GENERALIZED SZEGO THEORY IN FREQUENCY-ANALYSIS

A sequence of positive Borel measures with infinite support on the uni t circle is given, converging in the weak star topology to a measure w ith support consisting of a finite number n(0) of mass points. The seq uences of monic rational functions orthogonal with respect to inner pr oducts determined by the measures are studied, with the focus being on the asymptotic behavior of the zeros. The main result is twofold: For a fixed n > n(0), n(0) of the zeros (''interesting zeros'') tend to t he mass points of the limiting measure, while the remaining n - n(0) z eros (''uninteresting zeros'') may not converge at all. However, by co nsidering subsequences, limits may be obtained and the remaining zeros (and their limits) will be located in a closed disk \z\ less than or equal to R < 1, provided that a certain boundedness condition for the norm of the monic orthogonal functions is satisfied. As an application it is proved that this theory may be applied in frequency analysis, w here the interesting zeros tend to the frequency points while the unin teresting zeros are bounded away from the unit circle. (C) 1997 Academ ic Press.