Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval Articles uri icon

publication date

  • October 2014

start page

  • 33

end page

  • 63

volume

  • 186

International Standard Serial Number (ISSN)

  • 0021-9045

Electronic International Standard Serial Number (EISSN)

  • 1096-0430

abstract

  • We consider polynomials p(n)(omega)(x) that are orthogonal with respect to the oscillatory weight w (x) = e(iwx) on [-1, 1], where omega > 0 is a real parameter. A first analysis of p(n)(omega)(x) for large values of to was carried out in Asheim et al. (2014), in connection with complex Gaussian quadrature rules with uniform good properties in co. In this contribution we study the existence, asymptotic behavior and asymptotic distribution of the roots of p(n)(omega)(x) in the complex plane as n -> infinity. The parameter to grows with n linearly. The tools used are logarithmic potential theory and the S-property, together with the Riemann-Hilbert formulation and the Deift-Zhou steepest descent method.

keywords

  • orthogonal polynomials in the complex plane; strong asymptotics; zero distribution; logarithmic potential theory; s-property; riemann–hilbert problem; steepest descent method