Szegö’s extremum problem on the unit circle. (English) Zbl 0752.42015

Máté, Attila; Nevai, Paul; Totik, Vilmos

Ann. Math. (2) 134, No. 2, 433-453 (1991).

The Christoffel function \(\omega_ n(\mu,z)\) is the minimum of \({1\over 2\pi| P(z)|^ 2}\int^ \pi_{-\pi}| P(e^{i\theta})|^ 2d\mu(\theta)\) taken over all polynomials \(P\) of degree less than \(n\) for which \(P(z)\neq 0\). Szegö studied \(\omega_ n(\mu,0)\) for absolutely continuous measures \(\mu\) in 1915 and later, in 1922, he showed that \(\lim_{n\to\infty}n\omega_ n(\mu,e^{it})=\mu'(t)\), \(t\in[-\pi,\pi)\), assuming that \(\mu\) is absolutely continuous and \(\mu'>0\) is twice continuously differentiable. This result is very important for applications in orthogonal polynomials, probability theory and statistics (linear prediction) and other areas, e.g., it gives a useful and numerically adaptable method of computing the weight function for orthogonal polynomials.
The present paper substantially weakens the assumptions on the measure \(\mu\). Máté shows that \(\int^ \pi_{-\pi}\log\mu'(\theta)d\theta>- \infty\) is a sufficient condition in order that Szegö’s asymptotic formula holds for almost every \(t\in[-\pi,\pi)\). Nevai then shows how this result on the unit circle leads to a similar result about the asymptotic behaviour of the Christoffel function on the interval \([- 1,1]\). Finally, Totik improves the results on the unit circle and the real line: he shows that it is enough to assume that \(\mu\) is regular for the unit circle, and then \(\lim_{n\to\infty}n\omega_ n(\mu,e^{it})=\mu'(t)\) holds almost everywhere on every interval \(I\subset[-\pi,\pi)\) for which \(\int_ I\log\mu'(\theta)d\theta>- \infty\).
Unquestionably, one of the most important papers in the theory of orthogonal polynomials of the last decade.

Reviewer: W.Van Assche (Heverlee)

MSC:

42C05

Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Keywords:

Szegö theory; Christoffel function; orthogonal polynomials; Szegö’s asymptotic formula