Asymptotic Behavior of Orthogonal Polynomials Corresponding to Measure with Discrete Part Off the Unit Circle

✍ Scribed by X. Li; K. Pan

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 473 KB
• Volume: 79
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1994.1113

✦ Synopsis

For a positive measure (\mu) on the unit circle ((\Gamma)) in the complex plane, (m) points (z_{j}) off (\Gamma) and (m) positive numbers (A_{j}, j=1,2, \ldots, m), we investigate the asymptotic behavior of orthonormal polynomials (\Phi_{n}(z)) corresponding to (d_{\mu} / 2 \pi+) (\sum_{j=1}^{m} A_{j} \delta_{z}), where (\delta_{z}) denotes the unit measure supported at point (z). Our main result is the relative asymptotics of (\Phi_{n}(z)) with respect to the orthonormal polynomial corresponding to (d \mu /(2 \pi)) off and on (\Gamma). C 1994 Academic Press. inc.