On measures in frequency analysis

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:105

Article

Petersen, V

关键词: Szego polynomial;    frequency analysis;   

DOI  :  10.1016/S0377-0427(99)00026-6

来源: Elsevier

【 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 Szego polynomials. The Szego polynomials in question are orthogonal on the unit circle with respect to an inner product defined by a measure psi(N). The measure is constructed from the observed signal values x(N)(m): [GRAPHICS] This first method is called the N-process. Later there came a modification where the measure psi(N) was replaced by a new measure psi(N)((R)). This new method called the R-process, involves one additional parameter, but has certain convergence benefits. Very recently Njastad and Waadeland introduced and used the measure psi((T)) given by [GRAPHICS] Essential for the use in frequency analysis is the weak star convergence for T --> 1(-) of the absolutely continuous measure G(T)psi((T)) to a measure supported by a certain finite set on the unit circle for some positive G(T). For psi((T)) the function G(T) = 1 - T-2 works. The present paper is a report on measures psi((T)), obtained by inserting positive weights c(m) in d psi((T))/d theta. This means to study measures of a form given by [GRAPHICS] where the coefficients c(m) satisfy certain conditions. The main past 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.

【 授权许可】

Free