On measures in frequency analysis

摘要

The frequency analysis problem is to determine the unknown frequencies of a trigonometric signal with a sample of size N. Recently there has been established a method for determining those frequencies by using the asymptotic behaviour of the zeros of certain Szegö polynomials. 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 xN(m):dψNdθ=12π∑m=0N−1xN(m)e−imθ2.This first method is called the N-process. Later there came a modification where the measure ψN was replaced by a new measure ψN(R). This new method called the R-process, involves one additional parameter, but has certain convergence benefits. Very recently Njåstad and Waadeland introduced and used the measure ψ(T) given bydψ(T)dθ=12π∑m=0∞x(m)Tme−imθ2.Essential for the use in frequency analysis is the weak star convergence for T→1− of the absolutely continuous measure G(T)ψ(T) to a measure supported by a certain finite set on the unit circle for some positive G(T). For ψ(T) the function G(T)=1−T2 works. The present paper is a report on measures ψ(T), obtained by inserting positive weights cm in dψ(T)/dθ. This means to study measures of a form given bydψ(T)dθ=12π∑m=0∞x(m)cmTme−imθ2,where the coefficients cm satisfy certain conditions. The main part of the paper is to prove weak star convergence by a proper choice of G(T).