Journal of Mathematics

Journal of Mathematics / 2020 / Article

Research Article | Open Access

Volume 2020 |Article ID 7805730 | https://doi.org/10.1155/2020/7805730

Rong Liu, "The Zeros of Orthogonal Polynomials and Markov–Bernstein Inequalities for Jacobi-Exponential Weights on (−1,1)", Journal of Mathematics, vol. 2020, Article ID 7805730, 9 pages, 2020. https://doi.org/10.1155/2020/7805730

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

Academic Editor: Markos Koutras
Received18 Jan 2020
Revised26 Jun 2020
Accepted30 Jul 2020
Published24 Aug 2020

Abstract

Let , , , and where . We give the estimates of the zeros of orthogonal polynomials for the Jacobi-Exponential weight on . In addition, Markov–Bernstein inequalities for the weight are also obtained.

1. Introduction and Results

Let be a weight in , for which the moment problem possesses an unique solution. stands for the set of polynomials of degree at most n. is an usual (weighted) (quasi) norm on interval .

Assume that where is continuous. is an exponential weight on . Also, let andwhere is a generalized Jacobi weight on . The combination is called a Jacobi-exponential weight on . This paper deals with the zeros of orthogonal polynomials and Markov–Bernstein inequalities for Jacobi-exponential weights.

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 (see [1], Definition 1.7, p. 14). Given and a non-negative Borel measure with compact support in and total mass , we say thatis an exponential of a potential of mass . We denote the set of all such by .
We note that for ,

Definition 2 (see [1], p. 19). Let be a weight in . For , generalized Christoffel functions with respect to for are defined byFor , generalized Christoffel functions with respect to for are defined byMoreover, for the classical Christoffel function with respect to , we haveA function is said to be quasi-increasing (or quasidecreasing) if there exists such that

Definition 3. (see [1], pp. 10–12). Let . Assume that where satisfies the following properties:(a) and .(b) is nondecreasing in .(c)(d)The functionis quasidecreasing in and quasi-increasing in , respectively. Moreover, (e)There exists such that for ,Then, we write .(f)Furthermore, assume that there exist such that for all ,Then, we write .
In addition, let . Assume that there exist such that for all ,Then, we write .
For and , the Mhaskar–Rahmanov–Saff numbers are defined by the equationsPut for ,In 1994 and 2001, Levin and Lubinsky [1, 2] discussed orthogonal polynomials for exponential weights on and , respectively. Then, they [3, 4] dealt with exponential weights , in . Kasuga and Sakai [5] considered generalized Freud weights in . Recently, we discussed generalized Jacobi-exponential weights [6, 7], which centered on the distribution of zeros and the estimates of the generalized Christoffel functions, respectively. Shi [8] also considered Jacobi-exponential weights and subsequently dealt with a particular case on in [9].
For the weight on , its orthogonal polynomial has zeros , where
The estimates of the zeros [6] are based on the condition . In [6], we did not consider the case when and , which is different from . In this paper, we discuss orthogonal polynomials for generalized Jacobi-exponential weights in the case
Mastroianni and Totik in [10] gave the estimates of the spacing of zeros for doubling weights; in general, however, Jacobi-exponential weights are not doubling weights, so our main result (Theorem 4) cannot follow from it. The 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 and the real semiaxis to deal with Lagrange interpolation processes on corresponding interval, respectively.
We construct the following weight:Some corresponding notations for are also needed:In all that follows, denotes the open interval .

Theorem 1 (see [7], Theorem 1.7). Let and . Assume thatand for some constant satisfyingthe function is nondecreasing in .(a)Then there exists such that for and with , the relationuniformly holds.(b)Furthermore, there exists such that for and , the relationuniformly holds.

By specializing to of Theorem 1, we obtain estimates for the classical Christoffel functions.

Corollary 1. Assume that the conditions of Theorem 1 hold.

(a)Then, there exists such that for and with , the relationuniformly holds.(b)Furthermore, if , there exists such that for and , the relationuniformly holds.

Our results will mainly center on the zeros of orthogonal polynomials for Jacobi-exponential weights and Markov–Bernstein inequalities.

Theorem 2. Let , where is convex with and . Let , , . Assume that relation (16) is valid and is nondecreasing in . Then,

In particular, this holds for not identically vanishing polynomials of degree . For , (22) holds with replaced by .

Theorem 3. Let and . Assume that relation (16) is valid and is nondecreasing in .

(a)Let . Then, for and ,(b)Let and . Then, for and ,

Theorem 4. Let and . Assume that relation (16) is valid, is nondecreasing in , and

(a)Then, for large enough and ,(b)Furthermore, if , then for large enough and ,

Remark 1. By [7], (Lemma 2.12), for zeros with , the statement (a) of Theorem 4 is valid and can be replaced by .

Theorem 5. Assume that the assumptions of Theorem 2 hold. Then,

Theorem 6. Let . Assume that relation (16) is valid and is nondecreasing in .

(a)Then,(b)Furthermore, if , then for large enough ,

We prove Theorems 24 and Theorem 6 in Section 3, but first we need some auxiliary lemmas and the proofs of Corollary 1 and Theorem 5, which are presented in Section 2.

2. Auxiliary Lemmas

Lemma 1 (see [1], Theorem 4.1, p. 95). Let , where is convex with and . Let and . Then,

In particular, this holds for not identically vanishing polynomials of degree . For , (31) holds with replaced by .

Lemma 2 (see [1], Theorem 10.1, p. 293). Let .

(a)Let . Then, for and ,(b)Let and . Then, for and ,

Lemma 3 (see [7], Lemma 2.13). Let and . Assume that (16) is valid and is nondecreasing in . Then, .

Lemma 4. For fixed index , let . Let , satisfy

Then,

Proof. Following the argument in the proof of Lemma 2.5 in [6], we get (35) by replacing with

Lemma 5. Let and (25) be valid. Then, there exists such that for and for each index ,holds uniformly for .

Proof. By (1.55) in [1], we see that there exists such that for , , so we have for . Also, notice that , and (36) follows from Lemma 2.7 in [6].

Lemma 6. Let . Assume that relation (16) is valid and is nondecreasing in . Then, there exists such that for large enough,

Proof. By Lemma 3.11(a) in [1], for ,Fix ; for , we haveOn the other hand, using Definition 2 of , we obtainas and .
Thus, by (38), for large enough ,This yields (37).
Since the last lemma is based on the results of Corollary 1 and Theorem 5, we present the proofs of Corollary 1 and Theorem 5 first.

Proof of Corollary 1. It is the special case of Theorem 1 when we use (5) and the relation in from Lemma 9.7 [1]. We also see that for large enough.

Proof of Theorem 5. By Lemma 3, satisfies the conditions of . For , we have . Meanwhile, , so (28) follows directly from Theorem 1.9 in [6].

Lemma 7. Let and . Assume that relation (17) is valid and is nondecreasing in . Let be the fundamental polynomials of Lagrange Interpolation at the zeros satisfying . Then, for each index and large enough ,

Proof. Notice thatwhere is the reproducing kernel function. Applying the Cauchy–Schwarz inequality to , we obtainBy Lemma 6 and (28), we see . Now applying the Christoffel function bounds of Corollary 1 (a) and (b), it follows from the above relation thatAccording to the definition of ,and thenwhich by (2.23) in [7] for givesIt follows from (48) that for large enough ,as when ,Further, applying Theorem 5.7(b) in [1], we conclude for ,so thatand with a similar discussion, we also haveThis proves (42).

3. Proof of Theorems

3.1. Proof of Theorem 2

It is easy to check that is convex with and , so by considering Lemma 3, satisfies the assumptions about . Furthermore, for ,

Observe that

Then, applying Lemma 1, we obtain the results.

3.2. Proof of Theorem 3

(a)By Lemma 3, . For , we have . Thus, by (55), relation (23) follows from (32).(b)If , then with the similar discussion as (a) and using (33), we prove that the statement of (b) is valid. So, it is necessary to prove that if , then .

The properties of in Definition 3 hold for if 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 and . We will prove that the property of in Definition 3 also holds for .

By (2.38) in [7], we have

According to Definition 3,

Meanwhile, using Corollary 2.1 (a) in [7], for , we seewhere and and are shown in (2.29) and (1.23) in [7], respectively, and hence

Using this relation andwe obtain

By (2.30) and (2.35) in [7], we further get