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

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−1 x(m)e−imθ2. ial 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. l 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. 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. s 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.