# The Zeros of Orthogonal Polynomials and Markov–Bernstein Inequalities for Jacobi-Exponential Weights on (−1,1)

Authors: Rong Liu
Journal: Journal of Mathematics
 (2020)

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## Abstract

LetUx=∏i=1rx−tipi, 0<p<∞, −1=tr<tr−1<⋯<t2<t1=1,r≥2,pi>−1/p,i=1,2,…,r, and W=e−Qx where Q:−1,1⟶0,∞. We give the estimates of the zeros of orthogonal polynomials for the Jacobi-Exponential weight WU on −1,1. In addition, Markov–Bernstein inequalities for the weight WU are also obtained.


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## Body

## 1. Introduction and Results

Letw be a weight in I≔a,b,−∞≤a<0<b≤∞, for which the moment problem possesses an unique solution. Pn stands for the set of polynomials of degree at most n. ⋅LpI is an usual (weighted) Lp (quasi) norm on interval I.Assume thatW=e−Q where Q:I⟶0,∞ is continuous. W is an exponential weight on I. Also, let 0<p<∞,a≤tr<tr−1<⋯<t2<t1≤b,pi>−1/p,i=1,2,…,r, and(1)Ux=∏i=1rx−tipi,where U is a generalized Jacobi weight on I. The combination WU is called a Jacobi-exponential weight on I. This paper deals with the zeros of orthogonal polynomials and Markov–Bernstein inequalities for Jacobi-exponential weights.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=cλ or c≠cλ to indicate dependence on or independence of a parameter λ.Definition 1 (see [1], Definition 1.7, p. 14).
Givenc,t≥0 and a non-negative Borel measure ν with compact support in C and total mass ≤t, we say that(2)Pz≔cexp∫lnz−sdνs,is an exponential of a potential of mass ≤t. We denote the set of all such P by Pt.
We note that forP∈Pn, P∈Pt,t≥n.Definition 2 (see [1], p. 19).
Letw be a weight in I. For 0<p<∞, generalized Christoffel functions with respect to w for z∈C are defined by(3)λp,nw;z=infP∈PnPwLpIPzp.
Forp=∞, generalized Christoffel functions with respect to w for z∈C are defined by(4)λ∞,nw;z=infP∈PnPwL∞IPz.
Moreover, for the classical Christoffel functionλnw2;x with respect to w2, we have(5)λnw2;x=infP∈Pn−1∫IPw2tdtP2x=λ2,n−1w;x.
A functionf:c,d⟶0,∞ is said to be quasi-increasing (or quasidecreasing) if there exists C>0 such that fx≤or≥Cfy,c<x≤y<d.Definition 3.
(see [1], pp. 10–12). Let a<0<b. Assume that W=e−Q where Q:I⟶0,∞ satisfies the following properties:(a)
Q′∈CI and Q0=0.(b)
Q′ is nondecreasing in I.(c)
limx⟶a+Qx=limx⟶b−Qx=∞.(d)
The function(6)Tx≔xQ′xQx,x≠0,is quasidecreasing ina,0 and quasi-increasing in 0,b, respectively. Moreover, Tx≥Λ>1,x∈I\0.(e)
There existsϵ0∈0,1 such that for y∈I\0,(7)Ty∼Ty1−ϵ0Ty.Then, we writeW∈ℱ.(f)
Furthermore, assume that there existC,ϵ1>0 such that for all x∈I\0,(8)∫x−ϵ1x/TxxQ′s−Q′xs−x3/2ds≤CQ′xTxx1/2.
Then, we writeW∈ℱLip1/2.
In addition, letW∈ℱ. Assume that there exist C,ϵ1>0 such that for all x∈I\0,(9)∫x−ϵ1x/Txx+ϵ1x/TxQ′s−Q′xs−xds≤CQ′x.
Then, we writeW∈ℱDini.
ForW∈ℱ and t>0, the Mhaskar–Rahmanov–Saff numbers a−t≔a−tQ<0<at≔atQ are defined by the equations(10)t=1π∫a−tatxQ′xx−a−tat−x1/2dx,0=1π∫a−tatQ′xx−a−tat−x1/2dx.
Put fort>0,(11)Δt≔ΔtQ≔a−t,at,δt≔δtQ≔12at+a−t,(12)η±t≔η±tQ≔tTa±ta±tδt−2/3,(13)φtx≔φtQ;x≔x−a−2tx−a2ttx−a−t+a−tη−tx−at+atηt,x∈a−t,at,φtat,x∈at,b,φta−t,x∈a,a−t,JL,t≔JL,tQ≔a−t1+Lη−t,at1+Lηt,L>0,KL,t≔KL,tQ≔−1+L1+a−t,1−L1−at,L>1.
In 1994 and 2001, Levin and Lubinsky [1, 2] discussed orthogonal polynomials for exponential weights W2 on −1,1 and a,b,a<0<b, respectively. Then, they [3, 4] dealt with exponential weights x2αWx2,α>−1/2, in 0,b. Kasuga and Sakai [5] considered generalized Freud weights x2αWx2 in −∞,∞. Recently, we discussed generalized Jacobi-exponential weights UW[6, 7], which centered on the distribution of zeros and the estimates of the generalized Christoffel functions, respectively. Shi [8] also considered Jacobi-exponential weights UW and subsequently dealt with a particular case 1−x2ρe−Qx on −1,1 in [9].
For the weightUW on −1,1, its nth orthogonal polynomial pnUW2,x has zeros xknk=1n, where −1<xnn<xn−1,n<⋯<x2n<x1n<1.
The estimates of the zeros [6] are based on the condition a<tr<tr−1<⋯<t2<t1<b. In [6], we did not consider the case when a=tr and t1=b, which is different from a<tr,t1<b. In this paper, we discuss orthogonal polynomials for generalized Jacobi-exponential weights in the case −1=tr<tr−1<⋯<t2<t1=1.
Mastroianni and Totik in [10] gave the estimates of the spacing of zeros for doubling weights; in general, however, Jacobi-exponential weights UW are not doubling weights, so our main result (Theorem 4) cannot follow from it. The distribution of the zeros of orthogonal polynomials plays an important role in weighted approximation, for example, Mastroianni and Notarangelo [11, 12] applied the zeros for exponential weight on −1,1 and the real semiaxis to deal with Lagrange interpolation processes on corresponding interval, respectively.
We construct the following weight:(14)Q∗x≔Qx+p0qx,qx≔−ln1−x2,W∗x≔e−Q∗x,p0≔minp1,pr,0,U∗x≔1−x2−p0Ux.
Some corresponding notations forW∗x are also needed:(15)a±t∗≔a±tQ∗,η±t∗≔η±tQ∗,Δt∗≔ΔtQ∗≔a−t∗,at∗,δt∗≔δtQ∗,φt∗x≔φtQ∗;x,ρ∗≔ρU∗≔p1+pr−2p0+∑i=2r−1maxpi,0,JL,t∗≔JL,tQ∗,U¯tx≔∏i=1rx−ti+1tpi,Ut∗x=1−x21/2+1t−2p0U¯tx.
In all that follows,I denotes the open interval −1,1.Theorem 1 (see [7], Theorem 1.7).
LetW∈ℱLip1/2 and 0<p<∞. Assume that(16)x⟶lim0xQ′x=λ<Λ−12Λp0,p0≠0,and for some constant μ satisfying(17)2λp0<μ<1−1Λ,the function Q¯x=μQ′x+p0q′x is nondecreasing in I.(a)
Then there existsn0>0 such that for n≥n0 and x∈JL,n∗ with L>0, the relation(18)λp,nUW;x∼φn∗xU¯nxpWxp.uniformly holds.(b)
Furthermore, there existsn1>0 such that for n≥n1 and x∈I, the relation(19)λp,nUW;x≥Cφn∗xUn∗xpW∗xp,uniformly holds.By specializing top=2 of Theorem 1, we obtain estimates for the classical Christoffel functions.Corollary 1.
Assume that the conditions of Theorem1 hold.(a)
Then, there existsn0>0 such that for n≥n0 and x∈JL,n∗ with L>0, the relation(20)λnUW2;x∼φn∗xU¯nx2Wx2∼φn∗xUn∗x2W∗x2,uniformly holds.(b)
Furthermore, ifp0≤0, there exists C,n1>0 such that for n≥n1 and x∈I, the relation(21)λnUW2;x≥Cφn∗xU¯nx2Wx2,uniformly holds.Our results will mainly center on the zeros of orthogonal polynomials for Jacobi-exponential weightsUW and Markov–Bernstein inequalities.Theorem 2.
LetW=e−Qx, where Q:I⟶0,∞ is convex with Qa+=Qb−=∞ and Qx>Q0=0,x∈I\0. Let 0<p<∞, P∈Pt−ρ∗−2/p\0, pi≥0,i=2,…,r−1. Assume that relation (16) is valid and Q¯ is nondecreasing in I. Then,(22)PUWLpI\Δt∗<PUWLpΔt∗.In particular, this holds for not identically vanishing polynomialsP of degree ≤t−ρ∗−2/p. For p=∞, (22) holds with < replaced by ≤.Theorem 3.
Let0<p≤∞ and pi≥0,i=2,…,r−1. Assume that relation (16) is valid and Q¯ is nondecreasing in I.(a)
LetW∈ℱLip1/2. Then, for t≥1 and P∈Pt−ρ∗,(23)PUW′φt∗LpI≤CPUWLpI.(b)
Let0<α<1 and W∈ℱDini. Then, for t≥1 and P∈Pt−ρ∗,(24)PUW′φt∗LpΔαt∗≤CPUWLpI.Theorem 4.
LetW∈ℱLip1/2 and pi>−1/2,1≤i≤r. Assume that relation (16) is valid, Q¯ is nondecreasing in I, and(25)φt∗x=O1,t⟶∞.(a)
Then, for large enoughn and 1≤k≤n−1,(26)xkn−xk+1,n≤cφn∗xkn.(b)
Furthermore, ifr=2, then for large enough n and 1≤k≤n−1,(27)xkn−xk+1,n∼φn∗xkn.Remark 1.
By [7], (Lemma 2.12), for zeros xkn,xk+1,n∈KL,n with L>1, the statement (a) of Theorem 4 is valid and ≤ can be replaced by ∼.Theorem 5.
Assume that the assumptions of Theorem2 hold. Then,(28)a−n−ρ∗−1/2∗<xnn<xn−1,n<⋯<x2n<x1n<an+ρ∗+1/2∗.Theorem 6.
LetW∈ℱLip1/2. Assume that relation (16) is valid and Q¯ is nondecreasing in I.(a)
Then,(29)x1n≥an∗1−cηn∗,xnn≤a−n∗1−cη−n∗.(b)
Furthermore, ifpi≥0,i=2,…,r−1, then for large enough n,(30)1−x1nan∗∼ηn∗,1−xnna−n∗∼η−n∗.We prove Theorems2–4 and Theorem 6 in Section 3, but first we need some auxiliary lemmas and the proofs of Corollary 1 and Theorem 5, which are presented in Section 2.


## 2. Auxiliary Lemmas

Lemma 1 (see [1], Theorem 4.1, p. 95).
LetW=e−Qx, where Q:I⟶0,∞ is convex with Qa+=Qb−=∞ and Qx>Q0=0,x∈I\0. Let 0<p≤∞ and P∈Pt−2/p\0. Then,(31)PWLpI\Δt<PWLpΔt.In particular, this holds for not identically vanishing polynomialsP of degree ≤t−2/p. For p=∞, (31) holds with < replaced by ≤.Lemma 2 (see [1], Theorem 10.1, p. 293).
Let0<p≤∞.(a)
LetW∈ℱLip1/2. Then, for t≥1 and P∈Pt,(32)PW′φtLpI≤CPWLpI.(b)
LetW∈ℱDini and 0<α<1. Then, for t≥1 and P∈Pt,(33)PW′φtLpΔαt≤CPWLpI.Lemma 3 (see [7], Lemma 2.13).
LetI=−1,1 and W∈ℱLip1/2. Assume that (16) is valid and Q¯ is nondecreasing in I. Then, W∗∈ℱLip1/2.Lemma 4.
For fixed indexk,1≤k≤n−1, let Ik=xk+1,n,xkn. Let j,1≤j≤r, satisfy(34)minx∈Ikx−tj=min1≤i≤rminx∈Ikx−ti.Then,(35)∏i≠jxκn−tipi∼∏i≠jxκn−ti+1npi∼∏i≠jx−tipi,x∈Ik,κ=k,k+1.Proof.
Following the argument in the proof of Lemma 2.5 in [6], we get (35) by replacing δn/n with 1/n.Lemma 5.
LetW∈ℱ and (25) be valid. Then, there exists t0>0 such that for t>t0 and for each index j,2≤j≤r−1,(36)x−tj+1t∼x−tj+δtt∼x−tj+φtx,holds uniformly for x∈I.Proof.
By (1.55) in [1], we see that there exists t0>0 such that for t>t0, a±t∼1, so we have δt∼1 for t>t0. Also, notice that −1<tr−1<⋯<t2<1, and (36) follows from Lemma 2.7 in [6].Lemma 6.
LetW∈ℱLip1/2. Assume that relation (16) is valid and Q¯ is nondecreasing in I. Then, there exists L>0 such that for t large enough,(37)at+ρ∗+1/2∗≤at∗1+Lηt∗.Proof.
By Lemma 3.11(a) in [1], for t>0,(38)at+ρ∗+1/2∗at∗−1∼1Tat∗⋅ρ∗+1/2t.
Fixt0=ρ∗+1/23; for t≥t0, we have(39)1Tat∗⋅ρ∗+1/2t≤t−2/3T−1at∗.
On the other hand, using Definition2 of ηt∗, we obtain(40)ηt∗≥CT−1at∗t−2/3,as at∗∼δt∗ and Tat∗>1.
Thus, by (38), for large enough t,(41)at+ρ∗+1/2∗at∗−1≤Lηt∗.
This yields (37).
Since the last lemma is based on the results of Corollary1 and Theorem 5, we present the proofs of Corollary 1 and Theorem 5 first.Proof of Corollary 1.
It is the special case of Theorem1 when p=2 we use (5) and the relation φn∗∼φn−1∗ in I from Lemma 9.7 [1]. We also see that U¯n∼U¯n−1 for n large enough.Proof of Theorem 5.
By Lemma3, W∗ satisfies the conditions of W. For U∗=x+1pr−p0x−1p1−p0∏i=2r−1x−tipi, we have p1−p0≥0,pr−p0≥0. Meanwhile, pi≥0,i=2,…,r−1, so (28) follows directly from Theorem 1.9 in [6].Lemma 7.
LetW∈ℱLip1/2 and r=2. Assume that relation (17) is valid and Q¯ is nondecreasing in I. Let ℓjn∈Pn−1 be the fundamental polynomials of Lagrange Interpolation at the zeros pnUW2,x satisfying ℓjnxkn=δkj. Then, for each index j,1≤j≤n and large enough n,(42)ℓjnWUxWU−1xjn+ℓj+1,nWUxWU−1xj+1,n≤C,x∈Ij.Proof.
Notice that(43)ℓjnx=Knx,xjnKnxjn,xjn,where Knx,t=∑k=0npkxpkt is the nth reproducing kernel function. Applying the Cauchy–Schwarz inequality to Knx,t, we obtain(44)ℓjnWUxWU−1xjn≤Knx,xWU2xKnxjn,xjnWU2xjn1/2=λn−1WU2;xWU2xλn−1WU2;xjnWU2xjn1/2.
By Lemma6 and (28), we see Ij⊂JL,n∗,1≤j≤n. Now applying the Christoffel function bounds of Corollary 1 (a) and (b), it follows from the above relation that(45)ℓjnWUxWU−1xjn≤Cφn∗xjnφn∗x1/2⋅U¯nx−2U2xUn∗xjnW∗xjn−2WU2xjn,x∈Ij.
According to the definition ofW∗,(46)W∗x=1−x2p0Wx,and then(47)Un∗xW∗x=1−x2p01−x21/2+1n−2p0U¯nxWx,which by (2.23) in [7] for x∈JL,n∗ gives(48)Un∗xW∗x∼U¯nxWx.
It follows from (48) that for large enough n,(49)ℓjnWUxWU−1xjn≤Cφn∗xjnφn∗x1/2U¯nx−2U2xU¯nxjn−2U2xjn≤Cφn∗xjnφn∗x1/2,x∈Ij,as when r=2,(50)U¯nx−2U2xU¯nxjn−2U2xjn∼1.
Further, applying Theorem 5.7(b) in [1], we conclude for x∈Ij,(51)φn∗xjn∼φn∗x,so that(52)ℓjnWUxWU−1xjn≤C,x∈Ij,and with a similar discussion, we also have(53)ℓj+1,nWUxWU−1xj+1,n≤C,x∈Ij.
This proves (42).


## 3. Proof of Theorems

### 3.1. Proof of Theorem 2

It is easy to check thatQ∗:I⟶0,∞ is convex with Q∗a+=Q∗b−=∞ and Q∗x>Q∗0=0,x∈I\0, so by considering Lemma 3, W∗ satisfies the assumptions about W. Furthermore, for P∈Pt−ρ∗−2/p,(54)PU∗∈Pt−2/p.Observe that(55)UxWx=U∗xW∗x.Then, applying Lemma1, we obtain the results.


### 3.2. Proof of Theorem 3

(a)
By Lemma3, W∗∈ℱLip1/2. For P∈Pt−ρ∗, we have PU∗∈Pt. Thus, by (55), relation (23) follows from (32).(b)
IfW∗∈ℱDini, then with the similar discussion as (a) and using (33), we prove that the statement of (b) is valid. So, it is necessary to prove that if W∈ℱDini, then W∗∈ℱDini.The properties ofa−e in Definition 3 hold for W∗ if W∈ℱDini because of the same argument as in the proof of Lemma 2.13 in [7] since properties of (a)–(e) in Definition 3 are the same for both ℱLip1/2 and ℱDini. We will prove that the property of f in Definition 3 also holds for W∗.By (2.38) in [7], we have(56)S≔∫x−ϵ1x/T∗xx+ϵ1x/T∗xQ∗′s−Q∗′xs−xds≤∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxQ∗′s−Q∗′xs−xds=∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxQ∗′s−Q∗′xs−xds+p0∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxq′s−q′xs−xds=S1+S2.According to Definition3f,(57)S1≤CQ′x.Meanwhile, using Corollary 2.1 (a) in [7], for s∈x−ϵ1x/1−μTxx+ϵ1x/1−μTx, we see(58)11−s2−11−x2=s2−x21−x21−s2≤2s−x1−x21−s2≤2ϵ1x1−μTx1−x21−s2≤2ϵ1ϵ1−μ1−s2=121−s2,where ϵ1=ϵ1−μ/4 and ϵ and μ are shown in (2.29) and (1.23) in [7], respectively, and hence(59)11−s2≤21−x2.Using this relation and(60)q′x=2x1−x2,we obtain(61)S2=p0∫x−ϵ1x/1−μTxx+ϵ1x/1−μTx21+xs1−x21−s2ds≤8p01−x22∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxds≤Cx1−x2⋅11−x2Tx.By (2.30) and (2.35) in [7], we further get(62)S2≤Cq′x≤CQ′x.SubstitutingS1 and S2 into S gives(63)S≤CQ′x.Thus, by (2.35) in [7], we infer that(64)S≤CQ∗′x.This proves propertyf of Definition 3.


### 3.3. Proof of Theorem 4

(a) The proof is similar to Theorem 1.7 in [6], but we provide the details with modification. Denote by ℓknk=1n the fundamental polynomials of Lagrange interpolation at the zeros xknk=1n of the orthogonal polynomials pnWU2,x for the weight WU2.Recall (5); the infimum is actually attained when we take P to be ℓkn∈Pn−1 satisfying ℓknxjn=δkj. So, a classical Gauss quadrature formula for the weight WU2 is(65)λnWU2;xkn=∫Iℓkn2WU2.By Lemma 11.8 in [1], (pp. 320–321) and relation (55), we infer that(66)λnWU2;xknW∗xkn−2+λnWU2;xk+1,nW∗xk+1,n−2=∫Iℓknt2W∗xkn−2+ℓk+1,nt2W∗xk+1,n−2W∗t2U∗t2dt≥∫xk+1,nxknℓknt2W∗xkn−2+ℓk+1,nt2W∗xk+1,n−2W∗t2U∗t2dt≥12∫xk+1,nxknU∗t2dt.On the other hand, according to Lemma6 and Theorem 5, Ik⊂JL,n∗, so that by (20),(67)λnWU2;xknW∗xkn−2+λnWU2;xk+1,nW∗xk+1,n−2≤cφn∗xknUn∗xkn2+φn∗xk+1,nUn∗xk+1,n2.Letj=jk be j defined by (34); then by (51) and Lemma 4, we get(68)cφn∗xknUn∗xkn2+Un∗xk+1,n2≥∫xk+1,nxknU∗t2dt∼∫xk+1,nxknt−tj2pj∗dt∏i=1,i≠jrxkn−ti2pi∗,where pj∗=pjk∗,pi∗=pi, if 2≤i≤r−1, p1∗=p1−p0, and pr∗=pr−p0. Also, we have(69)Un∗xkn2+Un∗xk+1,n2∼xkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗∏i=1,i≠jrxkn−ti+1n2pi∗,and by (35), we further get(70)Un∗xkn2+Un∗xk+1,n2∼xkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗∏i=1,i≠jrxkn−ti2pi∗.By (68) and (70), we get the following relation after simplifying by ∏i=1,i≠jrxkn−ti2pi∗:(71)∫xk+1,nxknt−tj2pj∗dt≤cφn∗xknxkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗.In fact, forx,y∈Ik, using (2.8) in [6] and following the argument in the proof of Lemma 5 in [6], we can obtain 1−xp1−p0∼1−x+1/np1−p0∼1−yp1−p0∼1−y+1/np1−p0 and 1+xpr−p0∼1+x+1/npr−p0∼1+ypr−p0∼1+y+1/npr−p0, so (71) can be written as(72)∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+1n2pj+xk+1,n−tj+1n2pj,where 2≤j≤r−1.Further, by (36),(73)∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+φn∗xkn2pj+xk+1,n−tj+φn∗xkn2pj.By calculation from (73), we get(74)12pj+1xkn−tj2pj+1+σxk+1,n−tj2pj+1=∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+φn∗xkn2pj+xk+1,n−tj+φn∗xkn2pj,where(75)σ=1,tj∈Ik,−1,tj∉Ik.We distinguish two cases.Case 1. pj≥0. By Lemma 2.6 in [6], we assert that if p≥0, Bn≥An≥0, Cn≥0, σ=±1, and Bnp+1+σAnp+1≤CCnBn+Cnp+An+Cnp, then Bn+σAn≤cCn.Using this inequality, it follows from (74) that(76)xkn−xk+1,n≤cφn∗xkn.Case 2. −1/2<pj<0. By (74),(77)12pj+1xkn−tj2pj+1+σxk+1,n−tj2pj+1=∫xk+1,nxknt−tj2pjdt≤c0φn∗xknminφn∗xkn2pj,xk+1,n−tj2pj,xk,n−tj2pj.Case 2.1. tj∈Ik. Inequality (77) gives(78)xκn−tj2pj+1≤cφn∗xkn2pj+1,κ=k,k+1,which yields (76).Case 2.2. tj∉Ik. In this case, we distinguish two subcases. Suppose without loss of generality that xk+1,n>tj.Case 2.2.1. If xk+1,n−tj≥2c0φn∗xkn, where c0 is given by (77), then(79)∫xk+1,nxknt−tj2pjdt=∫xk+1,nxknt−tjt−tj2pj−1dt≥xk+1,n−tj∫xk+1,nxknt−tj2pj−1dt=xk+1,n−tj12pjxk+1,n−tj2pj−xkn−tj2pj≥c0φn∗xknpjxk+1,n−tj2pj−xkn−tj2pj,which by (77) gives(80)xk+1,n−tj2pj≤1−pj−1xkn−tj2pj≤2xkn−tj2pj.On the other hand, by (77)–(80),(81)c0φn∗xknxk+1,n−tj2pj≥∫xk+1,nxknt−tj2pjdt≥xkn−tj2pjxkn−xk+1,n≥12xk+1,n−tj2pjxkn−xk+1,n,and hence (76) follows.Case 2.2.2. xk+1,n−tj<2c0φn∗xkn. By (77),(82)c0φn∗xkn2pj+1≥12pj+1xkn−tj2pj+1−xk+1,n−tj2pj+1≥12pj+1xkn−tj2pj+1−2c0φn∗xkn2pj+1.So,xkn−tj≤cφn∗xkn and (76) follows.(b) Now, let us prove (27). We must prove that for some constant c>0 and n large enough, we have(83)xkn−xk+1,n≥cφn∗xkn,k=1,2,…,n−1.First, by our Markov–Bernstein inequality (23) and Lemma 7, we have that(84)ℓknWU′φn∗L∞IWU−1xkn≤C1ℓknWUL∞IWU−1xkn≤C.Then, by the mean value theorem, for someξ between xkn and xk+1,n,(85)1=ℓknWUxknWU−1xkn−ℓknWUxk+1,nWU−1xkn=ℓknWU′ξWU−1xknxkn−xk+1,n≤Cφn∗−1ξxkn−xk+1,n.Thus, by (51), we get the lower bound and finish the proof of (b).


### 3.4. Proof of Theorem 6

By Lemma3, W∗∈ℱLip1/2. Then, following the argument in the proof of Theorem 5, the statements of Theorem 6 follow directly from Theorem 1.10 in [6].


## 3.1. Proof of Theorem 2

It is easy to check thatQ∗:I⟶0,∞ is convex with Q∗a+=Q∗b−=∞ and Q∗x>Q∗0=0,x∈I\0, so by considering Lemma 3, W∗ satisfies the assumptions about W. Furthermore, for P∈Pt−ρ∗−2/p,(54)PU∗∈Pt−2/p.Observe that(55)UxWx=U∗xW∗x.Then, applying Lemma1, we obtain the results.


## 3.2. Proof of Theorem 3

(a)
By Lemma3, W∗∈ℱLip1/2. For P∈Pt−ρ∗, we have PU∗∈Pt. Thus, by (55), relation (23) follows from (32).(b)
IfW∗∈ℱDini, then with the similar discussion as (a) and using (33), we prove that the statement of (b) is valid. So, it is necessary to prove that if W∈ℱDini, then W∗∈ℱDini.The properties ofa−e in Definition 3 hold for W∗ if W∈ℱDini because of the same argument as in the proof of Lemma 2.13 in [7] since properties of (a)–(e) in Definition 3 are the same for both ℱLip1/2 and ℱDini. We will prove that the property of f in Definition 3 also holds for W∗.By (2.38) in [7], we have(56)S≔∫x−ϵ1x/T∗xx+ϵ1x/T∗xQ∗′s−Q∗′xs−xds≤∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxQ∗′s−Q∗′xs−xds=∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxQ∗′s−Q∗′xs−xds+p0∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxq′s−q′xs−xds=S1+S2.According to Definition3f,(57)S1≤CQ′x.Meanwhile, using Corollary 2.1 (a) in [7], for s∈x−ϵ1x/1−μTxx+ϵ1x/1−μTx, we see(58)11−s2−11−x2=s2−x21−x21−s2≤2s−x1−x21−s2≤2ϵ1x1−μTx1−x21−s2≤2ϵ1ϵ1−μ1−s2=121−s2,where ϵ1=ϵ1−μ/4 and ϵ and μ are shown in (2.29) and (1.23) in [7], respectively, and hence(59)11−s2≤21−x2.Using this relation and(60)q′x=2x1−x2,we obtain(61)S2=p0∫x−ϵ1x/1−μTxx+ϵ1x/1−μTx21+xs1−x21−s2ds≤8p01−x22∫x−ϵ1x/1−μTxx+ϵ1x/1−μTxds≤Cx1−x2⋅11−x2Tx.By (2.30) and (2.35) in [7], we further get(62)S2≤Cq′x≤CQ′x.SubstitutingS1 and S2 into S gives(63)S≤CQ′x.Thus, by (2.35) in [7], we infer that(64)S≤CQ∗′x.This proves propertyf of Definition 3.


## 3.3. Proof of Theorem 4

(a) The proof is similar to Theorem 1.7 in [6], but we provide the details with modification. Denote by ℓknk=1n the fundamental polynomials of Lagrange interpolation at the zeros xknk=1n of the orthogonal polynomials pnWU2,x for the weight WU2.Recall (5); the infimum is actually attained when we take P to be ℓkn∈Pn−1 satisfying ℓknxjn=δkj. So, a classical Gauss quadrature formula for the weight WU2 is(65)λnWU2;xkn=∫Iℓkn2WU2.By Lemma 11.8 in [1], (pp. 320–321) and relation (55), we infer that(66)λnWU2;xknW∗xkn−2+λnWU2;xk+1,nW∗xk+1,n−2=∫Iℓknt2W∗xkn−2+ℓk+1,nt2W∗xk+1,n−2W∗t2U∗t2dt≥∫xk+1,nxknℓknt2W∗xkn−2+ℓk+1,nt2W∗xk+1,n−2W∗t2U∗t2dt≥12∫xk+1,nxknU∗t2dt.On the other hand, according to Lemma6 and Theorem 5, Ik⊂JL,n∗, so that by (20),(67)λnWU2;xknW∗xkn−2+λnWU2;xk+1,nW∗xk+1,n−2≤cφn∗xknUn∗xkn2+φn∗xk+1,nUn∗xk+1,n2.Letj=jk be j defined by (34); then by (51) and Lemma 4, we get(68)cφn∗xknUn∗xkn2+Un∗xk+1,n2≥∫xk+1,nxknU∗t2dt∼∫xk+1,nxknt−tj2pj∗dt∏i=1,i≠jrxkn−ti2pi∗,where pj∗=pjk∗,pi∗=pi, if 2≤i≤r−1, p1∗=p1−p0, and pr∗=pr−p0. Also, we have(69)Un∗xkn2+Un∗xk+1,n2∼xkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗∏i=1,i≠jrxkn−ti+1n2pi∗,and by (35), we further get(70)Un∗xkn2+Un∗xk+1,n2∼xkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗∏i=1,i≠jrxkn−ti2pi∗.By (68) and (70), we get the following relation after simplifying by ∏i=1,i≠jrxkn−ti2pi∗:(71)∫xk+1,nxknt−tj2pj∗dt≤cφn∗xknxkn−tj+1n2pj∗+xk+1,n−tj+1n2pj∗.In fact, forx,y∈Ik, using (2.8) in [6] and following the argument in the proof of Lemma 5 in [6], we can obtain 1−xp1−p0∼1−x+1/np1−p0∼1−yp1−p0∼1−y+1/np1−p0 and 1+xpr−p0∼1+x+1/npr−p0∼1+ypr−p0∼1+y+1/npr−p0, so (71) can be written as(72)∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+1n2pj+xk+1,n−tj+1n2pj,where 2≤j≤r−1.Further, by (36),(73)∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+φn∗xkn2pj+xk+1,n−tj+φn∗xkn2pj.By calculation from (73), we get(74)12pj+1xkn−tj2pj+1+σxk+1,n−tj2pj+1=∫xk+1,nxknt−tj2pjdt≤cφn∗xknxkn−tj+φn∗xkn2pj+xk+1,n−tj+φn∗xkn2pj,where(75)σ=1,tj∈Ik,−1,tj∉Ik.We distinguish two cases.Case 1. pj≥0. By Lemma 2.6 in [6], we assert that if p≥0, Bn≥An≥0, Cn≥0, σ=±1, and Bnp+1+σAnp+1≤CCnBn+Cnp+An+Cnp, then Bn+σAn≤cCn.Using this inequality, it follows from (74) that(76)xkn−xk+1,n≤cφn∗xkn.Case 2. −1/2<pj<0. By (74),(77)12pj+1xkn−tj2pj+1+σxk+1,n−tj2pj+1=∫xk+1,nxknt−tj2pjdt≤c0φn∗xknminφn∗xkn2pj,xk+1,n−tj2pj,xk,n−tj2pj.Case 2.1. tj∈Ik. Inequality (77) gives(78)xκn−tj2pj+1≤cφn∗xkn2pj+1,κ=k,k+1,which yields (76).Case 2.2. tj∉Ik. In this case, we distinguish two subcases. Suppose without loss of generality that xk+1,n>tj.Case 2.2.1. If xk+1,n−tj≥2c0φn∗xkn, where c0 is given by (77), then(79)∫xk+1,nxknt−tj2pjdt=∫xk+1,nxknt−tjt−tj2pj−1dt≥xk+1,n−tj∫xk+1,nxknt−tj2pj−1dt=xk+1,n−tj12pjxk+1,n−tj2pj−xkn−tj2pj≥c0φn∗xknpjxk+1,n−tj2pj−xkn−tj2pj,which by (77) gives(80)xk+1,n−tj2pj≤1−pj−1xkn−tj2pj≤2xkn−tj2pj.On the other hand, by (77)–(80),(81)c0φn∗xknxk+1,n−tj2pj≥∫xk+1,nxknt−tj2pjdt≥xkn−tj2pjxkn−xk+1,n≥12xk+1,n−tj2pjxkn−xk+1,n,and hence (76) follows.Case 2.2.2. xk+1,n−tj<2c0φn∗xkn. By (77),(82)c0φn∗xkn2pj+1≥12pj+1xkn−tj2pj+1−xk+1,n−tj2pj+1≥12pj+1xkn−tj2pj+1−2c0φn∗xkn2pj+1.So,xkn−tj≤cφn∗xkn and (76) follows.(b) Now, let us prove (27). We must prove that for some constant c>0 and n large enough, we have(83)xkn−xk+1,n≥cφn∗xkn,k=1,2,…,n−1.First, by our Markov–Bernstein inequality (23) and Lemma 7, we have that(84)ℓknWU′φn∗L∞IWU−1xkn≤C1ℓknWUL∞IWU−1xkn≤C.Then, by the mean value theorem, for someξ between xkn and xk+1,n,(85)1=ℓknWUxknWU−1xkn−ℓknWUxk+1,nWU−1xkn=ℓknWU′ξWU−1xknxkn−xk+1,n≤Cφn∗−1ξxkn−xk+1,n.Thus, by (51), we get the lower bound and finish the proof of (b).


## 3.4. Proof of Theorem 6

By Lemma3, W∗∈ℱLip1/2. Then, following the argument in the proof of Theorem 5, the statements of Theorem 6 follow directly from Theorem 1.10 in [6].



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*Source: 7805730-2020-08-24.xml*