# The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights

Authors: Rong Liu; Ying Guang Shi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper gives the estimates of the zeros of orthogonal polynomials for Jacobi-exponential weights.


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## Body

## 1. Introduction and Results

This paper deals with the zeros of orthogonal polynomials for Jacobi-exponential weights. Letw be a weight in I∶=(a,b),-∞≤a<0<b≤∞, for which the moment problem possesses a unique solution. Denote by N the set of positive integers. Pn stands for the set of polynomials of degree at most n.Assume thatW=e-Q where Q:I→[0,∞) is continuous. Also, let 0<p<∞,
(1.1)a≤tr<tr-1<⋯<t2<t1≤b,pi>-1p,i=1,2,…,r,U(x)=∏i=1r‍|x-ti|pi.The lettersc,C0,C1,… stand for positive constants independent of variables and indices, unless otherwise indicated, and their values may be different at different occurrences, even in subsequent formulas. Moreover, Cn~Dn means that there are two constants c1 and c2 such that c1≤Cn/Dn≤c2 for the relevant range of n. We write c=c(λ) or c≠c(λ) to indicate dependence on or independence of a parameter λ.Definition 1.1 (see [1, Definition 1.7, page 14]).
Givenc,t≥0 and a nonnegative Borel measure ν with compact support in C and total mass ≤t, one says that
(1.2)P(z)∶=cexp(∫‍ln|z-t|dν(t))
is an exponential of a potential of mass ≤t. One  denotes the set of all such P by 𝒫t.One  notes that, forP∈Pn,
(1.3)|P|∈𝒫t,t≥n.Definition 1.2 (see [1, page 19]).
Letw be a weight in I. For 0<p<∞, generalized Christoffel functions with respect to w for z∈C are defined by
(1.4)λp,n(w;z)=infP∈Pn(∥Pw∥Lp(I)|P(z)|)p.
For p=∞, generalized Christoffel functions with respect to w for z∈C are defined by
(1.5)λ∞,n(w;z)=infP∈Pn∥Pw∥L∞(I)|P(z)|.
Obviously, for the classical Christoffel functionλn(w2;x) with respect to w2, we have
(1.6)λn(w2;x)=λ2,n-1(w;x).
A functionf:(c,d)→(0,∞) is said to be  quasi-increasing (or quasi-decreasing) if there exists C>0 such that
(1.7)f(x)≤(or≥)Cf(y),c<x≤y<d.Definition 1.3 (see [1, pages 10–12]).
Leta<0<b. Assume that W=e-Q where Q:I→[0,∞) satisfies the following properties(a)
Q′∈C(I) and Q(0)=0.(b)
Q′ is nondecreasing in I.(c)
We have(1.8)limt→a+Q(t)=limt→b-Q(t)=∞.(d)
The function(1.9)T(t)∶=tQ′(t)Q(t),t≠0,is quasi-decreasing in (a,0) and quasi-increasing in (0,b), respectively. Moreover
(1.10)T(t)≥Λ>1,t∈I∖{0}.(e)
There existsϵ0∈(0,1) such that, for y∈I∖{0},
(1.11)T(y)~T(y[1-ϵ0T(y)]).Then we write W∈ℱ.(f)
In addition, assume that there existC,ϵ1>0 such that, for all x∈I∖{0},
(1.12)∫x-ϵ1|x|/T(x)x‍|Q′(t)-Q′(x)||t-x|3/2dt≤C|Q′(x)|[T(x)|x|]1/2.Then we write W∈ℱ(Lip(1/2)).
ForW∈ℱ and t>0, the Mhaskar-Rahmanov-Saff numbers a-t∶=a-t(Q)<0<at∶=at(Q) are defined by the equations
(1.13)t=1π∫a-tat‍xQ′(x)[(x-a-t)(at-x)]1/2dx,0=1π∫a-tat‍Q′(x)[(x-a-t)(at-x)]1/2dx.
Put for t>0,
(1.14)Δt∶=Δt(Q)∶=[a-t,at],δt∶=δt(Q)∶=12(at+|a-t|),η±t∶=η±t(Q)∶=[tT(a±t)|a±t|δt]-2/3,φt(x)∶=φt(Q;x)∶={|x-a-2t||x-a2t|t[|x-a-t|+|a-t|η-t][|x-at|+atηt],x∈[a-t,at],φt(at),x∈(at,b),φt(a-t),x∈(a,a-t),JL,t∶=JL,t(Q)∶=[a-t(1+Lη-t),at(1+Lηt)],L>0,KL,t∶=KL,t(Q)∶=[-1+L(1+a-t),1-L(1-at)],L>1.
Let
(1.15)Ut(x)∶=∏i=1r‍(|x-ti|+δtt)pi,ρ∶=ρ(U)∶=∑i=1r‍max{pi,0}.In 1994 and 2001, Levin and Lubinsky [1, 2] published their monographs on orthogonal polynomials for exponential weights W2. Then they [3, 4] discussed orthogonal polynomials for exponential weights x2αW(x)2,α>-1/2, in [0,b), since the results of [1, 2] cannot be applied to such weights. Kasuga and Sakai [5] considered generalized Freud weights |x|2αW(x)2 in (-∞,∞). Recently the second author [6] obtained the Lp Christoffel functions for Jacobi-exponential weights UW, which are the combination of the two best important weights: Jacobi weight and the exponential weight, and restricted range inequalities.Theorem 1.4 (see [6, Theorem 1.1]).
LetW∈ℱ(Lip(1/2)),L>0, and 0<p<∞. Assume that
(1.16)limt→∞|a-t|at=γ,0<γ<∞.
Then there exists n0>0 such that, for n≥n0 and x∈JL,n, the relation
(1.17)λp,n(UW;x)~φn(x)Un(x)pW(x)p
uniformly holds.Theorem 1.5 (see [6, Theorem 1.2]).
LetW=e-Q(x), where Q:I→[0,∞) is convex with Q(a+)=Q(b-)=∞ and Q(x)>Q(0)=0,x∈I∖{0}. Let 0<p≤∞. Assume that relation (1.16) is valid. Then there exist C,t0>0 such that, for t≥t0 and P∈𝒫t-ρ-2/p,
(1.18)∥PUW∥Lp(I)≤C∥PUW∥Lp(Δt),∥PUtW∥Lp(I)≤C∥PUtW∥Lp(Δt).Theorem 1.6 (see [6, Theorem 1.3]).
LetW∈ℱ(Lip(1/2)),L>0, and 0<p<∞. Assume that relation (1.16) is valid. Then there exist C,t0>0 such that, for t≥t0 and P∈𝒫t,
(1.19)∥PUW∥Lp(I)≤C∥PUW∥Lp([a-t(1-Lη-t),at(1-Lηt)]).In this paper we discuss the zeros of orthogonal polynomials for Jacobi-exponential weightsUW and restricted range inequalities.Theorem 1.7.
LetW∈ℱ(Lip(1/2)). Assume that (1.16) is valid, and
(1.20)a<tr<⋯<t1<b,(1.21)φt(x)=O(1),t→∞.
Then
(1.22)xkn-xk+1,n≤cφn(xkn),k=1,2,…,n-1.Theorem 1.8.
LetW=e-Q(x), where Q:I→[0,∞) is convex with Q(a+)=Q(b-)=∞ and Q(x)>Q(0)=0,x∈I∖{0}. Let 0<p≤∞. Assume that all pi are positive and relation (1.16) is valid. Then there exist t0>0 such that, for t≥t0 and P∈𝒫t-ρ-2/p,
(1.23)∥PUW∥Lp(I∖Δt)≤∥PUW∥Lp(Δt).Theorem 1.9.
Let the assumptions of Theorem1.8 prevail. Then
(1.24)x1n<an+ρ+1/2,(1.25)xnn>a-n-ρ-1/2.Theorem 1.10.
LetW∈ℱ(Lip (1/2)). Then
(1.26)x1n≥an(1-cηn),(1.27)xnn≤a-n(1-cη-n).
If all pi≥0, then
(1.28)1-x1nan~ηn,(1.29)1-xnna-n~η-n.Here we should point out that our main result (Theorem1.7) cannot follow from [7] given by Mastroianni and Totik, because in general Jacobi-exponential weights UW are not doubling weights, although Jacobi weights U are doubling weights. A doubling weight means that the measure of a twice enlarged interval is less than a constant times the measure of the original interval. For example, for W(t)=exp(-t2), by L'Hospital rule
(1.30)limd→∞∫d/25d/2‍exp(-t2)dt∫d2d‍exp(-t2)dt=limd→∞exp(-(5d/2)2)-exp(-(d/2)2)exp(-(2d)2)-exp(-d2)=limd→∞exp(3d2/4)-exp(-21d2/4)1-exp(-3d2)=∞,
which implies that W(t)=exp(-t2) is not a doubling weight.We will give some auxiliary lemmas in Section2 and the proofs of Theorems 1.7–1.10 in Section 3, respectively.


## 2. Auxiliary Lemmas

Lemma 2.1 (Levin and Lubinsky [1, Lemma 3.5, pages 71-72]).
LetW∈ℱ. Then for fixed L>1 and uniformly for t>0,
(2.1)aLt~at.
Moreover, there existsτ0>0 such that, for t≥τ≥τ0, the inequalities
(2.2)1≤δtδτ≤c(tτ)1/Λ
hold.Lemma 2.2 (Shi [6]).
LetW∈ℱ. Then, for large enough t,
(2.3)a2t≥at(1+ηt).Lemma 2.3.
LetI=(-1,1),W∈ℱ, and L>1. Then, for x∈KL,t,
(2.4)φt(x)~1t[(a2t-x)(x-a-2t)]1/2,(2.5)φt(x)≤cδtt.Proof.
By the same argument as that of [8, (2.25)] we can prove (2.4). By (2.4) and (2.1) for x∈KL,t,
(2.6)φt(x)≤ct·12(a2t-a-2t)≤cδtt.Lemma 2.4.
LetW∈ℱ. Then, for x∈I,
(2.7)φt(x)≤cδtt2/3T(at)1/6.Proof.
By the definition ofφt it is enough to prove (2.7) for x∈Δt. Without loss of generality we can assume that 0≤x≤at. By Lemma 3.11(b) in  [1, page 81] for t>0,
(2.8)|a2tat-1|~1T(at).
By Lemma 2.12 in [8], (2.3), (2.1), and (2.8),
(2.9)S(x)=a2t-xat(1+ηt)-x·a-2t-xa-t(1+η-t)-x≤a2t-atatηt·a-2ta-t(1+η-t)≤ca2t/at-1ηt≤cηtT(at).
By (1.63) in [1, page 15],
(2.10)ηtT(at)≥t-2/3T(at)1/3
and hence
(2.11)S(x)≤ct2/3T(at)-1/3.
Thus
(2.12)φt(x)=[(a2t-x)(x-a-2t)]1/2tS(x)1/2≤cδtt[t2/3T(at)-1/3]1/2=cδtt2/3T(at)1/6.LetIk=[xk+1,n,xkn],dk=xkn-xk+1,n,k=1,2,…,n-1. Let, for n≥n0 and d∶=min1≤i≤r-1(ti-ti+1),
(2.13)max1≤k≤n-1dk≤d4.Lemma 2.5.
For fixed indexk,  1≤k≤n-1, let j,1≤j≤r, satisfy
(2.14)minx∈Ik|x-tj|=min1≤i≤rminx∈Ik|x-ti|.
Then
(2.15)∏i≠j‍|xκn-ti|pi~∏i≠j‍(|xκn-ti|+δnn)pi~∏i≠j‍|x-ti|pi,x∈Ik,κ=k,k+1.Proof.
We give the proof of (2.15) for κ=k only, the proof of (2.15) for κ=k+1 being similar.
We claim that, fori≠j,
(2.16)|xkn-ti|≥38d.
In fact, suppose without loss of generality that xkn≥tj. It is enough to show (2.16) for i=j-1. Because |xkn-tj+1|≥tj-tj+1≥d.
Iftj∈Ik then by (2.13)
(2.17)|xkn-xk+1,n|≤d4≤tj-1-tj≤|tj-1-xkn|+|xkn-tj|
and hence
(2.18)|xk+1,n-tj|≤|xkn-tj-1|;
if tj∉Ik then by (2.14)
(2.19)|xk+1,n-tj|=minx∈Ik|x-tj|≤minx∈Ik|x-tj-1|=tj-1-xkn,
which again implies (2.18). Then by (2.18)
(2.20)d≤|tj-1-tj|≤|tj-1-xkn|+|xkn-xk+1,n|+|xk+1,n-tj|≤2|xkn-tj-1|+dk≤2|xkn-tj-1|+14d
and hence |xkn-tj-1|≥3d/8. This proves (2.16).
With the help of (2.16) for x∈Ik and i≠j,
(2.21)|x-ti|≤|xkn-ti|+|x-xkn|≤|xkn-ti|+d4≤53|xkn-ti|,|x-ti|≥|xkn-ti|-|x-xkn|≥|xkn-ti|-d4≥13|xkn-ti|.
Hence
(2.22)|x-ti|~|xkn-ti|.
Furthermore, by (2.2) with τ=1(2.23)δtt≤cδ1t1/Λ-1=o(1),t→∞.
So for i≠j,
(2.24)|xkn-ti|~|xkn-ti|+δnn.
This proves (2.15).By the same argument as that of Lemma7.2.7 in [9, page 157] replacing 1/n by Cn, we can get its extension.Lemma 2.6.
Letp≥0,Bn≥An≥0,Cn≥0,σ=±1, and
(2.25)Bnp+1+σAnp+1≤CCn[(Bn+Cn)p+(An+Cn)p].
Then
(2.26)Bn+σAn≤cCn.Lemma 2.7.
LetW∈ℱ. Let (1.16), (1.20), and (1.21) prevail. Then there exists t0>0 such that, for t≥t0 and for each index j,1≤j≤r,
(2.27)|x-tj|+δtt~|x-tj|+φt(x)
holds uniformly for x∈I.Proof.
Let0<ϵ<min{b-t1,tr-a} and Δ=[tr-ϵ,t1+ϵ]. We separate two cases.
Case1 (x∈Δ). In this case by (2.3) and (2.1),
(2.28)φt(x)≥1t[(a2t-x)(x-a-2t)]1/2≥1t[(a2t-t1-ϵ)(tr-ϵ-a-2t)]1/2≥cδtt
which coupled with (2.5) gives
(2.29)φt(x)~δtt.
Hence (2.27) follows.
Case2 (x∉Δ). In this case by (2.23),
(2.30)|x-tj|≥ϵ≥cδtt
and by (1.21)
(2.31)|x-tj|≥ϵ≥cφt(x).
Again (2.27) follows.Corollary 2.8.
LetW∈ℱ. Let (1.16) and (1.20) prevail. If
(2.32)δtt2/3T(at)1/6=O(1),t→∞
then (2.27) holds.
In particular, ifΛ≥3/2 then (2.32), (1.21), and (2.27) hold.Proof.
By (2.7) relation (2.32) implies (1.21). Then by Lemma 2.7 relation (2.27) is valid.
In particular, ifΛ≥3/2 then by (2.2) with τ=τ0 relation (2.32) is valid and hence (1.21) and (2.27) hold.


## 3. Proof of Theorems

### 3.1. Proof of Theorem1.7

Denote byℓkn's the fundamental polynomials based on the zeros xkn's. By Theorem 1.4 and Lemma 11.8 in [8, pages 320-321]
(3.1)λn(WU;xkn)W(xkn)-2+λn(WU;xk+1,n)W(xk+1,n)-2=∫I‍[ℓkn(t)2W(xkn)-2+ℓk+1,n(t)2W(xk+1,n)-2]W(t)2U(t)2dt≥∫xk+1,nxkn‍[ℓkn(t)2W(xkn)-2+ℓk+1,n(t)2W(xk+1,n)-2]W(t)2U(t)2dt≥12∫xk+1,nxkn‍U(t)2dt.
On the other hand, by Theorem 1.4,
(3.2)λn(WU;xkn)W(xkn)-2+λn(WU;xk+1,n)W(xk+1,n)-2≤c[φn(xkn)Un(xkn)2+φn(xk+1,n)Un(xk+1,n)2].
Then for φ-n(xkn)∶=max{φn(xkn),φn(xk+1,n)},
(3.3)∫xk+1,nxkn‍U(t)2dt≤cφ-n(xkn)[Un(xkn)2+Un(xk+1,n)2].Letj be defined by (2.14). Using Lemma 2.5 it follows from (3.3) that
(3.4)∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn)[(|xkn-tj|+δnn)2pj+(|xk+1,n-tj|+δnn)2pj].
Further, by (2.27),
(3.5)∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn){[|xkn-tj|+φ-n(xkn)]2pj+[|xk+1,n-tj|+φ-n(xkn)]2pj}.
By calculation from (3.5) we get
(3.6)12pj+1[|xkn-tj|2pj+1+σ|xk+1,n-tj|2pj+1]=∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn){[|xkn-tj|+φ-n(xkn)]2pj+[|xk+1,n-tj|+φ-n(xkn)]2pj},
where
(3.7)σ={1,tj∈Ik,-1,tj∉Ik.
We separate two cases.Case1 (pj≥0). Using Lemma 2.6 it follows from (3.6) that
(3.8)xkn-xk+1,n≤cφ-n(xkn).Case2 (pj<0). Suppose without loss of generality that xk+1,n>tj for the case when tj∉Ik. By (3.6),
(3.9)12pj+1[|xkn-tj|2pj+1+σ|xk+1,n-tj|2pj+1]=∫xk+1,nxkn‍|t-tj|2pjdt≤c0φ-n(xkn)min{φ-n(xkn)2pj,|xk+1,n-tj|2pj}.Subcase 2.1 (tj∈Ik). Inequality (3.9) gives
(3.10)|xκn-tj|2pj+1≤cφ-n(xkn)2pj+1,κ=k,k+1
which yields (3.8).Subcase 2.2 (tj∉Ik). In this case we distinguish two subcases.(1)
|xk+1,n-tj|≥2c0φ-n(xkn), where c0 is given by (3.9). In this case
(3.11)∫xk+1,nxkn‍(t-tj)2pjdt=∫xk+1,nxkn‍(t-tj)(t-tj)2pj-1dt≥(xk+1,n-tj)∫xk+1,nxkn‍(t-tj)2pj-1dt=(xk+1,n-tj)12|pj|[(xk+1,n-tj)2pj-(xkn-tj)2pj]≥c0φ-n(xkn)|pj|[(xk+1,n-tj)2pj-(xkn-tj)2pj],which by (3.9) gives
(3.12)(xk+1,n-tj)2pj≤(1-|pj|)-1(xkn-tj)2pj≤2(xkn-tj)2pj.
On the other hand, by (3.9) and (3.12),
(3.13)c0φ-n(xkn)(xk+1,n-tj)2pj≥∫xk+1,nxkn‍(t-tj)2pjdt≥(xkn-tj)2pj(xkn-xk+1,n)≥12(xk+1,n-tj)2pj(xkn-xk+1,n)
and hence (3.8) follows.(2)
|xk+1,n-tj|<2c0φ-n(xkn). By (3.9),
(3.14)c0φ-n(xkn)2pj+1≥12pj+1[(xkn-tj)2pj+1-(xk+1,n-tj)2pj+1]≥12pj+1[(xkn-tj)2pj+1-(2c0φ-n(xkn))2pj+1].So xkn-tj≤cφ-n(xkn) and (3.8) follows.Finally, applying Theorem5.7(b) in [1, page 125] we conclude φ-n(xkn)~φn(xkn) and hence (1.22) follows from (3.8).


### 3.2. Proof of Theorem1.8

ForP∈𝒫t-ρ-2/p, we have PU∈𝒫t-2/p and hence apply Theorem 1.8 in [1, page 15] to obtain (1.23).


### 3.3. Proof of Theorem1.9

Use the same argument as that of Theorem 11.1 in [1, page 313].


### 3.4. Proof of Theorem1.10

We give the proofs of (1.26) and (1.28) only, the proofs of (1.27) and (1.29) being similar.First let us prove (1.26). Choose α,β>1 so that
(3.15)1α+1β=1,2βpi>-1,i=1,…,r.
Let Ln denote the linear map of Δn onto [-1,1]. By Lemma 11.7 in [1, page 318] there exists yn∈Δn such that
(3.16)Ln(yn)=cos2πm,m=m(n),
and for large enough n and Rn∈Pn-2m such that
(3.17)Rn(x)W(x)1/α≥C1,x∈[0,yn],(3.18)∥RnW1/α∥L∞(I)≤C2.
Using (11.7) in [1, page 318] in the form
(3.19)1-x1nan=minP∈Pn-1∫I‍(1-x/an)(PUW)2(x)dx∫I‍(PUW)2(x)dx.
Again choose [1, page 319]
(3.20)P(x)=Rn(x)Vm,cos(2π/m)(Ln(x))2∈Pn-2.
Applying Theorem 1.5 and (3.18), and using the same argument as that in [1, pages 319-320], we can get
(3.21)∫I‍(1-xan)(PUW)2(x)dx≤c∫Δn‍(1-xan)(PUW)2(x)dx=c∫Δn‍[(1-xan)(P(x)W(x)1/α)2][(U(x)W(x)1/β)2]dx≤c{∫Δn‍[(1-xan)(P(x)W(x)1/α)2]αdx}1/α{∫Δn‍[(U(x)W(x)1/β)2]βdx}1/β≤c{∫Δn‍[(1-xan)(P(x)W(x)1/α)2]αdx}1/α≤c{∫Δn‍[(1-xan)Vm,cos(2π/m)(Ln(x))4]αdx}1/α=cδnan{∫Δn‍[(1-Ln(x))Vm,cos(2π/m)(Ln(x))4]αdx}1/α=cδn2an{∫-11‍[(1-t)Vm,cos(2π/m)(t)4]αdt}1/α≤cδn2anm4{∫-∞∞‍[(1+|v|)min{1,c|v|}4]αdv}1/α≤canηn2.On the other hand, by (3.17),
(3.22)∫I‍(PUW)2(x)dx≥∫yn(1-C1ηn)yn‍(PUW)2(x)dx≥∫yn(1-C1ηn)yn‍Vm,cos(2π/m)(Ln(x))4U(x)2dx.
By (1.20) for large enough n, we have
(3.23)U(x)≥c>0,x∈[yn(1-C1ηn),yn].
Hence (3.22) implies
(3.24)∫I‍(PUW)2(x)dx≥c∫yn(1-C1ηn)yn‍Vm,cos(2π/m)(Ln(x))4dx=cδn∫cos(2π/m)-C1ynηn/δncos(2π/m)‍Vm,cos(2π/m)(t)4dt.
But in [1, page 320] the following estimate is given:
(3.25)δn∫cos(2π/m)-C1ynηn/δncos(2π/m)‍Vm,cos(2π/m)(t)4dt≥canηn.
Substituting this estimate into (3.24) gives
(3.26)∫I‍(PUW)2(x)dx≥canηn
which coupled with (3.21) yields (1.26).Next let us prove (1.28). We already know that
(3.27)an(1-cηn)≤x1n<an+ρ+1/2=an(1+o(ηn)),
by (1.26) and (1.24). We must prove that, for some c1>0, and n large enough, we have
(3.28)x1n<an(1-c1ηn).We use the idea for the proof of Corollary13.4(b) in [1, pages 380-381] with modification. By the same argument as that proof with A=an+ρ+1/2(1-εηn) instead, applying Theorem 1.8 we obtain
(3.29)1-x1nA=λn((UW)2,x1n)-1∫I‍(1-xA)(ℓ1nUW)(x)2dx,(3.30)∫I‍(1-xA)(ℓ1nUW)(x)2dx=∫aA‍|1-xA|(ℓ1nUW)(x)2dx-∫Ab‍|1-xA|(ℓ1nUW)(x)2dx≥∫aA‍|1-xA|(ℓ1nUW)(x)2dx-∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx-∫Δn+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx≥-2∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx,
where ℓ1n denotes the fundamental polynomial of Lagrange interpolation based on the zeros of the nth orthogonal polynomial with respect to the weight (UW)2.But(3.31)∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx≤(an+ρ+1/2A-1)∫I‍(ℓ1nUW)(x)2dx=(an+ρ+1/2A-1)λn((UW)2,x1n)
which, coupled with (3.30) and (3.29), gives
(3.32)1-x1nA≥-cεηn.
Thus
(3.33)x1nan=x1nAAan+ρ+1/2an+ρ+1/2an≤(1+cεηn)(1-εηn)(1+o(ηn))<1-c1ηn,
for n large enough, provided ε>0 is small enough.


## 3.1. Proof of Theorem1.7

Denote byℓkn's the fundamental polynomials based on the zeros xkn's. By Theorem 1.4 and Lemma 11.8 in [8, pages 320-321]
(3.1)λn(WU;xkn)W(xkn)-2+λn(WU;xk+1,n)W(xk+1,n)-2=∫I‍[ℓkn(t)2W(xkn)-2+ℓk+1,n(t)2W(xk+1,n)-2]W(t)2U(t)2dt≥∫xk+1,nxkn‍[ℓkn(t)2W(xkn)-2+ℓk+1,n(t)2W(xk+1,n)-2]W(t)2U(t)2dt≥12∫xk+1,nxkn‍U(t)2dt.
On the other hand, by Theorem 1.4,
(3.2)λn(WU;xkn)W(xkn)-2+λn(WU;xk+1,n)W(xk+1,n)-2≤c[φn(xkn)Un(xkn)2+φn(xk+1,n)Un(xk+1,n)2].
Then for φ-n(xkn)∶=max{φn(xkn),φn(xk+1,n)},
(3.3)∫xk+1,nxkn‍U(t)2dt≤cφ-n(xkn)[Un(xkn)2+Un(xk+1,n)2].Letj be defined by (2.14). Using Lemma 2.5 it follows from (3.3) that
(3.4)∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn)[(|xkn-tj|+δnn)2pj+(|xk+1,n-tj|+δnn)2pj].
Further, by (2.27),
(3.5)∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn){[|xkn-tj|+φ-n(xkn)]2pj+[|xk+1,n-tj|+φ-n(xkn)]2pj}.
By calculation from (3.5) we get
(3.6)12pj+1[|xkn-tj|2pj+1+σ|xk+1,n-tj|2pj+1]=∫xk+1,nxkn‍|t-tj|2pjdt≤cφ-n(xkn){[|xkn-tj|+φ-n(xkn)]2pj+[|xk+1,n-tj|+φ-n(xkn)]2pj},
where
(3.7)σ={1,tj∈Ik,-1,tj∉Ik.
We separate two cases.Case1 (pj≥0). Using Lemma 2.6 it follows from (3.6) that
(3.8)xkn-xk+1,n≤cφ-n(xkn).Case2 (pj<0). Suppose without loss of generality that xk+1,n>tj for the case when tj∉Ik. By (3.6),
(3.9)12pj+1[|xkn-tj|2pj+1+σ|xk+1,n-tj|2pj+1]=∫xk+1,nxkn‍|t-tj|2pjdt≤c0φ-n(xkn)min{φ-n(xkn)2pj,|xk+1,n-tj|2pj}.Subcase 2.1 (tj∈Ik). Inequality (3.9) gives
(3.10)|xκn-tj|2pj+1≤cφ-n(xkn)2pj+1,κ=k,k+1
which yields (3.8).Subcase 2.2 (tj∉Ik). In this case we distinguish two subcases.(1)
|xk+1,n-tj|≥2c0φ-n(xkn), where c0 is given by (3.9). In this case
(3.11)∫xk+1,nxkn‍(t-tj)2pjdt=∫xk+1,nxkn‍(t-tj)(t-tj)2pj-1dt≥(xk+1,n-tj)∫xk+1,nxkn‍(t-tj)2pj-1dt=(xk+1,n-tj)12|pj|[(xk+1,n-tj)2pj-(xkn-tj)2pj]≥c0φ-n(xkn)|pj|[(xk+1,n-tj)2pj-(xkn-tj)2pj],which by (3.9) gives
(3.12)(xk+1,n-tj)2pj≤(1-|pj|)-1(xkn-tj)2pj≤2(xkn-tj)2pj.
On the other hand, by (3.9) and (3.12),
(3.13)c0φ-n(xkn)(xk+1,n-tj)2pj≥∫xk+1,nxkn‍(t-tj)2pjdt≥(xkn-tj)2pj(xkn-xk+1,n)≥12(xk+1,n-tj)2pj(xkn-xk+1,n)
and hence (3.8) follows.(2)
|xk+1,n-tj|<2c0φ-n(xkn). By (3.9),
(3.14)c0φ-n(xkn)2pj+1≥12pj+1[(xkn-tj)2pj+1-(xk+1,n-tj)2pj+1]≥12pj+1[(xkn-tj)2pj+1-(2c0φ-n(xkn))2pj+1].So xkn-tj≤cφ-n(xkn) and (3.8) follows.Finally, applying Theorem5.7(b) in [1, page 125] we conclude φ-n(xkn)~φn(xkn) and hence (1.22) follows from (3.8).


## 3.2. Proof of Theorem1.8

ForP∈𝒫t-ρ-2/p, we have PU∈𝒫t-2/p and hence apply Theorem 1.8 in [1, page 15] to obtain (1.23).


## 3.3. Proof of Theorem1.9

Use the same argument as that of Theorem 11.1 in [1, page 313].


## 3.4. Proof of Theorem1.10

We give the proofs of (1.26) and (1.28) only, the proofs of (1.27) and (1.29) being similar.First let us prove (1.26). Choose α,β>1 so that
(3.15)1α+1β=1,2βpi>-1,i=1,…,r.
Let Ln denote the linear map of Δn onto [-1,1]. By Lemma 11.7 in [1, page 318] there exists yn∈Δn such that
(3.16)Ln(yn)=cos2πm,m=m(n),
and for large enough n and Rn∈Pn-2m such that
(3.17)Rn(x)W(x)1/α≥C1,x∈[0,yn],(3.18)∥RnW1/α∥L∞(I)≤C2.
Using (11.7) in [1, page 318] in the form
(3.19)1-x1nan=minP∈Pn-1∫I‍(1-x/an)(PUW)2(x)dx∫I‍(PUW)2(x)dx.
Again choose [1, page 319]
(3.20)P(x)=Rn(x)Vm,cos(2π/m)(Ln(x))2∈Pn-2.
Applying Theorem 1.5 and (3.18), and using the same argument as that in [1, pages 319-320], we can get
(3.21)∫I‍(1-xan)(PUW)2(x)dx≤c∫Δn‍(1-xan)(PUW)2(x)dx=c∫Δn‍[(1-xan)(P(x)W(x)1/α)2][(U(x)W(x)1/β)2]dx≤c{∫Δn‍[(1-xan)(P(x)W(x)1/α)2]αdx}1/α{∫Δn‍[(U(x)W(x)1/β)2]βdx}1/β≤c{∫Δn‍[(1-xan)(P(x)W(x)1/α)2]αdx}1/α≤c{∫Δn‍[(1-xan)Vm,cos(2π/m)(Ln(x))4]αdx}1/α=cδnan{∫Δn‍[(1-Ln(x))Vm,cos(2π/m)(Ln(x))4]αdx}1/α=cδn2an{∫-11‍[(1-t)Vm,cos(2π/m)(t)4]αdt}1/α≤cδn2anm4{∫-∞∞‍[(1+|v|)min{1,c|v|}4]αdv}1/α≤canηn2.On the other hand, by (3.17),
(3.22)∫I‍(PUW)2(x)dx≥∫yn(1-C1ηn)yn‍(PUW)2(x)dx≥∫yn(1-C1ηn)yn‍Vm,cos(2π/m)(Ln(x))4U(x)2dx.
By (1.20) for large enough n, we have
(3.23)U(x)≥c>0,x∈[yn(1-C1ηn),yn].
Hence (3.22) implies
(3.24)∫I‍(PUW)2(x)dx≥c∫yn(1-C1ηn)yn‍Vm,cos(2π/m)(Ln(x))4dx=cδn∫cos(2π/m)-C1ynηn/δncos(2π/m)‍Vm,cos(2π/m)(t)4dt.
But in [1, page 320] the following estimate is given:
(3.25)δn∫cos(2π/m)-C1ynηn/δncos(2π/m)‍Vm,cos(2π/m)(t)4dt≥canηn.
Substituting this estimate into (3.24) gives
(3.26)∫I‍(PUW)2(x)dx≥canηn
which coupled with (3.21) yields (1.26).Next let us prove (1.28). We already know that
(3.27)an(1-cηn)≤x1n<an+ρ+1/2=an(1+o(ηn)),
by (1.26) and (1.24). We must prove that, for some c1>0, and n large enough, we have
(3.28)x1n<an(1-c1ηn).We use the idea for the proof of Corollary13.4(b) in [1, pages 380-381] with modification. By the same argument as that proof with A=an+ρ+1/2(1-εηn) instead, applying Theorem 1.8 we obtain
(3.29)1-x1nA=λn((UW)2,x1n)-1∫I‍(1-xA)(ℓ1nUW)(x)2dx,(3.30)∫I‍(1-xA)(ℓ1nUW)(x)2dx=∫aA‍|1-xA|(ℓ1nUW)(x)2dx-∫Ab‍|1-xA|(ℓ1nUW)(x)2dx≥∫aA‍|1-xA|(ℓ1nUW)(x)2dx-∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx-∫Δn+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx≥-2∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx,
where ℓ1n denotes the fundamental polynomial of Lagrange interpolation based on the zeros of the nth orthogonal polynomial with respect to the weight (UW)2.But(3.31)∫Aan+ρ+1/2‍|1-xA|(ℓ1nUW)(x)2dx≤(an+ρ+1/2A-1)∫I‍(ℓ1nUW)(x)2dx=(an+ρ+1/2A-1)λn((UW)2,x1n)
which, coupled with (3.30) and (3.29), gives
(3.32)1-x1nA≥-cεηn.
Thus
(3.33)x1nan=x1nAAan+ρ+1/2an+ρ+1/2an≤(1+cεηn)(1-εηn)(1+o(ηn))<1-c1ηn,
for n large enough, provided ε>0 is small enough.



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*Source: 386359-2012-11-18.xml*