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

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.

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 |