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Zeros of Sobolev orthogonal polynomials on the unit circle

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Abstract

In this contribution, we study the sequences of orthogonal polynomials with respect to the Sobolev inner product

$$ \langle{f,g}\rangle_{S}:=\int_{\mathbb{T}} f(z) \overline{g(z)} d\mu(z) + \lambda f^{(j\,)}(\alpha)\overline{g^{(j\,)}(\alpha)}, $$

where μ is a nontrivial probability measure supported on the unit circle, α ∈ ℂ, \(\lambda \in {\mathbb{R}}_+\backslash\{0\}\), and j ∈ ℕ. In particular, we analyze the behavior of their zeros when n and λ tend to infinity, respectively. We also provide some numerical examples to illustrate the behavior of these zeros with respect to α.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Ave. Universidad 30, Leganés, Spain

    K. Castillo & F. Marcellán

  2. Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, México

    L. E. Garza

Authors
  1. K. Castillo
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  2. L. E. Garza
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  3. F. Marcellán
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Correspondence to K. Castillo.

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Dedicated to Professor Claude Brezinski and Professor Sebastiano Seatzu on the occasion of their 70th birthday.

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Castillo, K., Garza, L.E. & Marcellán, F. Zeros of Sobolev orthogonal polynomials on the unit circle. Numer Algor 60, 669–681 (2012). https://doi.org/10.1007/s11075-012-9594-6

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