Orthogonality and asymptotics of Pseudo-Jacobi polynomials for non-classical parameters

JOURNAL OF APPROXIMATION THEORY

卷:178

Article

Jordaan, K.¹  Tookos, F.² 

[1] Univ Pretoria, Dept Math & Appl Math, ZA-0002 Pretoria, South Africa

[2] Helmholtz Zentrum Munchen, Inst Computat Biol, Neuherberg, Germany

关键词: Orthogonal polynomials;    Quasi-orthogonal polynomials;    Jacobi polynomials with complex parameters;    Pseudo-Jacobi polynomials;    Zeros;   

DOI  :  10.1016/j.jat.2013.10.003

来源: Elsevier

【 摘 要 】

The family of general Jacobi polynomials P-n((alpha,beta)) where beta epsilon C can be characterised by complex (nonHermitian) orthogonality relations (cf. Kuijlaars et al. (2005)). The special subclass of Jacobi polynomials P-n((alpha,beta)) where alpha, beta epsilon R are classical and the real orthogonality, quasi-orthogonality as well as related properties, such as the behaviour of the n real zeros, have been well studied. There is another special subclass of Jacobi polynomials P-n((alpha,beta)) with alpha, beta epsilon C, beta = (alpha) over bar which are known as Pseudo-Jacobi polynomials. The sequence of Pseudo-Jacobi polynomials {P-n(alpha,(alpha) over bar)}(n=0)(infinity) is the only other subclass in the general Jacobi family (beside the classical Jacobi polynomials) that has n real zeros for every n = 0, 1, 2, ... for certain values of alpha epsilon C. For some parameter ranges Pseudo-Jacobi polynomials are fully orthogonal, for others there is only complex (non-Hermitian) orthogonality. We summarise the orthogonality and quasiorthogonality properties and study the zeros of Pseudo-Jacobi polynomials, providing asymptotics, bounds and results on the monotonicity and convexity of the zeros. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free