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

Let w be a weight in I ≔ (a, b), − ∞≤ a< 0< b≤∞, for which themoment problem possesses an unique solution. Pn stands for the set of polynomials of degree at most n. ‖·‖Lp(I) is an usual (weighted) Lp (quasi) norm on interval I. Assume that W � 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


Introduction and Results
Let w be a weight in I ≔ (a, b), − ∞ ≤ a < 0 < b ≤ ∞, for which the moment problem possesses an unique solution. P n stands for the set of polynomials of degree at most n. ‖·‖ L p (I) is an usual (weighted) L p (quasi) norm on interval I.
Assume that W � e − Q where Q: I ⟶ [0, ∞) is continuous. W is an exponential weight on I. Also, let 0 < p < ∞, a ≤ t r < t r− 1 < · · · < t 2 < t 1 ≤ b,p i > − 1/p, i � 1, 2, . . . , r, and where U is a generalized Jacobi weight on I. e combination WU is called a Jacobi-exponential weight on I. is paper deals with the zeros of orthogonal polynomials and Markov-Bernstein inequalities for Jacobi-exponential weights. e letters c, C 0 , C 1 , . . . 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 c 1 and c 2 such that c 1 ≤ C n /D n ≤ c 2 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 (see [1], Definition 1.7, p. 14). Given c, t ≥ 0 and a non-negative Borel measure ] with compact support in C and total mass ≤t, we say that is an exponential of a potential of mass ≤t. We denote the set of all such P by P t . We note that for P ∈ P n , |P| ∈ P t , t ≥ n.
Definition 2 (see [1], p. 19). Let w be a weight in I. For 0 < p < ∞, generalized Christoffel functions with respect to w for z ∈ C are defined by For p � ∞, generalized Christoffel functions with respect to w for z ∈ C are defined by Moreover, for the classical Christoffel function λ n (w 2 ; x) with respect to w 2 , we have Definition 3. (see [1], pp. [10][11][12]. Let a < 0 < b. Assume that W � e − Q where Q: I ⟶ [0, ∞) satisfies the following properties: is quasidecreasing in (a, 0) and quasi-increasing in en, we write W ∈ F.
e estimates of the zeros [6] are based on the condition a < t r < t r− 1 < · · · < t 2 < t 1 < b. In [6], we did not consider the case when a � t r and t 1 � b, which is different from a < t r , t 1 < b. In this paper, we discuss orthogonal polynomials for generalized Jacobi-exponential weights in the case − 1 � t r < t r− 1 < · · · < t 2 < t 1 � 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 ( eorem 4) cannot follow from it. e 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: Some corresponding notations for W * (x) are also needed: In all that follows, I denotes the open interval (− 1, 1).
Theorem 1 (see [7], eorem 1.7). Let W ∈ F(Lip(1/2)) and 0 < p < ∞. Assume that and for some constant μ satisfying (a) en there exists n 0 > 0 such that for n ≥ n 0 and x ∈ J * L,n with L > 0, the relation uniformly holds. (b) Furthermore, there exists n 1 > 0 such that for n ≥ n 1 and x ∈ I, the relation uniformly holds.
By specializing to p � 2 of eorem 1, we obtain estimates for the classical Christoffel functions. (a) en, there exists n 0 > 0 such that for n ≥ n 0 and x ∈ J * L,n with L > 0, the relation uniformly holds. (b) Furthermore, if p 0 ≤ 0, there exists C, n 1 > 0 such that for n ≥ n 1 and x ∈ I, the relation uniformly holds.
Our results will mainly center on the zeros of orthogonal polynomials for Jacobi-exponential weights UW and Markov-Bernstein inequalities.

Theorem 5.
Assume that the assumptions of eorem 2 hold. en,

Theorem 6. Let W ∈ F(Lip(1/2)). Assume that relation (16) is valid and Q is nondecreasing in I.
(a) en, (b) Furthermore, if p i ≥ 0, i � 2, . . . , r − 1, then for large enough n, We prove eorems 2-4 and eorem 6 in Section 3, but first we need some auxiliary lemmas and the proofs of Corollary 1 and eorem 5, which are presented in Section 2.

Auxiliary Lemmas
In particular, this holds for not identically vanishing polynomials P of degree ≤t − 2/p. For p � ∞, (31) holds with < replaced by ≤ .

Lemma 4. For fixed index
Proof. Following the argument in the proof of Lemma 2.5 in [6], we get (35) by replacing δ n /n with 1/n. □ Lemma 5. Let W ∈ F and (25) be valid. en, there exists t 0 > 0 such that for t > t 0 and for each index j, 2 ≤ j ≤ r − 1, holds uniformly for x ∈ I.

□
Proof of Corollary 1. It is the special case of eorem 1 when p � 2 we use (5) and the relation φ * n ∼ φ * n− 1 in I from Lemma 9.7 [1]. We also see that U n ∼ U n− 1 for n large enough. □ Proof of eorem 5. By Lemma 3, W * satisfies the conditions of W.
r − 1, so (28) follows directly from eorem 1.9 in [6]. □ Lemma 7. Let W ∈ F(Lip(1/2)) and r � 2. Assume that relation (17) is valid and Q is nondecreasing in I. Let ℓ jn ∈ P n− 1 be the fundamental polynomials of Lagrange Interpolation at the zeros p n ((UW) 2 , x) satisfying ℓ jn (x kn ) � δ kj . en, for each index j, 1 ≤ j ≤ n and large enough n, Proof. Notice that where K n (x, t) � n k�0 p k (x)p k (t) is the n th reproducing kernel function. Applying the Cauchy-Schwarz inequality to K n (x, t), we obtain By Lemma 6 and (28), we see I j ⊂ J * L,n , 1 ≤ j ≤ n. Now applying the Christoffel function bounds of Corollary 1 (a) and (b), it follows from the above relation that According to the definition of W * , and then which by (2.23) in [7] for x ∈ J * L,n gives It follows from (48) that for large enough n, as when r � 2, Further, applying eorem 5.7(b) in [1], we conclude for so that and with a similar discussion, we also have is proves (42).
e properties of (a) − (e) in Definition 3 hold for W * if W ∈ F(Dini) 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 F(Lip(1/2)) and F(Dini). We will prove that the property of (f) in Definition 3 also holds for W * .

Proof of eorem 4.
(a) e proof is similar to eorem 1.7 in [6], but we provide the details with modification. Denote by ℓ kn n k�1 the fundamental polynomials of Lagrange interpolation at the zeros x kn n k�1 of the orthogonal polynomials p n ((WU) 2 , x) for the weight (WU) 2 .