Szegö’s extremum problem on the unit circle


It is shown that the Christoffel functions arising from the Szegö extremum problem associated with a finite positive Borel measure on the interval $[-\pi,\pi)$ satisfy
\lim_{n\to \infty} n\omega_n (\mu,e^{it}) = \mu'(t) \qquad \mathrm{for} \quad \mathrm{a.e. } \, t\in [-\pi,\pi)
whenever $\mu$ belongs to the Szegö class; that is, $\log \mu’ \in L^1$. Some implications of this result are discussed.


Attila Máté

Paul Nevai

Vilmos Totik