The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights

Abstract

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

1. Introduction and Results

This paper deals with the zeros of orthogonal polynomials for Jacobi-exponential weights. Let 𝑤 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. 𝐏_𝑛 stands for the set of polynomials of degree at most 𝑛.

Assume that 𝑊=𝑒^−𝑄 where 𝑄∶𝐈→[0,∞) is continuous. Also, let 0<𝑝<∞, 𝑎≤𝑡_𝑟<𝑡_𝑟−1<⋯<𝑡₂<𝑡₁𝑝≤𝑏,_𝑖>−1𝑝𝑈(𝑥)=,𝑖=1,2,…,𝑟,_𝑟𝑖=1𝑥−𝑡_𝑖^𝑝𝑖. (1.1)

The letters 𝑐,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, 𝐶_𝑛∼𝐷_𝑛 means that there are two constants 𝑐₁ and 𝑐₂ such that 𝑐₁≤𝐶_𝑛/𝐷_𝑛≤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]). Given 𝑐,𝑡≥0 and a nonnegative Borel measure 𝜈 with compact support in 𝐂 and total mass ≤𝑡, one says that 𝑃(𝑧)∶=𝑐expln|𝑧−𝑡|𝑑𝜈(𝑡) (1.2) is an exponential of a potential of mass ≤𝑡. One denotes the set of all such 𝑃 by 𝒫_𝑡.

One notes that, for 𝑃∈𝐏_𝑛, |𝑃|∈𝒫_𝑡,𝑡≥𝑛. (1.3)

Definition 1.2 (see [1, page 19]). Let 𝑤 be a weight in 𝐈. For 0<𝑝<∞, generalized Christoffel functions with respect to 𝑤 for 𝑧∈𝐂 are defined by 𝜆_𝑝,𝑛(𝑤;𝑧)=inf_{𝑃∈𝐏𝑛}‖𝑃𝑤‖_{𝐿𝑝(𝐈)}|𝑃(𝑧)|^𝑝. (1.4) For 𝑝=∞, generalized Christoffel functions with respect to 𝑤 for 𝑧∈𝐂 are defined by 𝜆_∞,𝑛(𝑤;𝑧)=inf_{𝑃∈𝐏𝑛}‖𝑃𝑤‖_{𝐿∞(𝐈)}|𝑃(𝑧.)| (1.5)
Obviously, for the classical Christoffel function 𝜆_𝑛(𝑤²;𝑥) with respect to 𝑤², we have 𝜆_𝑛𝑤²;𝑥=𝜆_2,𝑛−1(𝑤;𝑥). (1.6)
A function 𝑓∶(𝑐,𝑑)→(0,∞) is said to be quasi-increasing (or quasi-decreasing) if there exists 𝐶>0 such that 𝑓(𝑥)≤(or≥)𝐶𝑓𝑦,𝑐<𝑥≤𝑦<𝑑. (1.7)

Definition 1.3 (see [1, pages 10–12]). Let 𝑎<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 lim_𝑡→𝑎+𝑄(𝑡)=lim_{𝑡→𝑏−}𝑄(𝑡)=∞. (1.8)(d) The function 𝑇(𝑡)∶=𝑡𝑄^′(𝑡)𝑄(𝑡),𝑡≠0, (1.9)is quasi-decreasing in (𝑎,0) and quasi-increasing in (0,𝑏), respectively. Moreover 𝑇(𝑡)≥Λ>1,𝑡∈𝐈⧵{0}. (1.10)(e) There exists 𝜖₀∈(0,1) such that, for 𝑦∈𝐈⧵{0}, 𝑇𝑦𝑦𝜖∼𝑇1−⁰𝑇𝑦. (1.11)Then we write 𝑊∈ℱ.(f) In addition, assume that there exist 𝐶,𝜖₁>0 such that, for all 𝑥∈𝐈⧵{0}, ^𝑥_{𝑥−𝜖1|𝑥|/𝑇(𝑥)}𝑄^′(𝑡)−𝑄^′(𝑥)|𝑡−𝑥|^3/2𝑑𝑡≤𝐶𝑄^′(𝑥)𝑇(𝑥)|𝑥|^1/2. (1.12)Then we write 𝑊∈ℱ(Lip(1/2)).
For 𝑊∈ℱ and 𝑡>0, the Mhaskar-Rahmanov-Saff numbers 𝑎_−𝑡∶=𝑎_−𝑡(𝑄)<0<𝑎_𝑡∶=𝑎_𝑡(𝑄) are defined by the equations 1𝑡=𝜋^𝑎𝑡_𝑎−𝑡𝑥𝑄^′(𝑥)𝑥−𝑎_−𝑡𝑎_𝑡−𝑥^1/21𝑑𝑥,0=𝜋^𝑎𝑡_𝑎−𝑡𝑄^′(𝑥)𝑥−𝑎_−𝑡𝑎_𝑡−𝑥^1/2𝑑𝑥. (1.13) Put for 𝑡>0, Δ_𝑡∶=Δ_𝑡(𝑄)∶=𝑎_−𝑡,𝑎_𝑡,𝛿_𝑡∶=𝛿_𝑡(𝑄)1∶=2𝑎_𝑡+𝑎_−𝑡,𝜂_±𝑡∶=𝜂_±𝑡(𝑄)∶=𝑡𝑇𝑎_±𝑡𝑎_±𝑡𝛿_𝑡^−2/3,𝜑_𝑡(𝑥)∶=𝜑_𝑡(𝑄;𝑥)∶=𝑥−𝑎_−2𝑡𝑥−𝑎_2𝑡𝑡𝑥−𝑎_−𝑡+𝑎_−𝑡𝜂_−𝑡𝑥−𝑎_𝑡+𝑎_𝑡𝜂_𝑡,𝑥∈𝑎_−𝑡,𝑎_𝑡,𝜑_𝑡𝑎_𝑡,𝑥∈𝑎_𝑡,𝜑,𝑏_𝑡𝑎_−𝑡,𝑥∈𝑎,𝑎_−𝑡,𝐉_𝐿,𝑡∶=𝐉_𝐿,𝑡(𝑄)∶=𝑎_−𝑡1+𝐿𝜂_−𝑡,𝑎_𝑡1+𝐿𝜂_𝑡𝐊,𝐿>0,_𝐿,𝑡∶=𝐊_𝐿,𝑡(𝑄)∶=−1+𝐿1+𝑎_−𝑡,1−𝐿1−𝑎_𝑡,𝐿>1. (1.14) Let 𝑈_𝑡(𝑥)∶=^𝑟_𝑖=1𝑥−𝑡_𝑖+𝛿_𝑡𝑡^𝑝𝑖,𝜌∶=𝜌(𝑈)∶=^𝑟_𝑖=1max𝑝_𝑖,0. (1.15)

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 𝑥^2𝛼𝑊(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 𝐿_𝑝 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]). Let 𝑊∈ℱ(Lip(1/2)),𝐿>0, and 0<𝑝<∞. Assume that lim_𝑡→∞𝑎_−𝑡𝑎_𝑡=𝛾,0<𝛾<∞. (1.16) Then there exists 𝑛₀>0 such that, for 𝑛≥𝑛₀ and 𝑥∈𝐉_𝐿,𝑛, the relation 𝜆_𝑝,𝑛(𝑈𝑊;𝑥)∼𝜑_𝑛(𝑥)𝑈_𝑛(𝑥)^𝑝𝑊(𝑥)^𝑝 (1.17) uniformly holds.

Theorem 1.5 (see [6, Theorem 1.2]). Let 𝑊=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∈𝒫_{𝑡−𝜌−2/𝑝}, ‖𝑃𝑈𝑊‖_{𝐿𝑝(𝐈)}≤𝐶‖𝑃𝑈𝑊‖_{𝐿𝑝(Δ𝑡)},𝑃𝑈_𝑡𝑊_{𝐿𝑝(𝐈)}≤𝐶𝑃𝑈_𝑡𝑊_{𝐿𝑝(Δ𝑡)}. (1.18)

Theorem 1.6 (see [6, Theorem 1.3]). Let 𝑊∈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∈𝒫_𝑡, ‖𝑃𝑈𝑊‖_{𝐿𝑝(𝐈)}≤𝐶‖𝑃𝑈𝑊‖_{𝐿𝑝([𝑎−𝑡(1−𝐿𝜂−𝑡),𝑎𝑡(1−𝐿𝜂𝑡)])}. (1.19)

In this paper we discuss the zeros of orthogonal polynomials for Jacobi-exponential weights 𝑈𝑊 and restricted range inequalities.

Theorem 1.7. Let 𝑊∈ℱ(Lip(1/2)). Assume that (1.16) is valid, and 𝑎<𝑡_𝑟<⋯<𝑡₁𝜑<𝑏,_𝑡(𝑥)=𝑂(1),𝑡→∞. (1.20)(1.21) Then 𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛≤𝑐𝜑_𝑛𝑥_𝑘𝑛,𝑘=1,2,…,𝑛−1. (1.22)

Theorem 1.8. Let 𝑊=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 𝑝_𝑖 are positive and relation (1.16) is valid. Then there exist 𝑡₀>0 such that, for 𝑡≥𝑡₀ and 𝑃∈𝒫_{𝑡−𝜌−2/𝑝}, ‖𝑃𝑈𝑊‖_{𝐿𝑝(𝐈⧵Δ𝑡)}≤‖𝑃𝑈𝑊‖_{𝐿𝑝(Δ𝑡)}. (1.23)

Theorem 1.9. Let the assumptions of Theorem 1.8 prevail. Then 𝑥_1𝑛<𝑎_{𝑛+𝜌+1/2},𝑥_𝑛𝑛>𝑎_{−𝑛−𝜌−1/2}. (1.24)(1.25)

Theorem 1.10. Let 𝑊∈ℱ(Lip (1/2)). Then 𝑥_1𝑛≥𝑎_𝑛1−𝑐𝜂_𝑛,𝑥_𝑛𝑛≤𝑎_−𝑛1−𝑐𝜂_−𝑛. (1.26)(1.27) If all 𝑝_𝑖≥0, then 𝑥1−^1𝑛𝑎_𝑛∼𝜂_𝑛,𝑥1−^𝑛𝑛𝑎_−𝑛∼𝜂_−𝑛. (1.28)(1.29)

Here we should point out that our main result (Theorem 1.7) 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 lim_𝑑→∞∫^5𝑑/2_𝑑/2exp−𝑡²𝑑𝑡∫_𝑑^2𝑑exp−𝑡²𝑑𝑡=lim_𝑑→∞exp−(5𝑑/2)²−exp−(𝑑/2)²exp−(2𝑑)²−exp−𝑑²=lim_𝑑→∞exp3𝑑²/4−exp−21𝑑²/41−exp−3𝑑²=∞, (1.30) which implies that 𝑊(𝑡)=exp(−𝑡²) is not a doubling weight.

We will give some auxiliary lemmas in Section 2 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]). Let 𝑊∈ℱ. Then for fixed 𝐿>1 and uniformly for 𝑡>0, 𝑎_𝐿𝑡∼𝑎_𝑡. (2.1)
Moreover, there exists 𝜏₀>0 such that, for 𝑡≥𝜏≥𝜏₀, the inequalities 𝛿1≤^𝑡𝛿_𝜏𝑡≤𝑐𝜏^1/Λ (2.2) hold.

Lemma 2.2 (Shi [6]). Let 𝑊∈ℱ. Then, for large enough 𝑡, 𝑎_2𝑡≥𝑎_𝑡1+𝜂_𝑡. (2.3)

Lemma 2.3. Let 𝐈=(−1,1),𝑊∈ℱ, and 𝐿>1. Then, for 𝑥∈𝐊_𝐿,𝑡, 𝜑_𝑡(𝑥)∼1𝑡𝑎_2𝑡−𝑥𝑥−𝑎_−2𝑡^1/2,𝜑_𝑡(𝑥)≤𝑐𝛿_𝑡𝑡. (2.4)(2.5)

Proof. By the same argument as that of [8, (2.25)] we can prove (2.4). By (2.4) and (2.1) for 𝑥∈𝐊_𝐿,𝑡, 𝜑_𝑡(𝑥)≤𝑐𝑡⋅12𝑎_2𝑡−𝑎_−2𝑡≤𝑐𝛿_𝑡𝑡. (2.6)

Lemma 2.4. Let 𝑊∈ℱ. Then, for 𝑥∈𝐈, 𝜑_𝑡(𝑥)≤𝑐𝛿_𝑡𝑡^2/3𝑇𝑎_𝑡^1/6. (2.7)

Proof. By the definition of 𝜑_𝑡 it is enough to prove (2.7) for 𝑥∈Δ_𝑡. Without loss of generality we can assume that 0≤𝑥≤𝑎_𝑡. By Lemma 3.11(b) in [1, page 81] for 𝑡>0, 𝑎_2𝑡𝑎_𝑡∼1−1𝑇𝑎_𝑡. (2.8) By Lemma 2.12 in [8], (2.3), (2.1), and (2.8), 𝑆(𝑥)=𝑎_2𝑡−𝑥𝑎_𝑡1+𝜂_𝑡⋅𝑎−𝑥^−2𝑡−𝑥𝑎_−𝑡1+𝜂_−𝑡≤𝑎−𝑥_2𝑡−𝑎_𝑡𝑎_𝑡𝜂_𝑡⋅𝑎_−2𝑡𝑎_−𝑡1+𝜂_−𝑡𝑎≤𝑐^2𝑡/𝑎_𝑡−1𝜂_𝑡≤𝑐𝜂_𝑡𝑇𝑎_𝑡. (2.9) By (1.63) in [1, page 15], 𝜂_𝑡𝑇𝑎_𝑡≥𝑡^−2/3𝑇𝑎_𝑡^1/3 (2.10) and hence 𝑆(𝑥)≤𝑐𝑡^2/3𝑇𝑎_𝑡^−1/3. (2.11) Thus 𝜑_𝑡(𝑥)=𝑎_2𝑡−𝑥𝑥−𝑎_−2𝑡^1/2𝑡𝑆(𝑥)^1/2≤𝑐𝛿_𝑡𝑡𝑡^2/3𝑇𝑎_𝑡^−1/31/2=𝑐𝛿_𝑡𝑡^2/3𝑇𝑎_𝑡^1/6. (2.12)

Let 𝐈_𝑘=[𝑥_𝑘+1,𝑛,𝑥_𝑘𝑛],𝑑_𝑘=𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛,𝑘=1,2,…,𝑛−1. Let, for 𝑛≥𝑛₀ and 𝑑∶=min_{1≤𝑖≤𝑟−1}(𝑡_𝑖−𝑡_𝑖+1), max_{1≤𝑘≤𝑛−1}𝑑_𝑘≤𝑑4. (2.13)

Lemma 2.5. For fixed index 𝑘, 1≤𝑘≤𝑛−1, let 𝑗,1≤𝑗≤𝑟, satisfy min_{𝑥∈𝐈𝑘}𝑥−𝑡_𝑗=min_{1≤𝑖≤𝑟}min_{𝑥∈𝐈𝑘}𝑥−𝑡_𝑖. (2.14) Then _𝑖≠𝑗𝑥_𝜅𝑛−𝑡_𝑖^𝑝𝑖∼_𝑖≠𝑗𝑥_𝜅𝑛−𝑡_𝑖+𝛿_𝑛𝑛^𝑝𝑖∼_𝑖≠𝑗𝑥−𝑡_𝑖^𝑝𝑖,𝑥∈𝐈_𝑘,𝜅=𝑘,𝑘+1. (2.15)

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, for 𝑖≠𝑗, 𝑥_𝑘𝑛−𝑡_𝑖≥38𝑑. (2.16) In fact, suppose without loss of generality that 𝑥_𝑘𝑛≥𝑡_𝑗. It is enough to show (2.16) for 𝑖=𝑗−1. Because |𝑥_𝑘𝑛−𝑡_𝑗+1|≥𝑡_𝑗−𝑡_𝑗+1≥𝑑.
If 𝑡_𝑗∈𝐈_𝑘 then by (2.13) 𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛≤𝑑4≤𝑡_𝑗−1−𝑡_𝑗≤𝑡_𝑗−1−𝑥_𝑘𝑛+𝑥_𝑘𝑛−𝑡_𝑗 (2.17) and hence 𝑥_𝑘+1,𝑛−𝑡_𝑗≤𝑥_𝑘𝑛−𝑡_𝑗−1; (2.18) if 𝑡_𝑗∉𝐈_𝑘 then by (2.14) 𝑥_𝑘+1,𝑛−𝑡_𝑗=min_{𝑥∈𝐈𝑘}𝑥−𝑡_𝑗≤min_{𝑥∈𝐈𝑘}𝑥−𝑡_𝑗−1=𝑡_𝑗−1−𝑥_𝑘𝑛, (2.19) which again implies (2.18). Then by (2.18) 𝑑≤𝑡_𝑗−1−𝑡_𝑗≤𝑡_𝑗−1−𝑥_𝑘𝑛+𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛+𝑥_𝑘+1,𝑛−𝑡_𝑗≤2𝑥_𝑘𝑛−𝑡_𝑗−1+𝑑_𝑘≤2𝑥_𝑘𝑛−𝑡_𝑗−1+14𝑑 (2.20) and hence |𝑥_𝑘𝑛−𝑡_𝑗−1|≥3𝑑/8. This proves (2.16).
With the help of (2.16) for 𝑥∈𝐈_𝑘 and 𝑖≠𝑗, 𝑥−𝑡_𝑖≤𝑥_𝑘𝑛−𝑡_𝑖+𝑥−𝑥_𝑘𝑛≤𝑥_𝑘𝑛−𝑡_𝑖+𝑑4≤53𝑥_𝑘𝑛−𝑡_𝑖,𝑥−𝑡_𝑖≥𝑥_𝑘𝑛−𝑡_𝑖−𝑥−𝑥_𝑘𝑛≥𝑥_𝑘𝑛−𝑡_𝑖−𝑑4≥13𝑥_𝑘𝑛−𝑡_𝑖. (2.21) Hence 𝑥−𝑡_𝑖∼𝑥_𝑘𝑛−𝑡_𝑖. (2.22) Furthermore, by (2.2) with 𝜏=1𝛿_𝑡𝑡≤𝑐𝛿₁𝑡^1/Λ−1=𝑜(1),𝑡→∞. (2.23) So for 𝑖≠𝑗, 𝑥_𝑘𝑛−𝑡_𝑖∼𝑥_𝑘𝑛−𝑡_𝑖+𝛿_𝑛𝑛. (2.24) This proves (2.15).

By the same argument as that of Lemma 7.2.7 in [9, page 157] replacing 1/𝑛 by 𝐶_𝑛, we can get its extension.

Lemma 2.6. Let 𝑝≥0,𝐵_𝑛≥𝐴_𝑛≥0,𝐶_𝑛≥0,𝜎=±1, and 𝐵_𝑛^𝑝+1+𝜎𝐴_𝑛^𝑝+1≤𝐶𝐶_𝑛𝐵_𝑛+𝐶_𝑛^𝑝+𝐴_𝑛+𝐶_𝑛^𝑝. (2.25) Then 𝐵_𝑛+𝜎𝐴_𝑛≤𝑐𝐶_𝑛. (2.26)

Lemma 2.7. Let 𝑊∈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, 𝑥−𝑡_𝑗+𝛿_𝑡𝑡∼𝑥−𝑡_𝑗+𝜑_𝑡(𝑥) (2.27) holds uniformly for 𝑥∈𝐈.

Proof. Let 0<𝜖<min{𝑏−𝑡₁,𝑡_𝑟−𝑎} and Δ=[𝑡_𝑟−𝜖,𝑡₁+𝜖]. We separate two cases.
Case 1 (𝑥∈Δ). In this case by (2.3) and (2.1), 𝜑_𝑡(𝑥)≥1𝑡𝑎_2𝑡−𝑥𝑥−𝑎_−2𝑡^1/2≥1𝑡𝑎_2𝑡−𝑡₁−𝜖𝑡_𝑟−𝜖−𝑎_−2𝑡^1/2≥𝑐𝛿_𝑡𝑡 (2.28) which coupled with (2.5) gives 𝜑_𝑡(𝑥)∼𝛿_𝑡𝑡. (2.29) Hence (2.27) follows.
Case 2 (𝑥∉Δ). In this case by (2.23), 𝑥−𝑡_𝑗≥𝜖≥𝑐𝛿_𝑡𝑡 (2.30) and by (1.21) 𝑥−𝑡_𝑗≥𝜖≥𝑐𝜑_𝑡(𝑥). (2.31) Again (2.27) follows.

Corollary 2.8. Let 𝑊∈ℱ. Let (1.16) and (1.20) prevail. If 𝛿_𝑡𝑡^2/3𝑇𝑎_𝑡^1/6=𝑂(1),𝑡→∞ (2.32) then (2.27) holds.
In particular, if Λ≥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, if Λ≥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 Theorem 1.7

Denote by ℓ_𝑘𝑛's the fundamental polynomials based on the zeros 𝑥_𝑘𝑛's. By Theorem 1.4 and Lemma 11.8 in [8, pages 320-321] 𝜆_𝑛𝑊𝑈;𝑥_𝑘𝑛𝑊𝑥_𝑘𝑛⁻²+𝜆_𝑛𝑊𝑈;𝑥_𝑘+1,𝑛𝑊𝑥_𝑘+1,𝑛⁻²=_𝐈ℓ_𝑘𝑛(𝑡)²𝑊𝑥_𝑘𝑛⁻²+ℓ_𝑘+1,𝑛(𝑡)²𝑊𝑥_𝑘+1,𝑛⁻²𝑊(𝑡)²𝑈(𝑡)²≥𝑑𝑡_{𝑥𝑘𝑛𝑥𝑘+1,𝑛}ℓ_𝑘𝑛(𝑡)²𝑊𝑥_𝑘𝑛⁻²+ℓ_𝑘+1,𝑛(𝑡)²𝑊𝑥_𝑘+1,𝑛⁻²𝑊(𝑡)²𝑈(𝑡)²≥1𝑑𝑡2^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑈(𝑡)²𝑑𝑡. (3.1) On the other hand, by Theorem 1.4, 𝜆_𝑛𝑊𝑈;𝑥_𝑘𝑛𝑊𝑥_𝑘𝑛⁻²+𝜆_𝑛𝑊𝑈;𝑥_𝑘+1,𝑛𝑊𝑥_𝑘+1,𝑛⁻²≤𝑐𝜑_𝑛𝑥_𝑘𝑛𝑈_𝑛𝑥_𝑘𝑛²+𝜑_𝑛𝑥_𝑘+1,𝑛𝑈_𝑛𝑥_𝑘+1,𝑛². (3.2) Then for 𝜑_𝑛(𝑥_𝑘𝑛)∶=max{𝜑_𝑛(𝑥_𝑘𝑛),𝜑_𝑛(𝑥_𝑘+1,𝑛)}, ^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑈(𝑡)²𝑑𝑡≤𝑐𝜑_𝑛𝑥_𝑘𝑛𝑈_𝑛𝑥_𝑘𝑛²+𝑈_𝑛𝑥_𝑘+1,𝑛². (3.3)

Let 𝑗 be defined by (2.14). Using Lemma 2.5 it follows from (3.3) that ^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡≤𝑐𝜑_𝑛𝑥_𝑘𝑛𝑥_𝑘𝑛−𝑡_𝑗+𝛿_𝑛𝑛^2𝑝𝑗+𝑥_𝑘+1,𝑛−𝑡_𝑗+𝛿_𝑛𝑛^2𝑝𝑗. (3.4) Further, by (2.27), ^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡≤𝑐𝜑_𝑛𝑥_𝑘𝑛𝑥_𝑘𝑛−𝑡_𝑗+𝜑_𝑛(𝑥_𝑘𝑛)^2𝑝𝑗+𝑥_𝑘+1,𝑛−𝑡_𝑗+𝜑_𝑛(𝑥_𝑘𝑛)^2𝑝𝑗. (3.5) By calculation from (3.5) we get 12𝑝_𝑗𝑥+1^𝑘𝑛−𝑡_𝑗^2𝑝𝑗+1+𝜎𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗+1=^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡≤𝑐𝜑_𝑛𝑥_𝑘𝑛𝑥_𝑘𝑛−𝑡_𝑗+𝜑_𝑛(𝑥_𝑘𝑛)^2𝑝𝑗+𝑥_𝑘+1,𝑛−𝑡_𝑗+𝜑_𝑛(𝑥_𝑘𝑛)^2𝑝𝑗, (3.6) where 𝜎=1,𝑡_𝑗∈𝐈_𝑘,−1,𝑡_𝑗∉𝐈_𝑘. (3.7) We separate two cases.

Case 1 (𝑝_𝑗≥0). Using Lemma 2.6 it follows from (3.6) that 𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛≤𝑐𝜑_𝑛𝑥_𝑘𝑛. (3.8)

Case 2 (𝑝_𝑗<0). Suppose without loss of generality that 𝑥_𝑘+1,𝑛>𝑡_𝑗 for the case when 𝑡_𝑗∉𝐈_𝑘. By (3.6), 12𝑝_𝑗𝑥+1^𝑘𝑛−𝑡_𝑗^2𝑝𝑗+1+𝜎𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗+1=^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡≤𝑐₀𝜑_𝑛𝑥_𝑘𝑛min𝜑_𝑛𝑥_𝑘𝑛^2𝑝𝑗,𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗. (3.9)

Subcase 2.1 (𝑡_𝑗∈𝐈_𝑘). Inequality (3.9) gives 𝑥_𝜅𝑛−𝑡_𝑗^2𝑝𝑗+1≤𝑐𝜑_𝑛𝑥_𝑘𝑛^2𝑝𝑗+1,𝜅=𝑘,𝑘+1 (3.10) which yields (3.8).

Subcase 2.2 (𝑡_𝑗∉𝐈_𝑘). In this case we distinguish two subcases.(1)|𝑥_𝑘+1,𝑛−𝑡_𝑗|≥2𝑐₀𝜑_𝑛(𝑥_𝑘𝑛), where 𝑐₀ is given by (3.9). In this case ^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡=^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗𝑡−𝑡_𝑗^{2𝑝𝑗−1}≥𝑥𝑑𝑡_𝑘+1,𝑛−𝑡_𝑗^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^{2𝑝𝑗−1}=𝑥𝑑𝑡_𝑘+1,𝑛−𝑡_𝑗12𝑝_𝑗𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗−𝑥_𝑘𝑛−𝑡_𝑗^2𝑝𝑗≥𝑐₀𝜑_𝑛𝑥_𝑘𝑛𝑝_𝑗𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗−𝑥_𝑘𝑛−𝑡_𝑗^2𝑝𝑗, (3.11)which by (3.9) gives 𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗≤1−𝑝_𝑗⁻¹𝑥_𝑘𝑛−𝑡_𝑗^2𝑝𝑗≤2𝑥_𝑘𝑛−𝑡_𝑗^2𝑝𝑗. (3.12) On the other hand, by (3.9) and (3.12), 𝑐₀𝜑_𝑛𝑥_𝑘𝑛𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗≥^𝑥𝑘𝑛_{𝑥𝑘+1,𝑛}𝑡−𝑡_𝑗^2𝑝𝑗𝑑𝑡≥𝑥_𝑘𝑛−𝑡_𝑗^2𝑝𝑗𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛≥12𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗𝑥_𝑘𝑛−𝑥_𝑘+1,𝑛 (3.13) and hence (3.8) follows.(2)|𝑥_𝑘+1,𝑛−𝑡_𝑗|<2𝑐₀𝜑_𝑛(𝑥_𝑘𝑛). By (3.9), 𝑐₀𝜑_𝑛𝑥_𝑘𝑛^2𝑝𝑗+1≥12𝑝_𝑗𝑥+1^𝑘𝑛−𝑡_𝑗^2𝑝𝑗+1−𝑥_𝑘+1,𝑛−𝑡_𝑗^2𝑝𝑗+1≥12𝑝_𝑗𝑥+1^𝑘𝑛−𝑡_𝑗^2𝑝𝑗+1−2𝑐₀𝜑_𝑛𝑥_𝑘𝑛^2𝑝𝑗+1. (3.14)So 𝑥_𝑘𝑛−𝑡_𝑗≤𝑐𝜑_𝑛(𝑥_𝑘𝑛) and (3.8) follows.

Finally, applying Theorem 5.7(b) in [1, page 125] we conclude 𝜑_𝑛(𝑥_𝑘𝑛)∼𝜑_𝑛(𝑥_𝑘𝑛) and hence (1.22) follows from (3.8).

3.2. Proof of Theorem 1.8

For 𝑃∈𝒫_{𝑡−𝜌−2/𝑝}, we have 𝑃𝑈∈𝒫_{𝑡−2/𝑝} and hence apply Theorem 1.8 in [1, page 15] to obtain (1.23).

3.3. Proof of Theorem 1.9

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

3.4. Proof of Theorem 1.10

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 1𝛼+1𝛽=1,2𝛽𝑝_𝑖>−1,𝑖=1,…,𝑟. (3.15) Let 𝐿_𝑛 denote the linear map of Δ_𝑛 onto [−1,1]. By Lemma 11.7 in [1, page 318] there exists 𝑦_𝑛∈Δ_𝑛 such that 𝐿_𝑛𝑦_𝑛=cos2𝜋𝑚,𝑚=𝑚(𝑛), (3.16) and for large enough 𝑛 and 𝑅_𝑛∈𝐏_𝑛−2𝑚 such that 𝑅_𝑛(𝑥)𝑊(𝑥)^1/𝛼≥𝐶₁,𝑥∈0,𝑦_𝑛,𝑅_𝑛𝑊^1/𝛼_{𝐿∞(𝐈)}≤𝐶₂. (3.17)(3.18) Using (11.7) in [1, page 318] in the form 𝑥1−^1𝑛𝑎_𝑛=min_{𝑃∈𝐏𝑛−1}∫_𝐈1−𝑥/𝑎_𝑛(𝑃𝑈𝑊)²(𝑥)𝑑𝑥∫_𝐈(𝑃𝑈𝑊)²(𝑥).𝑑𝑥 (3.19) Again choose [1, page 319] 𝑃(𝑥)=𝑅_𝑛(𝑥)𝑉_{𝑚,cos(2𝜋/𝑚)}𝐿_𝑛(𝑥)²∈𝐏_𝑛−2. (3.20) Applying Theorem 1.5 and (3.18), and using the same argument as that in [1, pages 319-320], we can get _𝐈𝑥1−𝑎_𝑛(𝑃𝑈𝑊)²(𝑥)𝑑𝑥≤𝑐_Δ𝑛𝑥1−𝑎_𝑛(𝑃𝑈𝑊)²(𝑥)𝑑𝑥=𝑐_Δ𝑛𝑥1−𝑎_𝑛𝑃(𝑥)𝑊(𝑥)^1/𝛼2𝑈(𝑥)𝑊(𝑥)^1/𝛽2𝑑𝑥≤𝑐_Δ𝑛𝑥1−𝑎_𝑛𝑃(𝑥)𝑊(𝑥)^1/𝛼2𝛼𝑑𝑥^1/𝛼_Δ𝑛𝑈(𝑥)𝑊(𝑥)^1/𝛽2𝛽𝑑𝑥^1/𝛽≤𝑐_Δ𝑛𝑥1−𝑎_𝑛𝑃(𝑥)𝑊(𝑥)^1/𝛼2𝛼𝑑𝑥^1/𝛼≤𝑐_Δ𝑛𝑥1−𝑎_𝑛𝑉_{𝑚,cos(2𝜋/𝑚)}𝐿_𝑛(𝑥)^4𝛼𝑑𝑥^1/𝛼=𝑐𝛿_𝑛𝑎_𝑛Δ𝑛1−𝐿_𝑛(𝑥)𝑉_{𝑚,cos(2𝜋/𝑚)}𝐿_𝑛(𝑥)^4𝛼𝑑𝑥^1/𝛼=𝑐𝛿²_𝑛𝑎_𝑛¹₋₁(1−𝑡)𝑉_{𝑚,cos(2𝜋/𝑚)}(𝑡)^4𝛼𝑑𝑡^1/𝛼≤𝑐𝛿²_𝑛𝑎_𝑛𝑚^4∞_−∞|𝑣𝑐(1+|)min1,|𝑣|^4𝛼𝑑𝑣^1/𝛼≤𝑐𝑎_𝑛𝜂²_𝑛. (3.21)

On the other hand, by (3.17), _𝐈(𝑃𝑈𝑊)²(𝑥)𝑑𝑥≥^𝑦𝑛_{𝑦𝑛(1−𝐶1𝜂𝑛)}(𝑃𝑈𝑊)²(𝑥)≥𝑑𝑥_{𝑦𝑛𝑦𝑛(1−𝐶1𝜂𝑛)}𝑉_{𝑚,cos(2𝜋/𝑚)}𝐿_𝑛(𝑥)⁴𝑈(𝑥)²𝑑𝑥. (3.22) By (1.20) for large enough 𝑛, we have 𝑈(𝑥)≥𝑐>0,𝑥∈𝑦_𝑛1−𝐶₁𝜂_𝑛,𝑦_𝑛. (3.23) Hence (3.22) implies _𝐈(𝑃𝑈𝑊)²(𝑥)𝑑𝑥≥𝑐^𝑦𝑛_{𝑦𝑛(1−𝐶1𝜂𝑛)}𝑉_{𝑚,cos(2𝜋/𝑚)}𝐿_𝑛(𝑥)⁴𝑑𝑥=𝑐𝛿_𝑛()^{cos(2𝜋/𝑚)}_{cos2𝜋/𝑚−𝐶1𝑦𝑛𝜂𝑛/𝛿𝑛}𝑉_{𝑚,cos(2𝜋/𝑚)}(𝑡)⁴𝑑𝑡. (3.24) But in [1, page 320] the following estimate is given: 𝛿_𝑛()^{cos(2𝜋/𝑚)}_{cos2𝜋/𝑚−𝐶1𝑦𝑛𝜂𝑛/𝛿𝑛}𝑉_{𝑚,cos(2𝜋/𝑚)}(𝑡)⁴𝑑𝑡≥𝑐𝑎_𝑛𝜂_𝑛. (3.25) Substituting this estimate into (3.24) gives _𝐈(𝑃𝑈𝑊)²(𝑥)𝑑𝑥≥𝑐𝑎_𝑛𝜂_𝑛 (3.26) which coupled with (3.21) yields (1.26).

Next let us prove (1.28). We already know that 𝑎_𝑛1−𝑐𝜂_𝑛≤𝑥_1𝑛<𝑎_{𝑛+𝜌+1/2}=𝑎_𝑛1+𝑜𝜂_𝑛, (3.27) by (1.26) and (1.24). We must prove that, for some 𝑐₁>0, and 𝑛 large enough, we have 𝑥_1𝑛<𝑎_𝑛1−𝑐₁𝜂_𝑛. (3.28)

We use the idea for the proof of Corollary 13.4(b) in [1, pages 380-381] with modification. By the same argument as that proof with 𝐴=𝑎_{𝑛+𝜌+1/2}(1−𝜀𝜂_𝑛) instead, applying Theorem 1.8 we obtain 𝑥1−^1𝑛𝐴=𝜆_𝑛(𝑈𝑊)²,𝑥_1𝑛⁻¹_𝐈𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥,_𝐈𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥=^𝐴_𝑎𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥−^𝑏_𝐴𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥≥^𝐴_𝑎𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²−𝑑𝑥_{𝑎𝑛+𝜌+1/2𝐴}𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥−_{Δ𝑛+𝜌+1/2}𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥≥−2^{𝑎𝑛+𝜌+1/2}_𝐴𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²𝑑𝑥, (3.29)(3.30) where ℓ_1𝑛 denotes the fundamental polynomial of Lagrange interpolation based on the zeros of the 𝑛th orthogonal polynomial with respect to the weight (𝑈𝑊)².

But ^{𝑎𝑛+𝜌+1/2}_𝐴𝑥1−𝐴ℓ_1𝑛𝑈𝑊(𝑥)²≤𝑎𝑑𝑥_{𝑛+𝜌+1/2}𝐴−1_𝐈ℓ_1𝑛𝑈𝑊(𝑥)²𝑎𝑑𝑥=^{𝑛+𝜌+1/2}𝐴−1𝜆_𝑛(𝑈𝑊)²,𝑥_1𝑛 (3.31) which, coupled with (3.30) and (3.29), gives 𝑥1−^1𝑛𝐴≥−𝑐𝜀𝜂_𝑛. (3.32) Thus 𝑥_1𝑛𝑎_𝑛=𝑥_1𝑛𝐴𝐴𝑎_{𝑛+𝜌+1/2}𝑎_{𝑛+𝜌+1/2}𝑎_𝑛≤1+𝑐𝜀𝜂_𝑛1−𝜀𝜂_𝑛1+𝑜𝜂_𝑛<1−𝑐₁𝜂_𝑛, (3.33) for 𝑛 large enough, provided 𝜀>0 is small enough.

Acknowledgments

The authors thank the referee for carefully reading their paper, and making helpful suggestions and comments on improving their original paper. The research is supported in part by the National Natural Science Foundation of China (no. 11171100, no. 10871065, and no. 11071064) and by Hunan Provincial Innovation Foundation for Postgraduate.