## 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.

### MSC:

 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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