Generalized Szego theory in frequency analysis

[摘要] A sequence of positive Borel measures with infinite support on the unit circle is given, converging in the weak star topology to a measure with support consisting of a finite number n(0) of mass points. The sequences of monic rational functions orthogonal with respect to inner products 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 the mass points of the limiting measure, while the remaining n - n(0) zeros (''uninteresting zeros'') may not converge at all. However, by considering 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, where the interesting zeros tend to the frequency points while the uninteresting zeros are bounded away from the unit circle. (C) 1997 Academic Press.

[发布日期] 1997-02-01