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Erschienen in:
Friction Models in the Framework of Set-Valued and Convex Analysis Kapitel lesen Erstes Kapitel lesen

2021 | OriginalPaper | Buchkapitel

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

verfasst von : Kenier Castillo, Marisa de Souza Costa, Fernando Rodrigo Rafaeli

Erschienen in: Nonlinear Analysis and Global Optimization

Verlag: Springer International Publishing

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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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Vorheriges Kapitel Friction Models in the Framework of Set-Valued and Convex Analysis
Nächstes Kapitel A Review of Two Network Curvature Measures
Fußnoten
1
The Dirac measure δ y is a positive Radon measure whose support is the set {y}.
 
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
\(A^{c}:=\{x\in \mathbb {R}\,|\,x\not \in A\)} and Co(A) denotes the convex hull of A.
 
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
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]).
 
Literatur
1.
S. Ahmed, M.E. Muldoon, R. Spigler, Inequalities and numerical bound for zeros of ultraspherical polynomials. SIAM J. Math. Anal. 17, 1000–1007 (1986)MathSciNetCrossRef
2.
I. Area, D.K. Dimitrov, E. Godoy, V. Paschoa, Zeros of classical orthogonal polynomials of a discrete variable. Math. Comput. 82, 1069–1095 (2013)MathSciNetCrossRef
3.
K. Castillo, F.R. Rafaeli, On the discrete extension of Markov’s theorem on monotonicity of zeros. Electron. Trans. Numer. Anal. 44, 271–280 (2015)MathSciNetMATH
4.
K. Castillo, M.S. Costa, F.R. Rafaeli, On zeros of polynomials in best L p-approximation and inserting mass points (2017). arXiv:1706.06295
5.
K. Castillo, M.S. Costa, F.R. Rafaeli, On Markov’s theorem on zeros of orthogonal polynomials revisited. Appl. Math. Comput. 339, 390–397 (2018)MathSciNetMATH
6.
T.S. Chihara, An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978)MATH
7.
D.K. Dimitrov, On a conjecture concerning monotonicity of zeros of ultraspherical polynomials. J. Approx. Theory 85, 88–97 (1996)MathSciNetCrossRef
8.
D.K. Dimitrov, Monotonicity of zeros of polynomials orthogonal with respect to a discrete measure (2015). arXiv:1501.07235
9.
D.K. Dimitrov, F.R. Rafaeli, Monotonicity of zeros of Jacobi polynomials. J. Approx. Theory 149, 15–29 (2007)MathSciNetCrossRef
10.
D.K. Dimitrov, F.R. Rafaeli, Monotonicity of zeros of Laguerre polynomials. J. Comput. Appl. Math. 233, 699–702 (2009)MathSciNetCrossRef
11.
A. Elbert, P.D. Siafarikas, Monotonicity properties of the zeros of ultraspherical polynomials. J. Approx. Theory 97, 31–39 (1999)MathSciNetCrossRef
12.
G. Freud, Orthogonal Polynomials (Pergamon Press, Oxford, 1971)MATH
13.
E.J. Huertas, F. Marcellán, F.R. Rafaeli, Zeros of orthogonal polynomials generated by canonical perturbations of measures. Appl. Math. Comput. 218, 7109–7127 (2012)MathSciNetMATH
14.
E.K. Ifantis, P.D. Siafarikas, Differential inequalities and monotonicity properties of the zeros of associated Laguerre and Hermite polynomials. Ann. Numer. Math. 2, 1–4, 79–91 (1995)MathSciNetMATH
15.
M.E.H. Ismail, Monotonicity of zeros of orthogonal polynomials. q-Series and Partitions, ed. by D. Stanton (Springer, New York, 1989), pp. 177–190
16.
M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and Its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)
17.
M.E.H. Ismail, J. Letessier, Monotonicity of zeros of ultraspherical polynomials, in Orthogonal Polynomials and Their Applications, ed. by M. Alfaro, J.S. Dehesa, F.J. Marcellán, J.L. Rubio de Francia, J. Vinuesa. Lecture Notes in Mathematics, vol. 1329 (Springer, Berlin, 1988), pp. 329–330
18.
K. Jordaan, H. Wang, J. Zhou, Monotonicity of zeros of polynomials orthogonal with respect to an even weight function. Integral Transform. Spec. Funct. 25(9), 721–729 (2014)MathSciNetCrossRef
19.
R. Koekoek, P.A. Lesky, R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics (Springer, Berlin, 2010)
20.
T.H. Koornwinder, Orthogonal polynomials with weight function (1 − x)α(1 + x)β + Mδ(x + 1) + Nδ(x − 1). Canad. Math. Bull. 27, 205–214 (1984)MathSciNetCrossRef
21.
A.M. Krall, Orthogonal polynomials satisfying fourth order differential equations. Proc. Roy. Soc. Edinb. Sec. A. 87, 271–288 (1980/1981)MathSciNetCrossRef
22.
H.L. Krall, On orthogonal polynomials satisfying a certain fourth order differential equation. Pa. State Coll. Stud. 6, 24 (1940)MathSciNetMATH
23.
A. Kroó, F. Peherstorfer, On the zeros of polynomials of minimal L p-norm. Proc. Amer. Math. Soc. 101, 652–656 (1987)MathSciNetCrossRef
24.
A. Laforgia, A monotonic property for the zeros of ultraspherical polynomials. Proc. Amer. Math. Soc. 83, 757–758 (1981)MathSciNetCrossRef
25.
A. Laforgia, Monotonicity properties for the zeros of orthogonal polynomials and Bessel function, in Polynômes Orthogonaux et Applications, ed. by C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, A. Ronveaux. Lecture Notes in Mathematics, vol. 1171 (Springer, Berlin, 1985), pp. 267–277
26.
F. Marcellán, P. Maroni, Sur l’adjonction d’une masse de Dirac à une forme régulière et semi-classique. Ann. Mat. Pura Appl. 162(4), 1–22 (1992)MathSciNetCrossRef
27.
28.
P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and Their Applications, ed. by C. Brezinski, L. Gori, A. Ronveaux. Annals Comput. Appl. Math. vol. 9 (Baltzer, Basel, 1991), pp. 95–130
29.
P. Natalini, B. Palumbo, Some monotonicity results on the zeros of the generalized Laguerre polynomials. J. Comput. Appl. Math. 153, 355–360 (2003)MathSciNetCrossRef
30.
A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable (Springer, Berlin, 1991)CrossRef
31.
E.D. Rainville, Special Functions (Chelsea, Bronx, 1971)
32.
33.
G. Szegő, Orthogonal Polynomials. Amer. Math. Soc. Coll. Publ., vol. 23, 4th edn. (American Mathematical Society, Providence, 1975)
34.
E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University Press, Cambridge, 1927)MATH
Metadaten
Titel
A Survey on Markov’s Theorem on Zeros of Orthogonal Polynomials
verfasst von
Kenier Castillo
Marisa de Souza Costa
Fernando Rodrigo Rafaeli
Copyright-Jahr
2021
Buch
Nonlinear Analysis and Global Optimization

Print ISBN: 978-3-030-61731-8

Electronic ISBN: 978-3-030-61732-5

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-61732-5_2