# Structure Properties for Binomial Operators

Authors: Chungou Zhang; Shifen Wang
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this paper, we discuss binomial operators structure properties, such as moments representation, derivatives representation, and binary representation and introduce some applications in preservation.


---

## Body

## 1. Introduction

As an extension to the well-known Bernstein operators, binomial operators are defined as follows (see [1], [2], or [3]):(1)LnQfx=1bn1∑k=0nnkbkxbn-k1-xfkn,f∈C0,1;if bn(1)=0, then those operators are substituted by(2)LnQfx=1bnn∑k=0nnkbknxbn-kn-nxfkn,f∈C0,1,where (bn)n≥0 is a sequence of binomial polynomials; i.e., bn(x) is a polynomial of n degree satisfying(3)bnx+y=∑k=0nnkbkxbn-ky,n=0,1,2,⋯.Q is a delta operator (see Definition 2 below), which is determined by the sequence of binomial polynomials (bn)n≥0 uniquely.In order to explore the binomial operators approximation and preservation, in present paper we are to investigate some structural properties, since behaviors of an operator are strongly dependent on its structure. In view of the Bernstein operators(4)Bnfx=∑k=0nnkxk1-xn-kfkn,f∈C0,1,with the following structural properties: endpoints interpolation, moments representation, derivatives representation, difference representation, and binary representation (see [4–10]), so our attention to the binomial operators will also focus on these aspects. On the study of Bernstein type operators, here we also want to refer to literatures [11–16].In the next section, we introduce some primary concepts and results involved in this paper, which can be found in [1, 2, 17].


## 2. Notations and Preliminaries

LetΠn be the linear space of polynomials of degree at most n and Π the space of all polynomials; i.e., Π=⋃n≥0Πn. We denote by I the identity operator and D the derivative. For real number a, the shift operator Ea:Π→Π is defined by Eap(x)=p(x+a),x∈(-∞,+∞).Definition 1.
If a linear operatorT:Π→Π commutes with all shift operators, then it is called a shift-invariant operator; i.e., TEa=EaT for any a real number a.
From [17], we can find that if T1 and T2 are shift-invariant operators, then T1T2=T2T1.Definition 2.
A shift-invariant operator Q is called a delta operator iffQe1=const≠0, where ei=ei(t)=ti,i=0,1,⋯.
For delta operators, we have the following assertion.Theorem 3.
The following statements are equivalent:
(i)Q is a delta operator.
(ii) There exists a reversible shift-invariant operatorP such that Q=DP.
(iii) There exists a power seriesϕ(t)=Σk=0∞ck(tk/k!) with c0=0,c1≠0 such that Q=ϕ(D).
Every shift-invariant operator can always be represented by any one delta operator that is so-called “First Expansion Theorem” as below.Theorem 4.
LetT be a shift-invariant operator, and let Q be a delta operator with basic polynomials bn(x); then(5)T=∑k=0∞akk!Qkwith ak=[Tbk(x)]x=0.Definition 5.
LetK:Π→Π be defined as (Kh)(t)=th(t). For an operator T:Π→Π, its Pincherle derivative T′ is defined by T′=TK-KT.
From [17], it is known that if T is a shift-invariant operator, then also is T′.Definition 6.
Let Q be a delta operator. A polynomial sequence(bn)n≥0 is called the sequence of basic polynomials associated with Q iff(6)b0x=1,bn0=0,Qbnx=nbn-1x,n≥1.
It has been proved that every delta operator has a unique sequence of basic polynomials (see [17], Proposition 3]). Moreover, if (bn)n≥0 is the basic sequence of a delta operator Q, then (bn)n≥0 is a sequence of binomial polynomials, and the converse is also right.In this paper, we also need the following assertions. IfQ is a delta operator, then Q′-1 exists and the binomial operator LnQ has the following representation.Theorem 7.
The operatorLnQ can be represented in the form(7)LnQfx=∑k=0nk!nknk0,1n,⋯kn;fdk,nx,where(8)dk,nx=1bn1ΘkE1-xbn-kx,Θ=KQ′-1and [0,1/n,⋯k/n;f] is divided difference of the function f. Using this theorem, we can get the moments of LnQ immediately(9)LnQerx=1bn1∑k=0rk!nk0,1n⋯kn;erΘkE1-xbn-kx,r=0.1.2⋯.In particular, when r=0,1,2, we have(10)LnQe0x=1,LnQe1x=x,LnQe2x=x2+1-n-1nQ′-2bn-21bn1x1-x.
According to these moments and Korovkin theorem, it is easy to see that ifLnQ is positive and (n-1/n)(Q′-2bn-21/bn1)→1, then LnQ uniformly converges to the continuous function f.
For convenience, we denote all the positive operatorsLnQ by B. Let F be the set of all formal power series ϕ(t)=∑k=1∞dk(tk/k!) with d1>0,dk≥0; then the positivity of LnQ can be characterized as follows.Theorem 8.
LetLnQ be defined as before, and Q=ϕ(D); then LnQ∈B iff(11)ϕ-1t≔∑k=0∞cktkk!∈F.Therefore, if LnQ∈B, then bn(x) has nonnegative coefficients, where (bn)n≥0 is the sequence of basic polynomials associated with Q.WhenQ=D, which is the simplest delta operator with the basic polynomials bn(x)=xn,n=0,1,⋯, its corresponding binomial operator LnQ is Bernstein operator Bn; when Q=I-E-1, which is also a delta operator and called backward difference operator with the basic polynomials bn(x)=x(x+1)⋯(x+n-1),n=0,1,⋯, its corresponding binomial operator is so-called Stancu operator (see [1] or [2]).


## 3. Some Structure Properties of the OperatorLnQ

Similar to Bernstein operators, the binomial operatorsLnQ, on which some researches can be found in [18–24], also have the property of endpoint interpolation; i.e., (LnQf)(0)=f(0) and (LnQf)(1)=f(1).Applying[x0,x1,…,xk;f]=△hk/k!hk to Theorem 7, we have the difference representation of LnQ as follows:(12)LnQfx=1bn1∑k=0nnk△1/nkf0ΘkE1-xbn-kx.As is known to all, the derivative of Bernstein operators can be expressed by difference operators (see [4]):(13)Bnrf,x=nn-1⋯n-r+1∑k=0n-r△1/nrfknbn-rkx,but the derivatives of LnQ can not be expressed in this form; actually when r=1,2 they have the following representation.Proposition 9.
LetQ be a delta operator and (bn)n≥0 be the sequence of basic polynomial associated with Q; then(14)DLnQfx=1bn1∑k=0n-1nkbn-k′0∑l=0kklblxbk-l1-x·fn-k+ln-fln;D2LnQfx=1bn1∑k=1n-1nkbn-k′0∑l=0k-1klbk-l′0∑j=0lljbjxbl-j1-x·fn-l+jn-fk-l+jn-fn-k+jn+fjn.Proof.
By Theorem4, we have(15)D=∑l=0∞bl′0l!Ql;it follows that(16)bk′x=Dbkx=∑l=0kklbl′0bk-lx,so that(17)∑k=0nnkbk′xbn-k1-xfkn=∑k=1nnk∑l=0kklbl′0bk-lxbn-k1-xfkn=∑k=1nbk′0∑l=knnllkbl-kxbn-l1-xfln=∑k=1nnkbk′0∑l=0n-kn-klblxbn-k-l1-xfk+ln.In the same way, we have(18)∑k=0nnkbkxbn-k′1-xfkn=∑k=1nnkbk′0∑l=0n-kn-klblxbn-k-l1-xfln.Therefore, we get(19)DLnQfx=1bn1∑k=1nnkbk′xbn-k1-x-1bn1∑k=1nnkbkxbn-k′1-x=1bn1∑k=1nnkbk′0∑l=0n-kn-klblxbn-k-l1-x·fk+ln-fln=1bn1∑k=0n-1nkbn-k′0∑l=0kklblxbk-l1-x·fn-k+ln-fln.
An argument similar to the above one, it is no difficult to obtain the expression for the second derivative of the operatorsD2(LnQf)(x). The proof is completed.This proposition supplies us a new proof for the next results known (see [1]).Theorem 10.
LetLnQ∈B.
(i) Iff is increasing (decreasing) on [0,1], then it also is LnQf.
(ii) Iff is convex (concave) on [0,1], then it also is LnQf.Proof.
(i) It is trivial.
(ii) Without loss of generality, we may assumef is convex. Recall that if f is convex on [0,1], then for any x1,x2,x3,x4∈[0,1] and x1<x2,x3<x4 the following holds:(20)fx1-fx2x1-x2≤fx3-fx4x3-x4.Obviously,(21)jn<k-l+jn,n-k+jn<n-l+jn,j<k<l<n,and it follows that(22)fn-k+j/n-fj/n1-k/n≤fn-l+j/n-fk-l+j/n1-k/n,so that(23)fn-l+jn-fk-l+jn-fn-k+jn+fjn=1-knfn-l+j/n-fk-l+j/n1-k/n-fn-k+j/n-fj/n1-k/n≥0.
According to Theorem8, it follows that (bn)n≥0 has nonnegative coefficients, which mean bi(x)≥0 for any x∈[0,1] and bi′(0)≥0,i=0,1,⋯. These lead to D2(LnQf)≥0, which complete the proof.The binomial operators have binary representation similar to Bernstein operators as follows.Proposition 11.
IfLnQ is defined as before, then(24)LnQfx=1bn1∑k=0nnk∑l=0n-kBnklx,yfkn;LnQfy=1bn1∑k=0nnk∑l=0n-kBnklx,yfk+ln,where Bnkl=(n!/k!l!n-k-l!)bk(x)bl(y-x)bn-k-l(1-y),x<y.Proof.
By definition of sequence of binomial polynomials, we have(25)bn-k1-x=∑l=0n-kn-klbly-xbn-k-l1-y,bkx=∑l=0kklbly-xbk-lx,and thus(26)LnQfx=1bn1∑k=0nnkbkxbn-k1-xfkn=1bn1∑k=0n∑l=0n-kn!k!l!n-k-l!bkxbly-xbn-k-l1-yfkn=1bn1∑k=0n∑l=0n-kBnklx,yfkn,and (27)LnQfy=1bn1∑k=0nnkbkybn-k1-yfkn=1bn1∑k=0n∑l=0kn!k!l!n-k-l!bly-xbk-lxbn-k1-yfkn=1bn1∑k=0n∑l=0n-kn!k!l!n-k-l!bkxbly-xbn-k-l1-yfk+ln=1bn1∑k=0n∑l=0n-kBnklx,yfk+ln,where(28)Bnkl=n!k!l!n-k-l!bkxbly-xbn-k-l1-y.This completes the proof.Using this proposition, we not only can prove Theorem 2.2(i) but also show the following result, which can be found in [3].Theorem 12.
IfLnQ∈B and f∈LipMα(0<α≤1), then LnQf∈LipMα, where(29)LipMα=f∈C0,1:∀x,y∈0,1,∃M>0,s.t.fy-fx≤My-xα.Proof.
First, we need the following two identities:(30)1bn1∑k=0nnk∑l=0n-kBnklx,y=1,1bn1∑k=0nnk∑l=0n-kBnklx,yln=y-x.The first one can be derived easily by the definition of the sequence of binomial polynomials; the rest only need to prove the second one. In fact, by the definition of Bnkl(x,y) and exchange the order of sum, we have(31)1bn1∑k=0nnk∑l=0n-kBnklx,yln=1bn1∑l=0nnlbly-xln∑k=0n-ln-lkbkxbn-l-k1-y=1bn1∑l=0nnlbly-xbn-lx+1-yln=y-x.
Now, we turn to the proof of this theorem. Since, for anyx<y and 0<α≤1, there is(32)fy-fx≤My-xα,by the identities above and LnQ∈B; we obtain(33)LnQfy-LnQfx≤1bn1∑k=0n∑l=0n-kBnklx,yfk+ln-fkn≤M1bn1∑k=0n∑l=0n-kBnklx,ylnα≤M1bn1∑k=0n∑l=0n-kBnklx,ylnα=My-xα,which means that LnQf∈LipMα. The proof of the theorem is now complete.In this paper, we discuss structure properties of the binomial operators, present the moments representation, the derivatives representation, and the binary representation of these operators, and introduce some applications in preservation. Here authors also wish to express their gratitude to the reviewers for the careful review and suggestions for improvement of this manuscript.



---

*Source: 5483451-2019-03-10.xml*