Asymptotics for orthogonal polynomials defined by a recurrence relation

Mate, Attila; Nevai, Paul; Тотик, Вилмос

doi:10.1007/bf01890033

Cited by 46 publications

(29 citation statements)

References 6 publications

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“…Therefore we find from the results of Máté and Nevai [13], Máté et al [14], Geronimo and Van Assche [8] and Van Assche [19] that the polynomials {p n,N (x)} ∞ n=−1 are a sequence of orthonormal polynomials with respect to a positive measure N for which…”

Section: Statement Of the Resultsmentioning

confidence: 98%

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Aptekarev

Geronimo

Assche

2008

Journal of Approximation Theory

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We consider orthogonal polynomials {P n,N , n = 0, 1, 2, . . .}, where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit W = lim N→∞ W 1/N N and link this limit to an external field for an equilibrium problem in logarithmic potential theory.

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Section: Statement Of the Resultsmentioning

confidence: 98%

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Aptekarev

Geronimo

Assche

2008

Journal of Approximation Theory

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“…The reason for the recent interest in Turán determinants is that for the orthogonal polynomials {p n } ∞ n=0 in a subclass of the class M (0, 1) the quantities ∆ n (p; x) converge uniformly on the compact subsets of (−1, 1) to 2(1 − x 2 ) 1/2 /(πα (x)), where α (x) is the absolutely continuous part of the measure, with respect to which the p n are orthogonal [5,6,7,12,19].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Higher order Turán inequalities

Dimitrov

1998

Proc. Amer. Math. Soc.

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Abstract. The celebrated Turán inequalitieswhere Pn(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ 2 n − γ n−1 γ n+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the Laguerre-Pólya class, ψ(x) = ∞ n=0 γnx n /n!.

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“…Using (1.10) and (1.13) by standard methods [13,22,[24][25][26][27]35,36], one can obtain the main result. | k,j − k+N,j | < ∞.…”

Section: Generalized Trace Formula and Asymptotics Of The Averaged Tumentioning

confidence: 96%

“…Put k = 1(k ∈ Z + ), then we get the following analog of Turan's determinant [12,13,16,22,24,26,32,35,36]: …”

Section: Generalized Trace Formula and Asymptotics Of The Averaged Tumentioning

confidence: 99%

Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product

Осиленкер

2006

Journal of Approximation Theory

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Let be a finite positive Borel measure supported on [−1, 1] and introduce the discrete Sobolev-type inner productwhere the mass points a k belong to [−1, 1], and M k,i > 0(i = 0, 1, . . . , N k ). In this paper, we obtain generalized trace formula and asymptotics of the averaged Turan determinant for the Sobolev-type orthogonal polynomials. Asymptotics of the recurrence coefficients for symmetric Gegenbauer-Sobolev orthogonal polynomials is obtained. Trace formula and asymptotics of Turan's determinant for Gegenbauer-Sobolev orthogonal polynomials are also given.

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Asymptotics for orthogonal polynomials defined by a recurrence relation

Cited by 46 publications

References 6 publications

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

Higher order Turán inequalities

Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product

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