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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 386359, 17 pages
The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Hunan, Changsha 410081, China
Received 13 July 2012; Revised 14 October 2012; Accepted 19 October 2012
Academic Editor: Patricia J. Y. Wong
Copyright © 2012 Rong Liu and Ying Guang Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 be a weight in , for which the moment problem possesses a unique solution. Denote by the set of positive integers. stands for the set of polynomials of degree at most .
Assume that where is continuous. Also, let ,
The letters 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, means that there are two constants and such that for the relevant range of . We write or to indicate dependence on or independence of a parameter .
Definition 1.1 (see [1, Definition 1.7, page 14]). Given , and a nonnegative Borel measure with compact support in and total mass , one says that is an exponential of a potential of mass . One denotes the set of all such by .
One notes that, for ,
Definition 1.2 (see [1, page 19]). Let be a weight in . For , generalized Christoffel functions with respect to for are defined by
For , generalized Christoffel functions with respect to for are defined by
Obviously, for the classical Christoffel function with respect to , we have
A function is said to be quasi-increasing (or quasi-decreasing) if there exists such that
Definition 1.3 (see [1, pages 10–12]). Let . Assume that where satisfies the following properties(a) and .(b) is nondecreasing in .(c) We have
(d) The function
is quasi-decreasing in and quasi-increasing in , respectively. Moreover
(e) There exists such that, for ,
Then we write .(f) In addition, assume that there exist such that, for all ,
Then we write (Lip(1/2)).
For and , the Mhaskar-Rahmanov-Saff numbers are defined by the equations Put for , Let
In 1994 and 2001, Levin and Lubinsky [1, 2] published their monographs on orthogonal polynomials for exponential weights . Then they [3, 4] discussed orthogonal polynomials for exponential weights , in , since the results of [1, 2] cannot be applied to such weights. Kasuga and Sakai  considered generalized Freud weights in . Recently the second author  obtained the Christoffel functions for Jacobi-exponential weights , 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 , and . Assume that Then there exists such that, for and , the relation uniformly holds.
In this paper we discuss the zeros of orthogonal polynomials for Jacobi-exponential weights and restricted range inequalities.
Theorem 1.7. Let . Assume that (1.16) is valid, and Then
Theorem 1.8. Let , where is convex with and . Let . Assume that all are positive and relation (1.16) is valid. Then there exist such that, for and ,
Theorem 1.9. Let the assumptions of Theorem 1.8 prevail. Then
Theorem 1.10. Let ( (1/2)). Then If all , then
Here we should point out that our main result (Theorem 1.7) cannot follow from  given by Mastroianni and Totik, because in general Jacobi-exponential weights are not doubling weights, although Jacobi weights 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 , by L'Hospital rule which implies that is not a doubling weight.
2. Auxiliary Lemmas
Lemma 2.1 (Levin and Lubinsky [1, Lemma 3.5, pages 71-72]). Let . Then for fixed and uniformly for ,
Moreover, there exists such that, for , the inequalities hold.
Lemma 2.2 (Shi ). Let . Then, for large enough ,
Lemma 2.3. Let , and . Then, for ,
Lemma 2.4. Let . Then, for ,
Proof. By the definition of it is enough to prove (2.7) for . Without loss of generality we can assume that . By Lemma in [1, page 81] for , By Lemma 2.12 in , (2.3), (2.1), and (2.8), By in [1, page 15], and hence Thus
Let . Let, for and ,
Lemma 2.5. For fixed index , , let ,, satisfy Then
Proof. We give the proof of (2.15) for only, the proof of (2.15) for being similar.
We claim that, for , In fact, suppose without loss of generality that . It is enough to show (2.16) for . Because .
If then by (2.13) and hence if then by (2.14) which again implies (2.18). Then by (2.18) and hence . This proves (2.16).
With the help of (2.16) for and , Hence Furthermore, by (2.2) with So for , This proves (2.15).
By the same argument as that of Lemma in [9, page 157] replacing by , we can get its extension.
Lemma 2.6. Let , and Then
Proof. Let and . We separate two cases.
Case 1 (). In this case by (2.3) and (2.1), which coupled with (2.5) gives Hence (2.27) follows.
Case 2 (). In this case by (2.23), and by (1.21) Again (2.27) follows.
3. Proof of Theorems
3.1. Proof of Theorem 1.7
Case 2 (). Suppose without loss of generality that for the case when . By (3.6),
Subcase 2.2 (). In this case we distinguish two subcases.(1), where is given by (3.9). In this case which by (3.9) gives On the other hand, by (3.9) and (3.12), and hence (3.8) follows.(2). By (3.9), So and (3.8) follows.
3.2. Proof of Theorem 1.8
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
First let us prove (1.26). Choose so that Let denote the linear map of onto . By Lemma 11.7 in [1, page 318] there exists such that and for large enough and such that Using (11.7) in [1, page 318] in the form Again choose [1, page 319] Applying Theorem 1.5 and (3.18), and using the same argument as that in [1, pages 319-320], we can get
On the other hand, by (3.17), By (1.20) for large enough , we have Hence (3.22) implies But in [1, page 320] the following estimate is given: Substituting this estimate into (3.24) gives which coupled with (3.21) yields (1.26).
We use the idea for the proof of Corollary in [1, pages 380-381] with modification. By the same argument as that proof with instead, applying Theorem 1.8 we obtain where denotes the fundamental polynomial of Lagrange interpolation based on the zeros of the th orthogonal polynomial with respect to the weight .
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.
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- A. L. Levin and D. S. Lubinsky, “Christoffel functions and orthogonal polynomials for exponential weights on ,” Memoirs of the American Mathematical Society, no. 535, 1994.
- A. L. Levin and D. S. Lubinsky, “Orthogonal polynomials for exponential weights on ,” Journal of Approximation Theory, vol. 134, no. 2, pp. 199–256, 2005.
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- T. Kasuga and R. Sakai, “Orthonormal polynomials with generalized Freud-type weights,” Journal of Approximation Theory, vol. 121, no. 1, pp. 13–53, 2003.
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- Y. G. Shi, Power Orthogonal Polynomials, Nova Science Publishers, New York, NY, USA, 2006.