# Adjoining a Constant Function ton-Dimensional Chebyshev Space

Authors: Mansour Alyazidi-Asiry
Journal: Journal of Function Spaces
 (2016)

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## Abstract

This paper is concerned with extending a Chebyshev system ofn continuous nonconstant functions into a set of n+1 functions including a constant function. Necessary and sufficient conditions for the new set to be a Chebyshev system are discussed and some results are obtained.


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## Body

## 1. Introduction

Most of the material in this section and Section2 can be found in any standard book in approximation theory and related topics; see, for example, [1–5]. The finite set of functions g1,…,gn⊂Ca,b is called a Chebyshev system on a,b if it is linearly independent and Dg1,…,gnx1,…,xn=Detgi(xj)≠0, i,j=1,…,n, for all xjj=1n such that a≤x1<x2<⋯<xn≤b, and the n-dimensional subspace G=g1,…,gn of Ca,b will be called a Chebyshev subspace or Haar subspace. Using the continuity of the determinant, it can be shown that the sign of the determinant is constant (see [6]), so we will assume that the determinant is always positive throughout this paper (replace g1 by -g1 if necessary). If each gi is continuously differentiable function on [a,b], i=1,…,n and a≤x1≤x2≤⋯≤xn≤b, then the determinant D∗g1,…,gnx1,…,xn is defined as follows: (1)D∗g1,…,gnx1,…,xn=g1t1⋯gnt1g1′t1⋯gn′t1⋮⋮g1r1-1t1⋯gnr1-1t1⋮⋮g1tp⋯gntp⋮⋮g1rp-1tp⋯gnrp-1tp,where xi is repeated ri times, i=1,…,p, a≤t1<t2<⋯<tp≤b, and t1,t2,…,tp=x1,x2,…,xn. The set of functions g1,…,gn is called an extended Chebyshev system on a,b if D∗g1,…,gnx1,…,xn>0, and the n-dimensional subspace G=g1,…,gn of Ca,b will be called an extended Chebyshev subspace.In this paper we will consider the following problem.IfG=g1,…,gn is a Chebyshev subspace of Ca,b such that 1∉G then what property must G have so that the subspace U=u0,u1,…,un is (n+1)-dimensional Chebyshev subspace of Ca,b, where u0=1, ui=gi, i=1,…,n? We will present some results in Section 3 which give a partial answer to this question.


## 2. Preliminary

LetG=g1,…,gn be a Chebyshev subspace of Ca,b and let xjj=1n+1 be a set of points such that a≤x1<x2<⋯<xn+1≤b, and then for any g∈G we have(2)0=Dg,g1,…,gnx1,…,xn+1=∑i=1n+1-1i+1Δigxi,where(3)Δ1=Dg1,…,gnx2,…,xn+1,Δn+1=Dg1,…,gnx1,…,xn,Δi=Dg1,…,gnx1,…,xi-1,xi+1,…,xn+1,i=2,…,n.Takingθi=Δi/∑j=1n+1Δj then θi>0 for all i=1,…,n+1 with ∑i=1n+1θi=1 and ∑i=1n+1-1iθig(xi)=0 for every g∈G.This discussion proves the existence part of the following lemma.Lemma 1.
LetG=g1,…,gn be a Chebyshev subspace of Ca,b and let xjj=1n+1 be a set of points such that a≤x1<x2<⋯<xn+1≤b, and then there exists a unique set of positive numbers θjj=1n+1 with ∑i=1n+1θi=1 such that ∑i=1n+1-1iθig(xi)=0 for every g∈G.Proof.
We only need to prove the uniqueness part of this lemma. Suppose that there are two sets of positive real numbersθjj=1n+1 and λjj=1n+1 with ∑i=1n+1θi=∑i=1n+1λi=1 such that(4)∑i=1n+1-1iθigxi=∑i=1n+1-1iλigxi=0for  everyg∈G.Since G is a Chebyshev subspace, then for each k∈2,…,n+1 there exists a unique function hk∈G such that hk(x1)=1 and hk(xl)=0,l∈2,…,n+1∖k (see [6]), and from (4) we have(5)-θ1+-1kθ2hkxk=0,-λ1+-1kλ2hkxk=0,where k=2,…,n+1.
Clearlyhk(xk)≠0, and therefore (5) yield(6)θkθ1=λkλ1,k=2,…,n+1.Hence (1/θ1)∑k=2n+1θk=(1/λ1)∑k=2n+1λk⇒(1-θ1)/θ1=(1-λ1)/λ1⇒θ1=λ1, and, by (6), θi=λi,  i=2,…,n+1, and the proof is complete.


## 3. The Main Result

We start this section by the following theorem.Theorem 2.
LetG=g1,…,gn, where gii=1n⊂C[a,b] is a Chebyshev system on [a,b]. Then g0=1,g1,…,gn is a Chebyshev system on [a,b] if and only if, for each set of points tii=0n such that a≤t0<t1<⋯<tn≤b and the corresponding set of positive real numbers θjj=0n with ∑i=0nθi=1 satisfying ∑i=0n-1iθig(ti)=0 for every g∈G, we have(7)∑i∈Iθi≠∑j∈Jθj,where I=i∈A:iisodd, A=0,1,…,n, and J=A∖I.Proof.
Lettii=0n be a set of points such that a≤t0<t1<⋯<tn≤b and the corresponding set of positive real numbers θjj=0n with ∑i=0nθi=1 satisfying ∑i=0n-1iθig(ti)=0 for every g∈G, where G=g1,…,gn is a Chebyshev subspace of C[a,b]. From Lemma 1  θi=Δi/d, where d=∑j=0nΔj and Δi=Dg1,…,gnt0,…,ti-1,ti+1,…,tn, i=0,…,n. Hence (8)Dg0,g1,…,gnt0,…,tn=Det⁡1g1t0⋯gnt01g1t1⋯gnt1⋮⋮⋯⋮1g1tn⋯gntn,=∑i=0n-1iΔi=∑i=0n-1idθi=d∑i∈Iθi-∑j∈Jθj≠0if and only if ∑i∈Iθi≠∑j∈Jθ, where I and J are as defined above, and the theorem is proved.Assumption 3.
Letgi∈C1[a,b] for all i=1,…,n and let G=g1,…,gn be a Chebyshev subspace of Ca,b. We say that G satisfies Assumption 3 if, for each nontrivial element g of G, g′ can have at most n-1 distinct zeros on (a,b). That is, if g′(xi)=0, a<x1<x2<⋯<xk<b, and k≥n, then g is identically zero.Theorem 4.
Supposegi∈C1[a,b] for all i=1,…,n and G=g1,…,gn is a Chebyshev subspace of Ca,b. If G satisfies Assumption 3, then g0=1,g1,…,gn is a Chebyshev system on [a,b].Proof.
If there is a functionh¯=α¯0+α¯1g1+⋯+α¯ngn such that h¯(ti)=0 at some set of points a≤t0<t1<⋯<tk≤b; then by Rolle’s theorem there exits a set of points xjj=1k, xj∈(tj-1,tj), such that h¯′xj=0=g¯′xj, j=1,…,k, where g¯=α¯1g1+⋯+α¯ngn∈G. So if k≥n then g¯=0, and hence h¯=α¯0⇒h¯=α¯0=0 and this shows that g0=1,g1,…,gn is a Chebyshev system on [a,b].Whenn=1, u∈C[a,b], and G=u is a Chebyshev subspace of C[a,b] if u(x)≠0 for all x∈[a,b]. For this special case we have the following results which can be found in [3].Proposition 5.
Letu∈C[a,b], and then H=1,u is a Chebyshev subspace of C[a,b] if and only if u is strictly monotonic function on [a,b].Remark 6.
Ifu is an even function on [-a,a], a>0, then 1,u is not a Chebyshev system on [-a,a], that is, since D1,u-t,t=0, ∀t∈0,a.Remark 7.
Assumption3 is not a necessary condition for Theorem 4 as the following example illustrates.Example 8.
Takeu=ex3 and then 1,u is a Chebyshev system on [-a,a], a>0 although u′(0)=0.Finally, we will give an example of a set of continuously differentiable functionsgii=1n which is an extended Chebyshev system on [a,b] and g0=1,g1,…,gn is a Chebyshev system on [a,b] but not an extended Chebyshev system.Example 9.
Letg1=x and g2=-cos⁡x, then D∗g1,g2t,t=tsin⁡t+cos⁡t>0, ∀t∈[0,π/2], and hence G=g1,g2 is an extended Chebyshev system on [0,π/2]. And if g∈G then g′ can have at most one zero on [0,π/2]; this means that G satisfies Assumption 3 and by Theorem 4  1,g1,g2 is a Chebyshev system on [0,π/2]. Taking v(x)=-π/2+x+cos⁡x, then v(π/2)=v′(π/2)=v′′(π/2)=0; that is, 1,g1,g2 is not an extended Chebyshev system on [0,π/2].



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