A note on orthogonal polynomials
An important and useful identity in the study of asymptotic properties of orthogonal polynomials is the one stating: \(\int_ 0^{2\pi} z^ k| s_ n(z)|^{-2} d\theta=\int_ 0^{2\pi} z^ k d\mu(\theta)\), \(z=e^{i\theta}\), \(| k|\leq n\), \(n=0,1,2,\dots\), where \(d\mu\) is a finite positive Borel measure on the interval \([0,2\pi]\) with support on an infinite set. The set \(\{s_ n(z)\}_{n=0}^ \infty\) is the unique system of orthonormal polynomials with respect to \(d\mu\) on the unit circle. In this note, the author establishes this identity in a simple and elementary manner.
scientific article; zbMATH DE number 168039
scientific article