Multiple zeros in frequency analysis: the T(r)-process

摘要

Recently, a method has been established for determining the n0 unknown frequencies ωj in a trigonometric signal by using Szegö polynomials; ρn(ψN;z). The Szegö polynomials in question are orthogonal on the unit circle with respect to an inner product defined by a measure ψN. The measure is constructed from the observed signal values x(m):dψN(θ)dθ=12π∑m=0N−1x(m)e−imθ2.Essential in the study is the asymptotic behavior of the zeros. If n⩾n0 then n0 of the zeros of each limiting polynomial will coincide with the frequency points e±iωj. The limiting polynomial is not unique. The remaining (n−n0) zeros are bounded away from the unit circle.Several modifications of this method have been developed. The modifications are of two main types: Modifying the measure by modifying the observed signal values or by modifying the moments.In the present paper, we will modify the measure and study measures of the formdψ(Tr)(θ)dθ=12π∑m=0∞x(m)Tmcme−imθ2,where T=1−d∈(0,1) and the coefficients cm satisfy certain conditions.In this situation, we find the rate at which certain Toeplitz determinants tend to zero, and prove that the limit of the absolute value of the reflection coefficients δβn0(Tr) for n=βn0 are limd→0|δβn0(Tr)|2=1. We also prove that the frequency points occur as zeros with a certain multiplicity in the limiting polynomials.