Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

1 Faculty 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 Boĺıvar, Apartado Postal 89000, Caracas 1080 A, Venezuela 3 Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, 28911 Madrid, Spain 4Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain


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 [13] the authors obtain the nth 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 [11], it could be obtained as a consequence of the boundedness of the multiplication operator M f (z) = zf (z). Thus, finding conditions to ensure the boundedness of M would provide important information about the crucial issue of determining the asymptotic behavior of Sobolev orthogonal polynomials (see, for instance [2,12,16,18,19,20,22,23]). The more general result on this topic is [2,Theorem 8.1] which characterizes in terms of equivalent norms in Sobolev spaces the boundedness of M for the classical diagonal norm ∥q∥ W N,p (µ0,µ1,...,µN ) := ) 1/p (see Theorem 2.2 below, which is [2,Theorem 8.1] in the case N = 1). The rest of the above mentioned papers provides conditions that ensure the equivalence of norms in Sobolev spaces, and consequently, the boundedness of M . Results related to non-diagonal Sobolev norms, may be found in [1,3,6,7,9,12,14,16]. Particularly, in [3,9,12,14,16] the authors establish the asymptotic behavior of orthogonal polynomials with respect to non-diagonal Sobolev inner products and the authors in [12] deal with the asymptotic behavior of extremal polynomials with respect to the following non-diagonal Sobolev norms.
Let P be the space of polynomials and µ be a finite Borel positive measure with compact support S(µ) consisting of infinitely many points in the complex plane; let us consider the diagonal matrix Λ := diag(λ j ), j = 0, . . . , N , with λ j positive µ-almost everywhere measurable functions, and U := (u jk ), 0 ≤ j, k ≤ N , a matrix of measurable functions such that the matrix U (x) = (u jk (x)), 0 ≤ j, k ≤ N is unitary µ-almost everywhere. If V := U ΛU * , where U * denotes the transpose conjugate of U (note that then V is a positive definite matrix µ-almost everywhere), and 1 ≤ p < ∞ we define the Sobolev norm on the space of polynomials P In [8, Chapter XIII] certain general conditions imposed on the matrix V are requested in order to guarantee the existence of an unitary representation with measurable entries.
If U is not the identity matrix µ-almost everywhere, then (1.1) defines a non-diagonal Sobolev norm in which the product of derivatives of different order appears. We say that q n (z) = z n +a n−1 z n−1 +· · ·+a 1 z +a 0 is an nth monic extremal polynomial with respect to the norm (1.1) if It is clear that there exists at least an nth monic extremal polynomial. Furthermore, it is unique if 1 < p < ∞. If p = 2, then the nth monic extremal polynomial is precisely the nth monic Sobolev orthogonal polynomial with respect to the inner product corresponding to (1.1).
In [12,Theorem 1] the authors showed that the zeros of the polynomials in {q n } n≥0 are uniformly bounded in the complex plane, whenever there exists a constant C such that λ j ≤ Cλ k , µ-almost everywhere for 0 ≤ j, k ≤ N . This property made possible to obtain the nth root asymptotic behavior of extremal polynomials (see [12,Theorems 2 and 6]). Although it is required compact support for µ, this is, certainly, a natural hypothesis: if S(µ) is not bounded, then we cannot expect to have zeros uniformly bounded, not even in the classical case (orthogonal polynomials in L 2 ), see [5].
For N = 1 and 1 ≤ p ≤ 2, in [17] is obtained an equivalent result to [12,Theorem 1], but stating hypothesis on the matrix V rather than on the diagonal matrix Λ that appears in its unitary factorization. In exchange for a certain loss of generality, a weaker hypothesis than the totally dominatedness for {λ j } 0≤j≤N is required. More precisely, the authors obtain the following result.
. In this paper we improve Theorem 1.1 in two directions: on the one hand, we enlarge the class of measures µ considered and, on the other hand, we prove our result for 1 ≤ p < ∞ (see Theorem 4.3). In order to describe the measures we will deal with, we introduce the definition of p-admissible pairs as follows: Given 1 ≤ p < ∞, we say that the pair (V, µ) is p-admissible if µ is a finite Borel measure which can be written as µ = µ 1 + µ 2 , its support S(µ) is a compact subset of the complex plane which contains infinitely many points and V is a positive definite matrix µ-almost everywhere with |b p | 2 ≤ (1 − ε 0 )a p c p , µ 1 -almost everywhere for some fixed 0 < ε 0 ≤ 1, the support S(µ 2 ) is contained in a finite union of rectifiable compact curves γ with ( c p/2 and dµ 2 /ds is the Radon-Nykodim derivative of µ 2 with respect to the Euclidean length in γ.
We want to make three remarks about this definition. First of all, since V = is a positive definite matrix µ-almost everywhere, V 2/p also has this property and hence, |b p | 2 < a p c p , µ-almost everywhere.
In order to obtain the best choice for µ 2 is the restriction of µ to γ. Note that the support of µ is an arbitrary compact set: we just require that S(µ 2 ) (the part of S(µ) in which V 2/p is about to be a degenerated quadratic form, when |b p | 2 is very close to a p c p ) is a union of curves.
Therefore, with the results on p-admissible pairs we complement and improve the study started in [17], where the case µ = µ 2 with 1 ≤ p ≤ 2 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 L 2 and this makes really difficult to find an explicit expression for the extremal polynomial of degree n. In this regard, Theorems 5.1 and 5.2 deduce the asymptotic behavior of extremal polynomials as an application of Theorems 4.2 and 4.3. More precisely, we obtain the nth root and the zero counting measure asymptotic both of those polynomials and their derivatives to any order. The study of the nth root asymptotic is a classical problem in the theory of orthogonal polynomials, see for instance, [11,12,13,25,26]. Furthermore, in Theorem 5.2 we find the following asymptotic relation: The main idea of [12,16,17] and this paper is to compare non-diagonal and diagonal norms.
When it comes to compare non-diagonal and diagonal norms, the reference [10] 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 [17] to the case of more derivatives is that there are too many entries in the matrix V and just a few relations to control them (see Lemma 3.5 and notice that some limits appearing in that Lemma do not provide any new information). In that case we have just three entries (a p , b p , c p ), but in the simple case of two derivatives (N = 2) we have a 11 a 12 a 13 a 12 a 22 a 23 a 13 a 23 a 33   , and we would need to control six functions (a 11 , a 12 , a 13 , a 22 , a 23 , a 33 ); in the general case with N derivatives, we would need to control (N + 1)(N + 2)/2 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 4.1 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.

Background and previous results
In what follows, given 1 ≤ p < ∞ we define for every polynomial f .
It is obviously much easier to deal with the norms ∥ · ∥ W 1,p (a p/2 p µ,c p/2 p µ) and ∥ · ∥ W 1,p (Dµ) than with the one ∥ · ∥ W 1,p (Vµ) . Therefore, one of our main goals is to provide weak hypotheses to guarantee the equivalence of these norms on the linear space of polynomials P (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 M f (z) = zf (z), where Regarding this issue, the following result is known. In what follows, we will fix a p-admissible pair (V, µ) with 1 ≤ p < ∞; then S(µ 2 ) is contained in a finite union of rectifiable compact curves γ in the complex plane; each of these connected components of γ is not required to be either simple or closed.

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 4.1 much more understandable.
The following result is well known. In what follows a p , b p and c p refer to the coefficients of the fixed matrix V 2/p .

Definition 3.4.
We say that {f n } n ⊂ P is an extremal sequence for p if, for every n, ∥f n ∥ L ∞ (µ2) = 1 and Lemma 3.5. If 1 ≤ p < ∞ and {f n } n is an extremal sequence for p, then: Proof. The case 1 ≤ p ≤ 2 is a consequence of Lemmas 3.5 and 3.6 in [17]. We deal now with the case p > 2. First note that we can rewrite the limit (3.5) in the definition 3.4 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 n 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 non-negative and less than or equal to 1 using, respectively, that |b p | 2 < a p c p µ 2 -almost everywhere and 2xy ≤ x 2 + y 2 . Thus, and (3.6) holds.
Given ε > 0, for each n let us define the following sets: Then there exists a constant 0 < C ε < 1, just depending on ε, such that Using this fact and (3.10), we have If we assume that lim inf Since for each n, we have ∫ then (3.12) implies that On the other hand, using (3.10) it is easy to deduce that (3.14) lim Consequently, (3.12), (3.13) and (3.14) give Therefore, (3.12), (3.15) and (3.16) give Similar arguments allow us to show From (3.17) and (3.18) we obtain (3.19) lim As a consequence of (3.19) we have In a similar way we obtain Since these inequalities hold for every ε > 0, we conclude that (3.9) holds. Applying now Lemma 3.3 we obtain (3.7).
Using Lemma 3.1, (3.9) and (3.20) we obtain that for every ε, η > 0 there exists N such that for every n ≥ N the following holds Then (3.8) follows from the previous inequalities, since ε, η > 0 are arbitrary.
This completes the proof. Definition 3.6. For each 0 < ε < 1, we define the sets A ε and A c ε as Lemma 3.7. If 1 ≤ p < ∞ and {f n } n is an extremal sequence for p and ε is small enough, then Proof. If 1 ≤ p ≤ 2, then the result follows from [17,Lemma 3.11]. For the case p > 2 it suffices to follow the proof of [17, Lemma 3.11] applying Lemmas 3.5, 3.7 and 3.9 to conclude the result.
Lemma 3.11. If 1 ≤ p < ∞ and {f n } n is an extremal sequence for p, then for every ε > 0 small enough with µ 2 (A c ε ) > 0 and for every t ∈ (0, 1) there exists N such that inf z∈A c ε |f n (z)| < t for every n ≥ N . Proof. If 1 ≤ p ≤ 2, then the result follows from [17,Lemma 3.12]. For the case p > 2 it is sufficient to follow the proof of [17, Lemma 3.12] applying Lemma 3.10 to conclude the result. , for every polynomial f .

Equivalent norms
Now we prove the announced result about the equivalence of norms for 1 ≤ p < ∞. Proof. The equivalence of the two first norms is straightforward, by Lemmas 3.1 and 3.2. We prove now the equivalence of the two last norms.
Let us prove that there exists a positive constant C := C(V, µ, p) such that , for every f ∈ P.
In order to prove the first inequality, C ∥f ∥ W 1,p (Dµ) ≤ ∥f ∥ W 1,p (Vµ) , note that If γ = ∅ (i.e. µ = µ 1 ), then we have finished the proof. Assume that γ ̸ = ∅ then we prove C ∥f ∥ W 1,p (Dµ2) ≤ ∥f ∥ W 1,p (Vµ2) , seeking for a contradiction. It is clear that it suffices to prove it when γ is connected, i.e. when γ is a rectifiable compact curve. Let us assume that there exists a sequence {f n } n ⊂ P such that This right-hand side of the inequality is positive, because |2b and hence and (4.21) also holds for p > 2.
If f n is constant for some n, then without loss of generality we can assume that f n is non-constant and ∥f n ∥ L ∞ (µ2) = 1 for every n. Then {f n } n is an extremal sequence for p. Applying Lemma 3.5, By Lemma 3.11, there exist {z n } n ⊂ S(µ 2 ) such that |f n (z n )| ≤ 1/2 for every n ≥ N 1 . Now, taking into account that ∥f n ∥ L ∞ (µ2) = 1 and that γ is connected, we can apply Lemma 3.13, and then (4.23) .
Let us fix ε small enough. On the one hand, by Lemma 3.10 it holds ∫ ( On the other hand, we have  Finally, we have the following particular consequence for Sobolev orthogonal polynomials.

Asymptotic of extremal polynomials
We start this section by setting some notation. Let Q n , ∥ · ∥ L 2 (µ) , cap(S(µ)) and ω S(µ) denote, respectively, the nth monic orthogonal polynomial with respect to L 2 (µ), the usual norm in the space L 2 (µ), the logarithmic capacity of S(µ) and the equilibrium measure of S(µ). Furthermore, in order to analyze the asymptotic behavior for extremal polynomials we will use a special class of measures, "regular measures", denoted by Reg and defined in [26]. In that work, the authors proved (see Theorem 3.1.1) that, for measures supported on a compact set of the complex plane, µ ∈ Reg if and only if lim n→∞ ∥Q n ∥ 1/n L 2 (µ) = cap(S(µ)). Finally, if z 1 , z 2 , . . . , z n denote the zeros, repeated according to their multiplicity, of a polynomial q whose degree is exactly n, and δ zj is the Dirac measure with mass one at the point z j ; the expression ν(q) := 1 n n ∑ j=1 δ zj , defines the normalized zero counting measure of q.
We can already state the first result in this section. Proof. Note that, in our context, the hypothesis removed with respect to [12,Theorem 2] are equivalent to the following two facts: on the one hand, the multiplication operator is bounded (see Theorem 2.2) and on the other hand, the norms of W 1,p (a p/2 p µ, c p/2 p µ) and W 1,p (Vµ) defined as in (2.2) are equivalent (see Theorem 4.2). With this in mind, we just need to follow the proof of [12, Theorem 2] to conclude the result.
In the following theorem, we use g Ω (z; ∞) to denote the Green's function for Ω with logarithmic singularity at ∞, where Ω is the unbounded component of the complement of S(µ). Notice that, if S(µ) is regular with respect to the Dirichlet problem, then g Ω (z; ∞) is continuous up to the boundary and it can be extended continuously to all C, with value zero on C \ Ω. Proof. Note that, in our context, the multiplication operator is bounded (see Theorem 2.2) and the norms of W 1,p (a p/2 p µ, c p/2 p µ) and W 1,p (Vµ) defined as in (2.2) are equivalent (see Theorem 4.2). This is the crucial fact in the proof of this theorem; once we know this, we just need to follow the proof given in [12,Theorem 6] point by point to conclude the result.