 2 and 3 enjoy the property of keeping the affine functions as fixed points. Relation (1.6) holds and it is easy to verify that Lαρe1=Λn[β]e1=e1.

Inspired by this fact, for certain results that we will establish in the next section, we will impose the following condition to be fulfilled

k=1λn,kIn,k(e1)=e1,n. (2.3)

This additional condition ensures the identity

2(Ln;x)=(Lne2)(x)x2,x0, (2.4)

and the quantity is strictly positive for x>0.

Remark 2.2.

Regarding the central moment of the second order for the operators presented above, we have the following relations

{2(Mn;x)=x(x+2)n,2(Lαρ;x)=ρ+1αρx,2(Λn[β];x)=1+(1β)2n(1β)x, (2.5)

see respectively [4, Lemma 1], [19, Eq. (2.1)], [2, Eq. (15)].

3. Results

Remark 3.1.

For each n, Ln maps continuously CB(+) into itself. Indeed, for fCB(+) we get |In,k(f)|f, k0, and based on relation (1.2), we deduce Lnff, where stands for the supremum norm on +.

Following the line of Ditzian and Totik [10, §1.2], we consider φ:++ an admissible weight function. In order to give an estimate of the approximation error, we use the weighted K-functional of second order fCB(+) defined as follows

K2,φ(f,t):=infg{fg+tφ2g′′:gACloc(+)},t>0, (3.1)

where gACloc(+) means that g is differentiable and g is absolutely continuous on every compact of +, [10, Eq. 2.1.1].

Theorem 3.2.

Let Ln, n, be defined by (1.5) such that (2.3) takes place. If φ is an admissible function such that φ2 is concave, then

|(Lnf)(x)f(x)|2K2,φ(f,(Lne2)(x)x22φ2(x)),x>0,

holds.

Proof. Let x>0 be fixed and g:+ be twice differentiable such that gACloc(+). Starting from Taylor’s expansion

g(u)=g(x)+g(x)(ux)+xug′′(t)(ut)𝑑t,u0,

and knowing that (2.3) holds, in other words Ln reproduces linear functions, we have

(Lng)(x)g(x)=Ln(xe0e1g′′(t)(e1t)𝑑t,x).

Since φ2 is concave, for every t=(1λ)u+λx, λ(0,1), we get

φ2(t)(1λ)φ2(u)+λφ2(x)λφ2(x)>0,

and consequently

|tu|φ2(u)=λ|xu|φ2(t)|xu|φ2(x).

It turns out that

|xug′′(u)(ut)𝑑t| φ2g′′|xu|tu|φ2(t)𝑑t|
φ2g′′|xu|xu|φ2(x)𝑑t|
=φ2g′′(xu)2φ2(x).

Applying the linear positive operator Ln and taking into account (2.4), we get

Ln(xe0e1g′′()(e1t)𝑑t;x)φ2g′′(Lne2)(x)x2φ2(x),

and further

|(Lnf)(x)f(x)| |Ln(fg;x)|+|g(x)f(x)|+|(Lng)(x)g(x)|
2fg+φ2g′′(Lne2)(x)x2φ2(x).

In the above we used Remark 3.1. At this point, taking the infimum over all g with gACloc(+), based on (3.1), we get the desired result.

In accordance with [10, Theorem 2.1.1], for some constants M>0 and t0, the following inequality

K2,φ(f,t)Mω2,φ(f,t), 0<tt0,

takes place, where

ω2,φ(f,t)=sup0htsupx±hφ(x)0|f(xφ(x)h)2f(x)+f(x+φ(x)h)|.

From Theorem 3.2 we deduce

Corollary 3.3.

Under the same assumptions of Theorem 3.2, the following inequality

|(Lnf)(x)f(x)|M~ω2,φ(f;1φ(x)(Lne2)(x)x2)

holds, where M~>0 is a certain constant.

Examining relation (1.2), on the basis of Korovkin’s first theorem [15], we can state

Theorem 3.4.

Let K+ be a compact interval and Ln, n, be defined by (1.5) such that (2.3) takes place. If

limnLne2=e2 uniformly on K,

then

limnLnf=f uniformly on K,

provided fC(+)(+).

For investigating other properties of our sequence, we need the following technical result.

Lemma 3.5.

Let Ln, n, be defined by (1.5). For any constant β(0,1] and function h:++, the following inequality

(Lnhβ)(x)λn,0(x)hβ(0)+((Lnh2)(x))β/2,x0, (3.2)

takes place.

Proof. We use Hölder’s inequality both for integrals and for sums. Considering r=2β1 in the relation 1/r+1/s=1, r>0, s>0, we can write

0μn,k(t)hβ(t)𝑑t (0μn,k(t)𝑑t)1/s(0μn,k(t)h2(t)𝑑t)1/r
=In,kβ/2(h2).

Further,

k=1λn,k(x)In,k(hβ) (k=1λn,k(x))1/s(k=1λn,k(x)In,k(h2))1/r
=(k=1λn,k(x)In,k(h2))β/2
=((Lnh2)(x)λn,0(x)h2(0))β/2((Lnh2)(x))β/2

and the conclusion occurs.

Next, we study the approximation properties for functions that are not necessarily bounded. We start with functions that satisfy a Hölder type condition with β(0,1] on E+, more precisely functions f:+ which verify the relation

|f(x)f(y)|Cβ,f|xy|β,(x,y)+×E, (3.3)

where Cβ,f is a positive constant depending only on β and f. Clearly, such functions belong to (+). For β>1 relation (3.3) leads us to constant functions on E, so we are not interested at this time.

We prove a relation between the local smoothness of functions and the local approximation.

Theorem 3.6.

Let Ln, n, be defined by (1.5). If f satisfies relation (3.3), then

|(Lnf)(x)f(x)|Cβ,f(λn,0(x)xβ+2β/2(Ln;x)+2dβ(x,E)), (3.4)

x0, where 2(Ln;x) is given at (1.7) and d(x,E) represents the distance between x and E.

Proof. It is clear that (3.3) holds for any x+ and yE¯, the closure of E in . Let (x,x0)+×E¯ such that

|xx0|=d(x,E):=inf{|xy|:yE}.

Since |ff(x)||ff(x0)|+|f(x0)f(x)| and Ln is a linear positive operator satisfying (1.6), we can write

|(Lnf)(x)f(x)| Ln(|ff(x0)|;x)+|f(x)f(x0)|
Ln(Cβ,f|e1x0|β;x)+Cβ,f|xx0|β. (3.5)

On the other hand, choosing in (3.2) h:=|φx|, see (1.7), we deduce

Ln(|e1x|β;x)λn,0(x)xβ+2β/2(Ln;x). (3.6)

Based on the elemental inequality |tx0|β|tx|β+|xx0|β, 0<β1, on the monotonicity property of the operator Ln as well as on relations (3.6) and (1.6), we get

Ln(|e1x0|β;x)λn,0(x)xβ+2β/2(Ln;x)+dβ(x,E).

Returning at (3), the proof is complete.

Remark 3.7.

In relation (3.4), in the particular cases xE¯ and E=+, the term d(x,E) vanishes.

Further, we discuss about the type of convergence of the sequence (Lnf)n1 to the continuous function f that it approximates. On compact intervals, if f is continuous, the uniform convergence of (Lnf)n1 to f takes place, see Theorem 3.4. On unbounded intervals, if f is continuous, usually only pointwise convergence occurs. We will indicate sufficient conditions which ensure the uniform convergence on unbounded intervals J+. For discrete approximation processes such an approach can be found, for example, in [3]. Similar results have been obtained for a long time ago by Jésus de la Cal and Javier Cárcamo [6] for families of operators of probabilistic type over non-compact intervals. The authors used representation of the operators in terms of appropriate stochastic processes.

Set UC(+) the space of all uniformly continuous real valued function on +. Let τ:++ be a continuous one-to-one function such that τ(0)=0. We define the following two functions

f=fτ1,τx(t)=|τ(t)τ(x)|,t0,x0, (3.7)

where fCB(+).

Theorem 3.8.

Let Ln, n, be defined by (1.5) such that λn,0(0)=1 and let the functions τx, f be given by (3.7).

If fUC(+) and there are two sequence of positive real numbers, (γ1,n)n1, (γ2,n)n1 that converge to zero satisfying the conditions

λn,0(x)(1+τ(x))γ1,n,xa, (3.8)
Ln(τx;x)γ2,n,xa, (3.9)

for a certain a0, then (Lnf)n1 converges uniformly to f on the interval [a,).

Proof. The main tool used to motivate our statement is modulus of smoothness associated to any bounded function g defined on + expressed by the following formula

ω(g;δ)=sup{|g(x)g(x′′)|:x,x′′+,|xx′′|δ},δ0. (3.10)

Its most prominent properties can be found, e.g., in [8, p. 40-44]. We recall three of them, useful for our proof. The function ω(g;) is nondecreasing,

ω(g;λδ)(1+λ)ω(g;δ),δ0 and λ0, (3.11)

and, if in addition g is uniformly continuous on +, then

limδ0+ω(g;δ)=0. (3.12)

We consider xa>0 arbitrarily fixed. Based on identities (1.5) and (3.10) we get

|(Lnf)(x)f(x)| =|Ln(fτ;x)f(τ(x))|
Ln(|fτf(τ(x))|;x)
=λn,0(x)|f(τ(0))f(τ(x))|
+k=1λn,k(x)In,k(fτf(τ(x)))
:=Ax+Bx. (3.13)

At first we evaluate Ax. If g is uniformly continuous function on +, then its growth on domain is at most affine, i.e., there exist c0,c1 positive constants such that |g(t)|c1t+c0 for all t+, see, e.g., [9, p. 48, Problème 4] or [11]. Because fUC(+), in view of this result, we can write

Axλn,0(x)(|f(0)|+|f(τ(x))|)c~λn,0(1+τ(x))c~γ1,n, (3.14)

where c~=max{2c0,c1}. We also used relation (3.8).

Next we evaluate Bx. Clearly, (Lnτx)(x)>0 for x0. The definition of modulus of smoothness and property (3.11) applied for

λ=|τ(t)τ(x)|/(Lnτx)(x)

allow us to write

|f(τ(t))f(τ(x))| ω(f;|τ(t)τ(x)|)
(1+τx(t)Ln(τx;x))ω(f;Ln(τx;x))
(1+τx(t)Ln(τx;x))ω(f;γ2,n).

In the above we also used the monotonicity of the function ω(f;) as well as the hypothesis specified at (3.9). Since f is bounded on the domain, consequently ω(f;) is well defined. Thus, we have

Bx(u1(x)+u2(x)Ln(τx;x))ω(f;γ2,n)2ω(f;γ2,n), (3.15)

where

{0u1(x):=1λn,0(x)1,0u2(x):=Ln(τx,x)τx(0)λn,0(x)Ln(τx,x).

Using the inequalities (3.14) and (3.15), relation (3) implies

|(Lnf)(x)f(x)|c~γ1,n+2ω(f;γ2,n),xa. (3.16)

Relation λn,0(0)=1 corroborated with (1.2) implies λn,k(0)=0, k, consequently Ln enjoys the interpolating property in x=0. Thus, relation (3.16) is true in the particular case a=0 as well.

Property (3.12) guarantees ω(f;γ2,n) tends to zero as n tends to infinity. Alongside (3.8), this statement leads to the completion of the proof. 

Example 3.9.

We apply Theorem 3.8 to Durrmeyer-Jain operators Λn[β], n1, where β[0,1). Let us choose a=1 and τ(x)=x. Taking in view (2.1) and (2.2), we can write

wβ(0;nx)(1+x)1n:=γ1,n,x1,

and

Λn[β](τx;x) =wβ(0;nx)x+k=1wβ(k;nx)0μn,k(t)|tx|t+x𝑑t
1n+1x21/2(Λn[β];x)=1+β~n,x1,

where β~=(1+(1β)21β)1/2, see (2.5).

The requirements of Theorem 3.8 being fulfilled, we deduce that (Λn[β]f)n1 converges uniformly to f on [1,) for any fUC(+).

Concluding remark. The results presented in this section have a unifying character in the sense that they can be applied to several particular classes of operators. As a weak point we admit that these results are not as spectacular as the ones that could be obtained for each particular case.

References

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[1] Abel, U., Karsli, H.,  Asymptotic expansions for Bernstein–Durrmeyer–Chlodovsky polynomials. Results Math. 73, Article 104 (2018)
[2] Agratini, O., On an approximation process of integral type. Appl. Math. Comput. 236, 195–201 (2014), MathSciNet MATH Google Scholar
[3] Agratini, O., Uniform approximation of some classes of linear positive operators expressed by series. Appl. Anal. 94(8), 1662–1669 (2015), MathSciNet Article Google Scholar
[4] Agrawal, P.N., Mohamed, A.J.: Linear combination of a new sequence of linear positive operators. Rev. Un. Mat. Argentina 42(2), 57–65 (2001), MathSciNet MATH Google Scholar
[5] Agrawal, P.N., Mohamed, A.J.: On L_{p} -approximation by a linear combination of a new sequence of linear positive operators. Turk. J. Math. 27, 389–405 (2003), Google Scholar
[6] de la Cal, J., Cárcamo, J., On uniform approximation by some classical Bernstein-type operators. J. Math. Anal. Appl. 279(2), 625–638 (2003), MathSciNet Article Google Scholar
[7] Derriennic, M.M., Sur l’approximation de fonctions intégrables sur [0,1] par des polynômes de Bernstein modifiés. J. Approx. Theory 31, 325–343 (1981), MathSciNet Article Google Scholar
[8] De Vore, R.A., Lorentz, G.G., Constructive Approximation. A Series of Comprehensive Studies in Mathematics, vol. 303. Springer, Berlin (1993), Google Scholar
[9] Dieudonné, J., Éléments d’Analyse. Tome 1: Fondements de l’Analyse Moderne. Gauthiers Villars, Paris (1968), MATH Google Scholar
[10] Ditzian, Z., Totik, V., Moduli of Smoothness. Springer Series in Computational Mathematics, vol. 9. Springer-Verlag, New York Inc., New York (1987), MATH Google Scholar
[11] Djebali, S., Uniform continuity and growth of real continuous functions. Int. J. Math. Educ. Sci. Technol. 32(5), 677–689 (2001),MathSciNet Article Google Scholar
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