Modification of a method using Szegö polynomials in frequency analysis: the V-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). Essential in the study is the asymptotic behavior of the zeros. If n⩾n0 then n0 of the zeros in the limit polynomial will tend to the frequency points e±iωj. The remaining (n−n0) are bounded away from the unit circle. Several modifications of this method are developed. The modifications are of two main types: Modifying the observed signal values or modifying the moments. In the present paper we will replace the moment sequence {μm(N)/N} by a new sequence {(μm(N)/N)Rm2}, where R∈(0,1). In this situation we prove the surprising result that a multiple of the n0 zeros tend to the frequency points. We also prove the rate at which certain Toeplitz determinants tend to zero.