Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)

Abstract Let Wα,ρ = xα(1 – x2)ρe–Q(x), where α > – 12 $\begin{array}{} \displaystyle \frac12 \end{array}$ and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights Wα,ρ2 $\begin{array}{} \displaystyle W^2_{\alpha, \rho} \end{array}$ and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.


Introduction and results
In this paper, for α > − , we set for which the moment problem possesses a unique solution, and discuss the orthogonal polynomials for the weight W α,ρ on [ , ). The main results tell us that adding an even factor ( − x ) ρ to the weight x α e − Q(x) , α > − , under su cient conditions for ρ and Q(x), its properties will be invariant. It is an important and meaningful extension to the case ρ = (we can see [1,2]).
Assume that where Q : I → [ , ∞) is continuous. All power moments for W exist. Such W is called an exponential weight on I. In the paper, for < p ≤ ∞, · L p(I) is the usual Lp (quasi) norm on the interval I. Levin and Lubinsky [3,4] discussed orthogonal polynomials for exponential weights W on [− , ] and (c, d), c < < d, respectively. Kasuga and Sakai [5] dealt with generalized Freud weights |x| α W(x) in (−∞, ∞). Liu and Shi [6] considered generalized Jacobi-Exponential weights UW, where U(x) is generalized Jacobi weights on (c, d), c < < d, and gave the estimates of the zeros of orthogonal polynomials for UW. Meanwhile, Shi [7] gave the estimates of the Lp Christo el functions for UW on (c, d). In [8], Liu and Shi got further estimations of the Lp Christo el functions for UW on [− , ]. In [9], Notarangelo stated analogues of the Mhaskar-Sa inequality for doubling-exponential weights on (− , ). The above references dealt with exponential weights on a real interval (c, d) containing in its interior. In [1,2], Levin and Lubinsky dealt with exponential weights x α W(x) , α > − / , in [ , d), since the results of [3,4] cannot be applied through such one-sided weights. All the results on one-sided case and two-sided case are useful in polynomial approximation. Mastroianni and Notarangelo [10,11] considered Lagrange interpolation processes based on the zeros for exponential weight on (− , ) and the real semiaxis, respectively.
Levin and Lubinsky [1,2] de ned an even weight corresponding to the one-sided weight. The weight is denoted that and for x ∈ I * , Throughout, c, C , C , . . . stand for positive constants independent of variables and indices, unless otherwise indicated and their values may be di erent at di erent occurrences, even in subsequent formulas. Moreover, Cn ∼ Dn means that there are two constants c and c such that c ≤ Cn /Dn ≤ c 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. Pn stands for the set of polynomials of degree at most n. √ xQ (x) ∈ C(I) with limit at and Q( ) = .
Then we write W ∈ L(C ). If also there exists a compact subinterval J of I * , and C > such that
For W ∈ L(C ) and t > , the Mhaskar-Rahmanov-Sa number < a t := a t (Q) is de ned by the equation Put for t > , We also need a modi cation of φ t , namely The orthogonal polynomial of degree n for W α,ρ is denoted by pn(W α,ρ , x) or just pn(x). Thus where γn = γn(W α,ρ ) > . The zeros of pn(x) are denoted by xnn < x n− ,n < · · · < x n < x n , and the corresponding fundamental polynomials of Lagrange interpolation are polynomials jn ∈ P n− . The classical Christo el function is Considering the factor ( − x ) ρ , we introduce the following weight Before stating our results, we need some corresponding notations, The following theorems are similar in spirit with their analogues for weights ( − x ) ρ e −Q(x) on two-sided intervals [12]. while the results of [12] cannot be applied to one-sided case. Furthermore, the formulation of the results are di erent, just as there are between the Laguerre and Hermite weights.
According to the above Theorem and applying Theorem 1.5 in [1] and Theorem 1.4 and Theorem 1.5 in [2], we gain the following Theorem 1.2. We also get the following Theorem 1.3, by using Theorem 1.2 and Theorem 1.3 in [1].

Theorem 1.2. Let α > − , W ∈ L(C ) and the other assumptions of Theorem 1.1 be valid. Assume that
There exist C , n > such that for n ≥ n and P ∈ Pn, Moreover, given r > , there exist C , n , ν > such that for n ≥ n and P ∈ Pn, Then for n ≥ , P ∈ Pn and for some C, .
(c) Let L > . Then uniformly for n ≥ and x ∈ [ ,ân( + Lηn)], Moreover, there exists C > such that uniformly for n ≥ and x ∈ I, (d) Then uniformly for n ≥ , (e) There exist C , C > such that for n ≥ and ≤ j ≤ n − , Furthermore, for each xed j and n, x jn is a non-decreasing function of α. and x jn − x j+ ,n ∼φn(x jn ), ≤ j < n.
If W ∈ L(C ), these estimates hold with ∼ replaced by ≤ C. (b) There exists n such that uniformly for n > n , ≤ j ≤ n, . .
(a) Then the relation of (1.10) holds.
We shall give some technical lemmas in Section 2 and the proofs of Theorem 1.1, Theorem 1.4 and Theorem 1.5 in Section 3. Proof. The case when ρ = is trivial. Let ρ ≠ .
(b) Given (2.2), using (2.12) it is shown that one of the following cases will hols: . (2.14) Proof. By a short calculation we gain

Proof of Theorem 1.1
Here let ρ ≠ . The theorem for the case when ρ = is trivial. In deed, the authors in [1,2] discussed the case when ρ = . We use the idea of Theorem 1.1 in [12] with modi cation, and according to Lemma 2.1 it is more easier to get µ since we only require ε > in formula (2.3).
We setQ With De nition 1.1 (a) for Q we haveQ ( ) = and by (2.4) which shows that √ xQ (x) is continuous in I, with limit at .
This proves the properties listed in De nition 1.1 (a) with Q replaced byQ.
(3.15) (b) Using (2.14) we have Then by (1.11) By (3.15) we see Λ = and hence coupling with the above relation we obtain that (1.6) and (1.7) are valid. Thus applying Theorem 1.1 we get W ∈ L(C +). By the statements of (a), we see Λ = and hence coupling with the above relation (1.6) and (1.7) are valid. Thus applying Theorem 1.1 we get W ∈ L(C +). 2