Abstract and Applied Analysis

Volume 2012 (2012), Article ID 386359, 17 pages

doi:10.1155/2012/386359

Research Article

The Zeros of Orthogonal Polynomials for Jacobi-Exponential Weights

Rong Liu and Ying Guang Shi

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.

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 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 [5] considered generalized Freud weights in . Recently the second author [6] 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.

Theorem 1.5 (see [6, Theorem 1.2]). Let , where is convex with and . Let . Assume that relation (1.16) is valid. Then there exist such that, for and ,

Theorem 1.6 (see [6, Theorem 1.3]). Let , and . Assume that relation (1.16) is valid. Then there exist such that, for and ,

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 [7] 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.

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 . Then for fixed and uniformly for ,

Moreover, there exists such that, for , the inequalities

hold.

Lemma 2.2 (Shi [6]). Let . Then, for large enough ,

Lemma 2.3. Let , and . Then, for ,

Proof. By the same argument as that of [8, (2.25)] we can prove (2.4). By (2.4) and (2.1) 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 [8], (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

Lemma 2.7. Let . Let (1.16), (1.20), and (1.21) prevail. Then there exists such that, for and for each index ,

holds uniformly for .

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.

Corollary 2.8. Let . Let (1.16) and (1.20) prevail. If

then (2.27) holds.

In particular, if 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 then by (2.2) with 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 's the fundamental polynomials based on the zeros 's. By Theorem 1.4 and Lemma 11.8 in [8, pages 320-321]

On the other hand, by Theorem 1.4,

Then for ,

Let be defined by (2.14). Using Lemma 2.5 it follows from (3.3) that

Further, by (2.27),

By calculation from (3.5) we get

where

We separate two cases.

Case 1 (). Using Lemma 2.6 it follows from (3.6) that

Case 2 (). Suppose without loss of generality that for the case when . By (3.6),

Subcase 2.1 (). Inequality (3.9) gives

which yields (3.8).

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.

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

3.2. Proof of Theorem 1.8

For , we have 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 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).

Next let us prove (1.28). We already know that

by (1.26) and (1.24). We must prove that, for some , and large enough, we have

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 .

But

which, coupled with (3.30) and (3.29), gives

Thus

for large enough, provided 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.

References

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