NDLI: Asymptotic Behavior and Zero Distribution of Polynomials Orthogonal with Respect to Bessel Functions

Content Provider	Springer Nature Link
Author	Kuijlaars, Ar B. J. Deaño, Alfredo Román, Pablo
Copyright Year	2015
Abstract	We consider polynomials $$P_n$$ orthogonal with respect to the weight $$J_{\nu }$$ on $$[0,\infty )$$ , where $$J_{\nu }$$ is the Bessel function of order $$\nu $$ . Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of $$P_n$$ are complex and accumulate as $$n \rightarrow \infty $$ near the vertical line $${{\mathrm{{\text {Re}}\,}}}z = \frac{\nu \pi }{2}$$ . We prove this fact for the case $$0 \le \nu \le 1/2$$ from strong asymptotic formulas that we derive for the polynomials $$P_n$$ 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 $$\nu \le 1/2$$ .
Ending Page	196
Page Count	44
Starting Page	153
File Format	PDF
ISSN	01764276
e-ISSN	14320940
Journal	Constructive Approximation
Issue Number	1
Volume Number	43
Language	English
Publisher	Springer US
Publisher Date	2015-07-17
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) Numerical Analysis Analysis Orthogonal polynomials Asymptotic representations in the complex domain Limiting zero distribution Bessel functions Bessel and Airy functions, cylinder functions, ${}_0F_1$ Riemann–Hilbert problems Other special orthogonal polynomials and functions
Content Type	Text
Resource Type	Article
Subject	Analysis Computational Mathematics

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