Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 628031, 11 pages http://dx.doi.org/10.1155/2013/628031 Research Article Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms Ana Portilla,1 Yamilet Quintana,2 José M. Rodríguez,3 and Eva Tourís4 1 Faculty Mathematics and Computer Science, St. Louis University (Madrid Campus), Avenida del Valle 34, 28003 Madrid, Spain Departamento de Matemáticas Puras y Aplicadas, Edificio Matemáticas y Sistemas (MYS), Universidad Simón Bolı́var, 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 4 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain 2 Correspondence should be addressed to Yamilet Quintana; Received 28 January 2013; Accepted 8 March 2013 Academic Editor: Józef Banaś 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. Let P be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm ‖ ⋅‖𝑊1,𝑝 (𝑉𝜇) , where the matrix 𝑉 and the measure 𝜇 constitute a 𝑝-admissible pair for 1 ≤ 𝑝 ≤ ∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to ‖ ⋅‖𝑊1,𝑝 (𝑉𝜇) , stating hypothesis on the matrix 𝑉 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 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 [2], it could be obtained as a consequence of the boundedness of the multiplication operator 𝑀𝑓(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, e.g., [3–13]). 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 𝑀 for the classical diagonal norm 𝑁 󵄩 (𝑘) 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑞󵄩󵄩𝑊𝑁,𝑝 (𝜇0 ,𝜇1 ,...,𝜇𝑁 ) := ( ∑ 󵄩󵄩󵄩󵄩𝑞 󵄩󵄩󵄩󵄩𝐿𝑝 (𝜇 ) ) 𝑘 𝑘=0 1/𝑝 (1) (see Theorem 3 below, which is [3, 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 nondiagonal Sobolev norms may be found in [5, 6, 14–19]. 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 P be the space of polynomials with complex coefficients and let 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 being positive 𝜇-almost everywhere measurable functions, and 𝑈 := (𝑢𝑗𝑘 ), 0 ≤ 𝑗, 𝑘 ≤ 𝑁, a matrix of measurable functions such that the matrix 𝑈(𝑥) = (𝑢𝑗𝑘 (𝑥)), 0 ≤ 𝑗, 𝑘 ≤ 𝑁 is unitary 𝜇-almost everywhere. If 𝑉 := 𝑈Λ𝑈∗ , where 𝑈∗ denotes the transpose conjugate of 𝑈 (note that then 𝑉 is a positive definite matrix 2 Journal of Function Spaces and Applications 𝜇-almost everywhere), and 1 ≤ 𝑝 < ∞ we define the Sobolev norm on the space of polynomials P 󵄩󵄩 󵄩󵄩 󸀠 (𝑁) 2/𝑝 󵄩󵄩𝑞󵄩󵄩𝑊𝑁,𝑝 (𝑉𝜇) := (∫ [ (𝑞, 𝑞 , . . . , 𝑞 ) 𝑉 ∗ 𝑝/2 × (𝑞, 𝑞󸀠 , . . . , 𝑞(𝑁) ) ] 𝑑𝜇) 1/𝑝 (2) 󸀠 := (∫ [ (𝑞, 𝑞 , . . . , 𝑞 (𝑁) 2/𝑝 ) 𝑈Λ ∗ 𝑝/2 × (𝑞, 𝑞󸀠 , . . . , 𝑞(𝑁) ) ] 𝑈 ∗ 𝑑𝜇) 1/𝑝 . In [20, 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 (2) defines a nondiagonal Sobolev norm in which the product of derivatives of different order appears. We say that 𝑞𝑛 (𝑧) = 𝑧𝑛 + 𝑎𝑛−1 𝑧𝑛−1 + ⋅ ⋅ ⋅ + 𝑎1 𝑧 + 𝑎0 is an 𝑛th monic extremal polynomial with respect to the norm (2) if 󵄩 󵄩 = inf {󵄩󵄩󵄩𝑞󵄩󵄩󵄩𝑊𝑁,𝑝 (𝑉𝜇) : 𝑞 (𝑧) = 𝑧𝑛 + 𝑏𝑛−1 𝑧𝑛−1 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 (2). In [5, Theorem 1] the authors showed that the zeros of the polynomials in {𝑞𝑛 }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 [5, 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 𝐿2 ); see [21]. Taking 𝑁 = 1, 1 ≤ 𝑝 ≤ 2 and setting up hypothesis on the matrix 𝑉 (see (4)) rather than on the diagonal matrix 𝜆, the authors of [22] the following equivalent result to [5, Theorem 1]. Theorem 1 (see [22, Theorem 4.3]). Let 𝛾 be a finite union of rectifiable compact curves in the complex plane, 𝜇 a finite Borel measure with compact support 𝑆(𝜇) = 𝛾, 𝑉 a positive definite matrix 𝜇-almost everywhere and 𝑏𝑝 𝑐𝑝 𝑝/2 𝑝/2 𝑝/2 In this paper we improve Theorem 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 19). 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 𝑝admissible if 𝜇 is a finite Borel measure which can be written as 𝜇 = 𝜇1 + 𝜇2 , 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 |𝑏𝑝 |2 ≤ (1−𝜀0 )𝑎𝑝 𝑐𝑝 , 𝜇1 -almost everywhere for some fixed 0 < 𝜀0 ≤ 1; the support 𝑆(𝜇2 ) is contained in a finite union of rectifiable 𝑝/2 compact curves 𝛾 with (𝑐𝑝 𝑑𝜇2 /𝑑𝑠) −1 ∈ 𝐿1/(𝑝−1) (𝛾) if 𝛾 ≠ 0, 𝑎𝑝 𝑏𝑝 𝑉2/𝑝 := ( 𝑏 𝑐 ) and 𝑑𝜇2 /𝑑𝑠 is the Radon-Nykodim derivative 𝑝 𝑝 −1 𝑝/2 In order to obtain (𝑐 (...truncated)