`Journal of Function Spaces and ApplicationsVolume 2013 (2013), Article ID 628031, 11 pageshttp://dx.doi.org/10.1155/2013/628031`
Research Article

Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

1Faculty Mathematics and Computer Science, St. Louis University (Madrid Campus), Avenida del Valle 34, 28003 Madrid, Spain
2Departamento de Matemáticas Puras y Aplicadas, Edificio Matemáticas y Sistemas (MYS), Universidad Simón Bolívar, Apartado Postal 89000, Caracas 1080 A, Venezuela

Received 28 January 2013; Accepted 8 March 2013

Copyright © 2013 Ana Portilla et al. 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

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm , where the matrix and the measure constitute a -admissible pair for . In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to , stating hypothesis on the matrix rather than on the diagonal matrix appearing in its unitary factorization.

1. Introduction

In the last decades the asymptotic behavior of Sobolev orthogonal polynomials has been one of the main topics of interest to investigators in the field. In [1] the authors obtain the th root asymptotic of Sobolev orthogonal polynomials when the zeros of these polynomials are contained in a compact set of the complex plane; however, the boundedness of the zeros of Sobolev orthogonal polynomials is an open problem, but as was stated in [2], it could be obtained as a consequence of the boundedness of the multiplication operator . Thus, finding conditions to ensure the boundedness of would provide important information about the crucial issue of determining the asymptotic behavior of Sobolev orthogonal polynomials (see, e.g., [313]). The more general result on this topic is [3, Theorem 8.1] which characterizes in terms of equivalent norms in Sobolev spaces the boundedness of for the classical diagonal norm (see Theorem 3 below, which is [3, Theorem 8.1] in the case ). The rest of the above mentioned papers provides conditions that ensure the equivalence of norms in Sobolev spaces, and consequently, the boundedness of .

Results related to nondiagonal Sobolev norms may be found in [5, 6, 1419]. Particularly, in [5, 6, 15, 18, 19] the authors establish the asymptotic behavior of orthogonal polynomials with respect to nondiagonal Sobolev inner products and the authors in [5] deal with the asymptotic behavior of extremal polynomials with respect to the following nondiagonal Sobolev norms.

Let be the space of polynomials with complex coefficients and let be a finite Borel positive measure with compact support consisting of infinitely many points in the complex plane; let us consider the diagonal matrix , with being positive -almost everywhere measurable functions, and , a matrix of measurable functions such that the matrix  , is unitary -almost everywhere. If  , where denotes the transpose conjugate of (note that then is a positive definite matrix -almost everywhere), and we define the Sobolev norm on the space of polynomials

In [20, Chapter XIII] certain general conditions imposed on the matrix are requested in order to guarantee the existence of an unitary representation with measurable entries.

If is not the identity matrix -almost everywhere, then (2) defines a nondiagonal Sobolev norm in which the product of derivatives of different order appears. We say that is an th monic extremal polynomial with respect to the norm (2) if

It is clear that there exists at least an th monic extremal polynomial. Furthermore, it is unique if . If , then the th monic extremal polynomial is precisely the th monic Sobolev orthogonal polynomial with respect to the inner product corresponding to (2).

In [5, Theorem 1] the authors showed that the zeros of the polynomials in are uniformly bounded in the complex plane, whenever there exists a constant such that , -almost everywhere for . This property made possible to obtain the th root asymptotic behavior of extremal polynomials (see [5, Theorems 2 and 6]). Although it is required compact support for , this is, certainly, a natural hypothesis: if is not bounded, then we cannot expect to have zeros uniformly bounded, not even in the classical case (orthogonal polynomials in ); see [21].

Taking , and setting up hypothesis on the matrix (see (4)) rather than on the diagonal matrix , the authors of [22] the following equivalent result to [5, Theorem  1].

Theorem 1 (see [22, Theorem 4.3]). Let be a finite union of rectifiable compact curves in the complex plane, a finite Borel measure with compact support , a positive definite matrix -almost everywhere and Assume that , , and the norms in and are equivalent on . Let be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded with the norm and the zeros of lie in the bounded disk .

In this paper we improve Theorem 1 in two directions: on the one hand, we enlarge the class of measures considered and, on the other hand, we prove our result for (see Theorem 19). In order to describe the measures we will deal with, we introduce the definition of -admissible pairs as follows: given , we say that the pair is -admissible if is a finite Borel measure which can be written as , its support is a compact subset of the complex plane which contains infinitely many points, and is a positive definite matrix  -almost everywhere with ,   -almost everywhere for some fixed ; the support is contained in a finite union of rectifiable compact curves with if , and is the Radon-Nykodim derivative of with respect to the Euclidean length in .

We want to make three remarks about this definition. First of all, since is a positive definite matrix -almost everywhere, also has this property and hence , -almost everywhere.

In order to obtain the best choice for is the restriction of to .

Note that the support of is an arbitrary compact set: we just require that (the part of in which is about to be a degenerated quadratic form, when is very close to ) is a union of curves.

Therefore, with the results on -admissible pairs we complement and improve the study started in [22], where the case with was considered.

Another interesting property which could be studied is the asymptotic estimate for the behavior of extremal polynomials because, in this setting, there does not exist the usual three-term recurrence relation for orthogonal polynomials in and this makes it really difficult to find an explicit expression for the extremal polynomial of degree . In this regard, Theorems 22 and 23 deduce the asymptotic behavior of extremal polynomials as an application of Theorems 18 and 19. More precisely, we obtain the th root and the zero counting measure asymptotic both of those polynomials and their derivatives to any order. The study of the th root asymptotic is a classical problem in the theory of orthogonal polynomials; see for instance, [1, 2, 5, 23, 24].

Furthermore, in Theorem 23 we find the following asymptotic relation: for any .

The main idea of [5, 6, 22] and this paper is to compare nondiagonal and diagonal norms.

When it comes to compare nondiagonal and diagonal norms, [25] is remarkable, since the authors show that symmetric Sobolev bilinear forms, like symmetric matrices, can be rewritten with a diagonal representation; unfortunately, the entries of these diagonal matrices are real measures, and we cannot use this representation since we need positive measures for the Sobolev norms.

Finally, we would like to note that the central obstacle in order to generalize the results given in this paper and [22] to the case of more derivatives is that there are too many entries in the matrix and just a few relations to control them (see Lemma 8 and notice that some limits appearing in that Lemma do not provide any new information). In that case we have just three entries , but in the simple case of two derivatives we have and we would need to control six functions ; in the general case with derivatives, we would need to control functions.

The outline of the paper is as follows. In Section 2 we provide some background and previous results on the multiplication operator and the location of zeros of extremal polynomials. We have devoted Section 3 to some technical lemmas in order to simplify the proof of Theorem 17 about the equivalence of norms; in fact, in these lemmas the hardest part of this proof is collected. In Section 4 we give the proof of that Theorem and in Section 5 we deduce some results on asymptotic of extremal polynomials.

2. Background and Previous Results

In what follows, given we define for every polynomial .

It is obviously much easier to deal with the norms and than with the one . Therefore, one of our main goals is to provide weak hypotheses to guarantee the equivalence of these norms on the linear space of polynomials (see Section 4).

In order to bound the zeros of polynomials, one of the most successful strategies has certainly been to bound the multiplication operator by the independent variable , where Regarding this issue, the following result is known.

Theorem 2 (see [5, Theorem 3]). Let be a finite Borel measure in with compact support let and . Let be a sequence of extremal polynomials with respect to (2). Then the zeros of lie in the disk .

It is also known the following simple characterization of the boundedness of .

Theorem 3 (see [3, Theorem 8.1]). Let be a finite Borel measure in with compact support; nonnegative measurable functions; and . Then the multiplication operator is bounded in if and only if the following condition holds:

It is clear that if there exists a constant such that -almost everywhere, then (9) holds. In [8, 13] some other very simple conditions implying (9) are shown.

In what follows, we will fix a -admissible pair with ; then is contained in a finite union of rectifiable compact curves in the complex plane; each of these connected components of is not required to be either simple or closed.

3. Technical Lemmas

For the sake of clarity and readability, we have opted for proving all the technical lemmas in this section. This makes the proof of Theorem 17 much more understandable.

The following result is well known.

Lemma 4. Let us consider . Then

Lemma 5 (see [22, Lemma 3.1]). Let us consider . Then (1) for every ; (2) for every .

Lemma 6 (see [22, Lemma 3.2]). Let and be two sequences of positive numbers. Then

In what follows , and refer to the coefficients of the fixed matrix .

Definition 7. We say that is an extremal sequence for if, for every , and

Lemma 8. If and is an extremal sequence for , then

Proof. The case is a consequence of [22, Lemmas 3.5 and 3.6]. We deal now with the case . First note that we can rewrite limit (12) in Definition 7 as the limit of the following product:
Since the limit of the product is 1, if we prove that the first, third, and fourth factors tend to 1 as tends to infinity, then the limit of the second factor must also be 1.
So, our problem is reduced to show
Again, we can rewrite the limit in the definition of extremal sequence as the limit of the following product:
The two factors above are nonnegative and less than or equal to 1 using, respectively, that -almost everywhere and . Thus, and (15) holds.
Given , for each let us define the following sets:
Let us consider the strictly decreasing function on . If , then . Consequently, if , then , and if , then and . Therefore,
Using this fact and (20), we have
If we assume that , then from the previous inequality we have , and this is a contradiction. Hence, and consequently,
Since for each , we have then (24) implies that
On the other hand, using (20) it is easy to deduce that
Consequently, (24), (26), and (27) give
Furthermore, since we obtain
Therefore, (24), (28), and (30) give
Similar arguments allow us to show
From (31) and (32) we obtain
As a consequence of (33) we have
In a similar way we obtain Since these inequalities hold for every , we conclude that (18) holds. Applying now Lemma 6 we obtain (16).
Using Lemma 4, (18), and (34) we obtain that for every there exists such that for every the following holds:
Then (17) follows from the previous inequalities, since are arbitrary.
This completes the proof.

Definition 9. For each , we define the sets and as

Lemma 10. If and is an extremal sequence for and is small enough, then

Remark 11. The statement of the lemma might seem strange, because we could have a priori ; however, the existence of the fundamental sequence implies .

Proof. If , then the result follows from [22, Lemma 3.8]. For the case it suffices to follow the proof of [22, Lemma 3.8] applying Lemma 8 to conclude the result.

Lemma 12. If , is an extremal sequence for and is small enough, then

Proof. If , then the result follows from [22, Lemma 3.10]. For the case it suffices to follow the proof of [22, Lemma 3.10] applying Lemmas 8 and 10 to conclude the result.

Lemma 13. If , is an extremal sequence for and is small enough, then

Proof. If , then the result follows from [22, Lemma 3.11]. For the case it suffices to follow the proof of [22, Lemma 3.11] applying Lemmas 8, 10, and 12 to conclude the result.

Lemma 14. If and is an extremal sequence for , then for every small enough with and for every there exists such that for every .

Proof. If , then the result follows from [22, Lemma 3.12]. For the case it is sufficient to follow the proof of [22, Lemma 3.12] applying Lemma 13 to conclude the result.

Definition 15. If is a continuous function on , we define the oscillation of on , and we denote it by , as

Lemma 16 (see [22, Lemma 3.14]). For , let us assume that is connected and , where is the Radon-Nykodim derivative of with respect to the Euclidean length in . (According to one’s notation, if then .) Then for every polynomial .

4. Equivalent Norms

Now we prove the announced result about the equivalence of norms for .

Theorem 17. Let one consider and a -admissible pair. Then the norms , , and defined as in (3) are equivalent on the space of polynomials .

Proof. The equivalence of the two first norms is straightforward, by Lemmas 4 and 5. We prove now the equivalence of the two last norms.
Let us prove that there exists a positive constant such that
Let us prove first the second inequality .
Note that ; therefore, for every polynomial it holds that
In order to prove the first inequality, , note that
If (i.e., ), then we have finished the proof. Assume that then we prove , seeking for a contradiction. It is clear that it suffices to prove it when is connected, that is, when is a rectifiable compact curve. Let us assume that there exists a sequence such that
If , then [22, Lemma 3.1] (with ) gives
This right-hand side of the inequality is positive, because -almost everywhere. This implies and hence
If , then since -almost everywhere. Therefore, or, equivalently, and (50) also holds for .
If is constant for some , then ; therefore, taking a subsequence if it is necessary, without loss of generality we can assume that is nonconstant and for every . Then is an extremal sequence for . Applying Lemma 8,
By Lemma 14, there exists such that for every . Now, taking into account that and that is connected, we can apply Lemma 16, and then for every , with .
Let us fix small enough. On the one hand, by Lemma 13 it holds that for every .
On the other hand, we have This implies Given any there exists with . Hence, for every . Therefore, , which is a contradiction with (54) and (55).

The following result is a direct consequence of Theorems 3 and 17.

Theorem 18. Let one consider and a -admissible pair. Then the multiplication operator is bounded in if and only if the following condition holds:

This latter theorem and Theorem 2 give the following result.

Theorem 19. Let one consider and a -admissible pair such that (59) takes place. Let be a sequence of extremal polynomials with respect to (2). Then the multiplication operator is bounded and the zeros of lie in the bounded disk .

In general, it is not difficult to check wether or not (59) holds. It is clear that if there exists a constant such that -almost everywhere, then (59) holds. In [8, 13] some other very simple conditions implying (59) are shown.

The following is a direct consequence of Theorem 19.

Corollary 20. Let one consider and a -admissible pair. Assume that ,-almost everywhere for some constant . Let be a sequence of extremal polynomials with respect to (2). Then the zeros of are uniformly bounded in the complex plane.

Finally, we have the following particular consequence for Sobolev orthogonal polynomials.

Corollary 21. Let be a -admissible pair. Assume that there exists a constant such that ,-almost everywhere. Let be the sequence of Sobolev orthogonal polynomials with respect to . Then the zeros of the polynomials in are uniformly bounded in the complex plane.

5. Asymptotic of Extremal Polynomials

We start this section by setting some notation. Let , , , and denote, respectively, the th monic orthogonal polynomial with respect to , the usual norm in the space , the logarithmic capacity of , and the equilibrium measure of . Furthermore, in order to analyze the asymptotic behavior for extremal polynomials we will use a special class of measures, “regular measures,” denoted by and defined in [24]. In that work, the authors proved (see Theorem 3.1.1) that, for measures supported on a compact set of the complex plane, if and only if

Finally, if denote the zeros, repeated according to their multiplicity, of a polynomial whose degree is exactly , and is the Dirac measure with mass one at the point , the expression defines the normalized zero counting measure of .

We can already state the first result in this section.

Theorem 22. Let one consider , a -admissible pair and the sequence of extremal polynomials with respect to . Assume that the following conditions hold: (i);(ii) is regular with respect to the Dirichlet problem; (iii)condition (59) takes place. Then, Furthermore, if the complement of is connected, then in the weak star topology of measures.

Proof. Note that, in our context, the hypothesis removed with respect to [5, Theorem 2] is equivalent to the following two facts: on the one hand, the multiplication operator is bounded (see Theorem 3), and on the other hand, the norms of and defined as in (3) are equivalent (see Theorem 18). With this in mind, we just need to follow the proof of [5, Theorem 2] to conclude the result.

In the following theorem, we use to denote the Green's function for with logarithmic singularity at , where is the unbounded component of the complement of . Notice that, if is regular with respect to the Dirichlet problem, then is continuous up to the boundary and it can be extended continuously to all , with value zero on .

Theorem 23 23. Let one consider , a -admissible pair and the sequence of extremal polynomials with respect to . Assume that the following conditions hold: (i);(ii) is regular with respect to the Dirichlet problem;(iii)condition (59) takes place. Then, for each , uniformly on compact subsets of . Furthermore, for each , uniformly on each compact subset of . Finally, if the complement of is connected, one has equality in (64) for all , except for a set of capacity zero, and uniformly on each compact subset of .

Proof. Note that, in our context, the multiplication operator is bounded (see Theorem 3) and the norms of and defined as in (3) are equivalent (see Theorem 18). This is the crucial fact in the proof of this theorem; once we know this, we just need to follow the proof given in [5, Theorem 6] point by point to conclude the result.

Acknowledgments

Ana Portilla and Eva Tourís are supported in part by a grant from Ministerio de Ciencia e Innovación (MTM 2009-12740-C03-01), Spain. Yamilet Quintana is supported in part by the Research Sabbatical Fellowship Program (2011-2012) from Universidad Simón Bolvar, Venezuela. Ana Portilla, José M. Rodríguez, and Eva Tourís are supported in part by two grants from Ministerio de Ciencia e Innovación (MTM 2009-07800 and MTM 2008-02829-E), Spain. José M. Rodríguez is supported in part by a grant from CONACYT (CONACYT-UAG I0110/62/10 FON.INST.8/10), México. This word is dedicated to Francisco Marcellán Español on his 60th birthday.

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