Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions

Deaño, Alfredo, Kuijlaars, Arno B. J., Román, Pablo (2015) Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions. Constructive Approximation, 43 . pp. 153-196. ISSN 0176-4276. (doi:10.1007/s00365-015-9300-8) (KAR id:51340)

PDF Author's Accepted Manuscript
Language: English
 Preview
Official URL
http://www.dx.doi.org/10.1007/s00365-015-9300-8

Abstract

We consider polynomials P_n orthogonal with respect to the weight J_? on [0,?), where J_? is the Bessel function of order ?. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n?? near the vertical line Rez=??2. We prove this fact for the case 0???1/2 from strong asymptotic formulas that we derive for the polynomials Pn in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for ??1/2.

Item Type: Article 10.1007/s00365-015-9300-8 Full text upload complies with journal requirements Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysisQ Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA351 Special functions Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Alfredo Deano Cabrera 02 Nov 2015 11:42 UTC 13 Jan 2020 10:24 UTC https://kar.kent.ac.uk/id/eprint/51340 (The current URI for this page, for reference purposes) https://orcid.org/0000-0003-1704-247X