Asymptotic behavior of Szegö polynomials

• Authors: Vigdis Brevik Petersen
• Venue: Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
• Year: 2005

A way of finding the unknown frequencies in a trigonometric signal is based upon the use of a certain family of measures on the unit circle, constructed from observations of the signal. The measure gives rise to an inner product, moments and orthogonal polynomials; Szegö polynomials. Asymptotic behavior of the zeros leads to the unknown frequencies. Several variations of this method have been presented. Two main approaches have been studied. One is to construct new modified measures, another to modify the moment in various ways. In both the modifications it is proved that several zeros tend to one and the same frequency point eiωj. An important question is whether there can be other zeros tending to the unit circle. If so, separation of the frequency points from the remaining zeros could be a problem. Here we prove that the limit of the zeros, not tending to the frequency points, are located inside the unit circle.