Zeros and Critical Points of Sobolev Orthogonal Polynomials

✍ Scribed by W. Gautschi; A.B.J. Kuijlaars

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 340 KB
• Volume: 91
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3097

✦ Synopsis

Using potential theoretic methods we study the asymptotic distribution of zeros and critical points of Sobolev orthogonal polynomials, i.e., polynomials orthogonal with respect to an inner product involving derivatives. Under general assumptions it is shown that the critical points have a canonical asymptotic limit distribution supported on the real line. In certain cases the zeros themselves have the same asymptotic limit distribution, while in other cases we can only ascertain that the support of a limit distribution lies within a specified set in the complex plane. One of our tools, which is of independent interest, is a new result on zero distributions of asymptotically extremal polynomials. Our results are illustrated by numerical computations for the case of two disjoint intervals. We also describe the numerical methods that were used.

1997 Academic Press 7 0 :=supp(+ 0 ), 7 1 :=supp(+ 1 ), 7 :=7 0 _ 7 1 .

(1.2)

If, as we assume, + 0 has infinite support, there exists a unique sequence of monic polynomials ? n , deg ? n =n, which is orthogonal with respect to the article no. AT963097 117