Weakly convergent sequences of functions and orthogonal polynomials (Q807912)

In 1982 \textit{E. A. Rakhmanov} [Mat. Sb., Nov. Ser. 118(160), 104-117 (1982; Zbl 0509.30028)] proved an important result for monic orthogonal polynomials \(\Phi_ n(z)\) on the unit circle: if the orthogonality measure satisfies \(\mu '(t)>0\) for almost every \(t\in [0,2\pi)\) then \(\Phi_ n(0)\to 0\) as \(n\to \infty.\) The numbers \(\Phi_ n(0)\) are known as reflection coefficients and completely determine the full sequence of polynomials \(\Phi_ n(z)\) via a recurrence formula. They coincide with the Schur parameters in the Schur algorithm and are crucial in Levinson's algorithm in signal processing. The present paper gives a simplified proof of Rakhmanov's theorem. Precise estimates of the reflection coefficients and some relevant consequences are given, such as the weak convergence of the orthogonal polynomials to the absolutely continuous part of the measure \(\mu\).