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A Survey on Markov’s Theorem on Zeros of Orthogonal Polynomials

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Nonlinear Analysis and Global Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 167))

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Abstract

This manuscript is an extended version of the paper by the same authors who appeared in Castillo et al. (Appl Math Comput 339:390–397, 2018). It briefly surveys a Markov’s result dating back to the end of the nineteenth century, which is related to zeros of orthogonal polynomials.

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Notes

  1. 1.

    The Dirac measure δ y is a positive Radon measure whose support is the set {y}.

  2. 2.

    In the case that ω(x, t) is an even function in an interval of the form (−a, a), it is well known that the zeros of the orthogonal polynomials are symmetric with respect to the origin, i.e., x k(t) = −x n−k+1(t), k = 1,  2, …, n.

  3. 3.

    \(A^{c}:=\{x\in \mathbb {R}\,|\,x\not \in A\)} and Co(A) denotes the convex hull of A.

  4. 4.

    Because of the symmetry of the zeros of \(P_{n}^{(\lambda )}(x)\), its negative zeros are increasing functions of λ, for λ > −1∕2.

  5. 5.

    One has the identity \(\displaystyle \frac {\Gamma '(z)}{\Gamma (z)}=-\gamma -\frac {1}{z}-\sum _{n=1}^{\infty }\left [\frac {1}{z+n}-\frac {1}{n} \right ]\), where γ is the Euler constant (see [34, Section 12.3], [31, Chapter 7]).

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Acknowledgments

Castillo is supported by the Center for Mathematics of the University of Coimbra, Grant No. UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Rafaeli is supported by the Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) Demanda Universal under Grant No. APQ-03782-18 and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

    Kenier Castillo

  2. FAMAT-UFU, Department of Mathematics, Federal University of Uberlândia, Uberlândia, Minas Gerais, Brazil

    Marisa de Souza Costa & Fernando Rodrigo Rafaeli

Authors
  1. Kenier Castillo
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  2. Marisa de Souza Costa
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  3. Fernando Rodrigo Rafaeli
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Correspondence to Fernando Rodrigo Rafaeli .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

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Castillo, K., Costa, M.d.S., Rafaeli, F.R. (2021). A Survey on Markov’s Theorem on Zeros of Orthogonal Polynomials. In: Rassias, T.M., Pardalos, P.M. (eds) Nonlinear Analysis and Global Optimization. Springer Optimization and Its Applications, vol 167. Springer, Cham. https://doi.org/10.1007/978-3-030-61732-5_2

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