Abstract

This note aims to highlight a general class of discrete linear and positive operators, focusing on some of their approximation properties. The construction includes a sequence of positive numbers ðknÞ, strictly decreasing. Imposing additional conditions on it, we set the convergence of the operators towards the identity operator and establish upper bounds for the error of approximation. A probabilistic approach is given. Also, in certain weighted spaces the study of its approximation properties is developed.

Authors

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

Keywords

Abdulofizov generalization; Balazs-Szabados operator; Korovkin theorem; modulus of smoothness.

Paper coordinates

O. Agratini, On a class of Bernstein-type rational functions, Numerical Functional Analysis and Optimization, 41 (2020) no. 4, pp. 483-494, https://doi.org/10.1080/01630563.2019.1664566

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On a class of Bernstein type rational functions

On a class of Bernstein type rational functions

Octavian Agratini Babeş-Bolyai University
Faculty of Mathematics and Computer Science
Str. Kogălniceanu, 1
400084 Cluj-Napoca, Romania
agratini@math.ubbcluj.ro
Abstract.

This note aims to highlight a general class of discrete linear and positive operators, focusing on some of their approximation properties. The construction includes a sequence of positive numbers (λn), strictly decreasing. Imposing additional conditions on it, we set the convergence of the operators towards the identity operator and establish upper bounds for the error of approximation. A probabilistic approach is given. Also, in certain weighted spaces the study of its approximation properties is developed.

Keywords and phrases: Linear positive operator, Korovkin theorem, modulus of smoothness, Balázs-Szabados operator, Abdulofizov generalization.

Mathematics Subject Classification: Primary: 41A25, Secondary: 47A63.

1. Introduction

The classical Korovkin theory is mainly connected with the approximation of continuous functions by linear operators having as essential ingredient the property of positivity. Considering such operators, say Ln (n), they have the approximation property on the normed space (S,S) if limnLnffS=0 for each fS.

In what follows, we consider linear positive operators of discrete type which approximate functions defined on +=[0,). In order to construct the proposed class of operators, we use a strictly decreasing positive sequence (λn)n1 with the property

limnλn=0. (1)

The operators investigated in this paper are designed as follows

(Lnf)(x)=1(1+λnx)nk=0n(nk)(λnx)kf(knλn),n,x0, (2)

where f is continuous function on + satisfying a certain growth condition. We also require

limnnλn= (3)

to be fulfilled.

The construction given at (2) is not new. With a slight modification, it appears in [7, Eq. (1.2)]. In this mentioned paper, K. Balázs studied the special case λn=n1/3. In time, another special case has been studied deeper

λn=nβ1,n, 0<β<1, (4)

and for these particular operators we use the notation Rn[β]. We point out some of the most outstanding results concerning these operators. Balázs and Szabados [8] gave weighted estimated and investigated the uniform convergence of Rn[β]. V. Totik [19] settled the saturation properties of Rn[β], proved a general convergence theorem for Rn[β] – like rational functions and obtained a Voronovskaja-type result. Biancamaria Della Vecchia [9] proved some preservation properties and asymptotic relations for Rn[β]. Further on, Ulrich Abel and B. Della Vecchia [4] obtained the complete asymptotic expansion for Rn[β], all coefficients being calculated explicitly. An integral generalization in Kantorovich sense was achieved in [5].

In Quantum Calculus, the q-analogues of Rn[β] operators were studied by O. Doğru [11]. The rational complex Rn[β] operators, 0<q2/3, appear in the monograph of S.G. Gal [13]. See, respectively, the papers of Ispir and Yildiz Ozkan [14], Mahmudov [17].

Returning at general form indicated by relation (2), in the recent paper [3] the authors showed that Ln, n, operators enjoy the variation detracting property. At this moment, we want to point out some features of Ln, n, operators. They are built using a finite sum. Other classes of discrete operators are expressed by series, making them harder to use in approximation of functions, see for example, Baskakov operators or Favard-Szász-Mirakjan operators. For each n, Ln uses an equidistant net Δn=(k/(nλn))k=0,n¯ with a flexible step, that is (nλn)1. It changes depending on an arbitrary sequence (λn)n1 satisfying the conditions (1) and (3).

Regarding the definition of Ln, n, operators, it is fair to mention that a more general form was proposed a long time ago by Abdulofizov [1]. The author considered the linear operators Sn, n, defined as follows

(Snf)(x)=k=0nqk,n(x)f(ξk,n),

where

qk,n(x)=(nk)(1+rn)kxk(rn(1+x))n(rnx)nk and ξk,n=knk+nrn1,k=0,n¯.

Instead of the net used in (2), here appears a net of more general nodes. Estimations of |(Snf)(x)f(x)| under various assumptions concerning f have been proved by Abdulofizov. Modifications of Bernstein operators by using nodes of the form (ξk,n)1kn can also be found in [2].

The approximation properties highlighted in our paper are different from those obtained in [1]. More precisely, we establish two variants of local upper bounds of the error of approximation, one of them including the distance d(x,E) between a certain x>0 and a subset E+. Also, we prove that our class of operators does not always form an approximation process in some weighted spaces. Further on, by using a weighted modulus of smoothness defined in 2004, we give both a local and global estimation of the rate of convergence. It is known that a more general class of sequences enjoys fewer specific properties than special cases of that. Particular cases narrow the approach but sometimes provide better approximation variants.

2. Preliminary results

Set 0={0}. For every m0 we shall denote by em the monomial defined as e0(t)=1 and em(t)=tm (m1), where t+.

Lemma 1. Let Ln, n, be defined by (2). For each x+, the following relations

(Lne0)(x)=1, (5)
(Lne1)(x)=x1+λnx, (6)
(Lne2)(x)=1(1+λnx)2(x2+xnλn) (7)

take place.

Proof. Since k=0n(nk)(λnx)k=(1+λnx)n, the first identity is evident. Further, we get

(Lne1)(x) =1(1+λnx)nk=1nknλn(nk)(λnx)k
=x(1+λnx)nk=1n(n1k1)(λnx)k1
=x1+λnx.

The last identity is obtained as follows

(Lne2)(x) =1(1+λnx)nk=1nk2n2λn2(nk)(λnx)k
=1(1+λnx)n(k=2nn1nλn2(n2k2)(λnx)k+1nλn2k=1n(n1k1)(λnx)k)
=n1nx2(1+λnx)2+xnλn(1+λnx),

which leads us to the identity (7).

Remarks. (i) Since Ln, n, are linear operators, relation (5) implies that they reproduce the constants, consequently Ln are of Markov type.

(ii) (Lnf)(0)=f(0), accordingly the operators interpolate functions in x=0.

We introduce the s-th order central moment of the operator Ln, s0, that is μn,s(x)=(Lnφx,s)(x), where

φx,s(t)=(tx)s,x0,t0. (8)

In view of the identities (5)-(7), by a simple computation, for any x0, we get

{μn,0(x)=1,μn,1(x)=λnx1+λnx,μn,2(x)=λn2x4(1+λnx)2+xnλn(1+λnx)2. (9)

We mention that in the particular case (4), for s and for all integers n>x1/(1β), the moments Rn[β]es and the central moments Rn[β]((x)s,) have been determined in the paper [4, Lemma 3.1, Lemma 3.3].

In the sequel we denote by CB(+) the Banach lattice of all real-valued bounded and continuous functions on + endowed with the natural order and the sup-norm ,

f=supx0|f(x)|.

If K is a compact interval (in our case K+), the same norm is valid in the space C(K).

We also recall the notion of modulus of smoothness associated with any bounded function h defined on +. For any δ0, it is given as follows

ωh(δ)ω(h;δ) =sup{|h(x)h(x′′)|:x,x′′+,|xx′′|δ}
=sup0hδsupx0|f(x+h)f(x)|. (10)

If hC(+) and supremum in (2) is taken only for (x,x′′)K×K, K+ compact interval, we can use the writing option ωK(h;).

The significant properties of ω(h;) are presented, e.g., in [16, pp. 43-46]. Among these, we emphasize that ωh is a non-decreasing function and

ω(h;μδ)(1+μ)ω(h;δ),(δ,μ)+×+. (11)

Modulus of smoothness is continuous at δ=0, i.e.

limδ0+ω(h;δ)=ω(h;0)=0, (12)

if and only if h is uniformly continuous on its domain, [10, page 40] can be consulted.

At the end of this section we present a probabilistic approach of the studied sequence. To accomplish this, we use a classical scheme that can be found, for example, in [6, Section 5.2]. We are fixing a probability space (Ω,,P) and a random scheme on +, Z:×+2(Ω), where 2(Ω) is the space of all real square-integrable random variables on Ω. Z(n,x) is discretely distributed taking the values k/(nλn), 0kn, such that

P{Z(n,x)=knλn}=pn,k(x), where pn,k(x)=(nk)(λnx)k(1+λnx)n.

Consequently, for fCB(+) we get

(Lnf)(x)=E(fZ(n,x))=ΩfZ(n,x)𝑑P=fPZ(n,x),

which leads us to relation (2). In the above E is the expected value operator and PZ(n,x) represents the distribution of Z(n,x) with respect to P. The above identities imply Lnff.

3. Results

We show that the sequence (Ln)n1 is an approximation process in a certain space, more precisely it takes place

Theorem 1. Let the operators Ln, n, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any compact interval K+, the following relation

limnLnf=f uniformly on K (13)

occurs, provided fC(+).

Proof. Based on the properties (1) and (3), relations (5)-(7) involve

limn(Lnej)(x)=ej(x),x+,j{0,1,2}. (14)

Further, we defined the lattice homomorphism TK:C(+)C(K) given by TK(f)=f|K for every fC(+). Identities (14) can be rewritten in following form

limnTK(Lnej)=TK(ej),j{0,1,2},

uniformly on K. The Bohman-Korovkin type criterion, see, e.g., [6, Theorem 4.1.4 (vi)], implies our statement (13).

The rate of convergence will be established by using modulus of smoothness defined by (2).

Theorem 2. Let the operators Ln, n, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any function fCB(+), we have

|(Lnf)(x)f(x)|(1+x4+x)ω(f;λn),x0, (15)

where

λn=max{λn2,(nλn)1}. (16)

Proof. In order to obtain easily this quantitative result we use the following inequality proved by Shisha and Mond [18], that says: if Λ is a linear positive operator defined on C(+), then one has

|(Λf)(x)f(x)||f(x)||(Λe0)(x)1|
+((Λe0)(x)+1δ(Λe0)(x)(Λφx,2)(x))ω(f;δ), (17)

for every fCB(+), x+ and δ>0. The function φx,2 was defined by (8). To achieve the condition ω(f;δ)< for every δ>0, the original relation was given on a compact interval. Since we considered the space CB(+), the inequality of Shisha and Mond holds on the unbounded interval +. The proof of (17) is mainly based on relation

|f(x)f(y)|ω(f;|xy|),

property (11) and Schwarz’ inequality.

Choosing Λ=Ln and taking in view (5) and (9), relation (17) becomes

|(Lnf)(x)f(x)|(1+1δμn,2(x))ω(f;δ),δ>0. (18)

On the other hand, from (9) we get

μn,2(x)λn2x4+xnλnλn(x4+x). (19)

Taking in (18) δ=λn, the proof is ended.

Remarks. (i) If fC(+), considering the compact [a,b]+, the upper bound of the error of approximation indicated by (15) turns into

Lnff(1+b4+b)ω[a,b](f;λn),

where the norm is taken on the compact [a,b].

(ii) In the light of the definition of λn and relations (1), (3), we infer the following two possibilities, preserving the condition that the sequence (λn)n1 to be strictly decreasing positive.

α) λnn1/3 and limnnλn=. In this case λn=(nλn)1.

β) λn>n1/3 and limnλn=0. Now, we have λn=λn2.

It is noted that in the above, the numerical sequence (n1/3)n1 appears. This sequence was used in the first study dedicated to Bernstein type rational functions, see K. Balázs’ work [7].

In the next step we prove a relation between the local smoothness of functions and the local approximation. For the completeness of the information, we recall that a continuous function f:+ is locally Lipα on E (0<α1, E+) if it satisfies the condition

|f(x)f(y)|Mf|xy|α,(x,y)+×E, (20)

where Mf is a constant depending only on f.

Also, the distance between x+ and E is denoted by d(x,E) and defined as follows

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

Theorem 3. Let Ln, n, be defined by (2), 0<α1 and E is a subset of +. If f is locally Lipα on E, then

|(Lnf)(x)f(x)|Mf((λn)α/2max{x2α,xα/2}+2dα(x,E)),x0,

takes place, where λn is given by (15).

Proof. At the beginning we recall a classical relation stemming from Hölder’s inequality. Considering r=2α1 in the relation 1/r+1/s=1, r>0, s>0, from (2) and (5) we obtain

Ln(hα;x)(Ln(h2;x))α/2,x0,

where α(0,1] and h0. Choosing h=|φx,1|, see (8), and using (19), we get

Ln(|e1xe0|α;x)μn,2α/2(x)(λn(x4+x))α/2,x0. (21)

By using the continuity of f, it is obvious that (20) holds for any x0 and yE¯, E¯ being the closure of the set E. Let (x,x0)+×E¯ be such that |xx0|=d(x,E). We can write

|ff(x)||ff(x0)|+|f(x0)f(x)|

and applying the operators Ln (linear and positive, consequently monotone), we have

|(Lnf)(x)f(x)| Ln(|ff(x0)|;x)+|f(x)f(x0)|
Ln(Mf|e1x0e0|α;x)+Mf(xx0)α. (22)

In the well known inequality (a+b)αaα+bα (a0, b0, 0<α1), we take a=|e1xe0|, b=|xx0|e0, consequently

|e1x0e0|α|e1xe0|α+(|xx0|e0)α.

By using (5) and (21) we can write

Ln(Mf|e1x0e0|α;x) Mf(Ln(|e1xe0|α;x)+|xx0|α)
Mf((λnx4)α/2+(λnx)α/2+|xx0|α).

Returning at (3), the conclusion of our theorem is proved.

Unfortunately this class of operators is not an approximation process in some weighted spaces. We present such a negative result, indicating first the general frame of our study. Let φ be a continuous and strictly increasing function defined on + such that limxφ(x)=. Set ρ=1+φ2. We consider the weighted space

Bρ(+)={f:+|f(x)|Mfρ(x),x0},

Mf being a positive constant depending only on f, and the following subspaces of Bρ(+)

Cρ(+)=C(+)Bρ(+),
Cρ0(+)={fCρ(+)limxf(x)ρ(x) exists and is finite}.

The usual norm of these spaces is ρ defined by

fρ=supx0|f(x)|ρ(x). (23)

For any fCρ(+), based on (23), we have

|(Lnf)(x)|fρ(Lnρ)(x),n1.

Hence, the operator Ln maps Cρ(+) into Cρ(+)Bρ(+) if and only if

(Lnρ)(x)Mρ(x),x0, (24)

where M is a constant.

We recall the following result due to Gadzhiev [12, Theorem 2] which says: a sequence (An)n1 of linear and positive operators acting from Cρ(+) to Bρ(+) satisfies

limnAnffρ=0 for every fCρ0(+), (25)

if and only if

limnAnφνφνρ=0,ν=0,1,2. (26)

We mention that in [12] the result was presented without ”only if” and considering the set . Clearly, to insert ”only if” is trivial and the use of the set + does not change the result.

As it seen above, the weight function ρ not only characterizes the growth of f at infinity but also defines the test functions in Korovkin type theorem.

Choosing a very simple weight namely φ:=e1, based on (5) and (7), we find that for Ln, n1, relation (24) is fulfilled. For example, we can consider M=1+(infn1(nλn))1. Trying to apply Gadzhiev’s result for our operators, we notice that (26) does not hold for ν=2. Indeed,

supx0|(Lne2)(x)x2|1+x2=supx0|nλn3x4+2nλn2x3x|nλn(1+x2)(1+λnx)21,

consequently limnLne2e2ρ0.

We can enunciate

Theorem 4. The sequence (Ln)n1 defined by (2) is not an approximation process in the space Cρ0(+), where ρ=1+e2.

It is worth mentioning that for this particular weight we could use the fact that {e0,e1,e2} is a strict Korovkin subset in Cρ0(+), see the monograph [6, Proposition 4.2.5, statement (6)].

For a more general weight, i.e. φ(x)=xm/2, m2 fixed, we give estimates of the error |(Lnf)(x)f(x)|, n, for functions f belonging to Cρ0(+). Clearly, this does not mean that the sequence is an approximation process on the mentioned space, the evaluation being pointwise. For our purpose we use a weighted modulus of smoothness defined as follows

Ωm(f;δ)=supx00<hδ|f(x+h)f(x)|1+(x+h)m,δ>0,fBρ(+). (27)

Obviously, Ωm(f;δ)2fρ, δ>0, fBρ(+), and it possesses the following main properties ([15])

{Ωm(f;) is a monotone increasing function,Ωm(f;λδ)(λ+1)Ωm(f;δ),δ>0,λ>0,limδ0+Ωm(f;δ)=0, for any fCρ0(+). (28)

Theorem 5. Let the operators Ln, n, be defined by (2) such that the requirements (1) and (3) are fulfilled. For any function fCρ0(+), ρ=1+em with m2, the following inequality

|(Lnf)(x)f(x)|C(1+xm)x4+xΩm(f;λn) (29)

takes place, where C is a constant depending only on m, Ωm(f;) and λn are defined by (27) and (16), respectively.

Proof. For x=0 the statement is evident. Let n, fCρ0(+) and x>0 be arbitrarily fixed.

Set θx,m(t)=1+(2x+t)m. We recall |φx,1(t)|=|tx|, see (8). By using definition of Ωm(f;) and properties (28), we can write

|f(t)f(x)| (1+(x+|tx|)m)(1+1δ|tx|)Ωm(f;δ)
θx,m(t)(1+1δ|φx,1(t)|)Ωm(f;δ).

Knowing that the operators are linear positive therefore monotone, and applying Cauchy inequality, we get

|(Lnf)(x)f(x)|Ln(|ff(x)|;x)
((Lnθx,m)(x)+1δ(Lnθx,m2)(x)(Lnφx,12)(x))Ωm(f;δ). (30)

Since LnemC1(1+em), C1 a certain constant depending on m, there are positive constants C2 and C3 depending also on m, such that

(Lnθx,m)(x)C2(1+xm)and((Lnθx,m2)(x))1/2C3(1+xm).

On the other hand, based on (9), we deduce

(Lnφx,12)(x)=μn,2(x)λn(x4+x).

Returning at (30) and choosing δ=λn, the result follows.

Remark. In the following, for m2 fixed, we set ρ1=1+em and ρ2=1+em+2. Clearly, Bρ1(+)Bρ2(+). Taking in view relation (29) we can write

Lnffρ2C~Ωm(f;λn),

for every fCρ1(+), where C~=Csupx0(1+xm)x4+x1+xm+2<. Due to the assumptions (1) and (3), we obtain limnλn=0 and the last property of Ωm(f;) at (28) implies

limnLnffρ2=0,fCρ10(+).

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