On measures in frequency analysis

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 Szeg o polynomials. The Szeg o polynomials in question are orthogonal on the unit circle with respect to an inner product de ned by a measure N . The measure is constructed from the observed signal values xN (m): d N d = 1 2 ∣∣∣∣ N−1 ∑ m=0 xN (m)e −im ∣∣∣∣ 2 : This rst method is called the N -process. Later there came a modi cation where the measure N was replaced by a new measure (R) N . This new method called the R-process, involves one additional parameter, but has certain convergence bene ts. Very recently Nj astad and Waadeland introduced and used the measure (T ) given by d (T ) d = 1 2 ∣∣∣∣ ∞ ∑ m=0 x(m)Tme−im ∣∣∣∣ 2 : Essential for the use in frequency analysis is the weak star convergence for T → 1− of the absolutely continuous measure G(T ) (T ) to a measure supported by a certain nite set on the unit circle for some positive G(T ). For (T ) the function G(T ) = 1 − T 2 works. The present paper is a report on measures (T , obtained by inserting positive weights cm in d (T =d . This means to study measures of a form given by d (T ) d = 1 2 ∣∣∣∣ ∞ ∑ m=0 x(m)cmT me−im ∣∣∣∣ 2 ; where the coe cients cm satisfy certain conditions. The main part of the paper is to prove weak star convergence by a proper choice of G(T ). c © 1999 Elsevier Science B.V. All rights reserved.