Asymptotics of polynomials orthogonal with respect to varying measures
This very important paper deals with orthogonal polynomials \(\phi_{n,m}(z)\) \((m=0,1,2,...)\) on the unit circle with respect to a measure \(d\sigma_ n(\theta)=d\sigma (\theta)/| W_ n(e^{i\theta})|^ 2\), where \(W_ n\) is a polynomial of degree n whose zeros lie on the closed unit disk \(\{\) \(| z| \leq 1\}\) and \(\sigma '>0\) almost everywhere on the unit circle. The asymptotic behaviour of the ratio \(\phi_{n,n+k+1}(z)/\phi_{n,n+k}(z)\) for \(n\to \infty\) is examined for \(| z| \geq 1\), the behaviour of the reflection coefficients and various strong and weak asymptotic results are given. The results are extended to complex measures on the unit circle. The transition to the real line is made and related asymptotic behaviour of orthogonal polynomials on [-1,1] with varying measures is given. The special case \(W_ n(z)=z^ n\) gives the famous results of E. A. Rakhmanov for orthogonal polynomials on the unit circle (and on the real line) with \(\sigma '>0\) almost everywhere. Power asymptotic behaviour of orthogonal polynomials with varying measures is given by \textit{G. López} [Lect. Notes Math. 1329, 255-260 (1988; Zbl 0652.42014)].
orthogonal polynomials with varying measures
scientific article; zbMATH DE number 4096123
scientific article