Szegő polynomials applied to frequency analysis
type: Item
instance of: scholarly article
title: Szegő polynomials applied to frequency analysis
zbMATH Open document ID: 0784.65105
DOI: 10.1016/0377-0427(93)90297-O
author: W. B. Jones
author: W. J. Thron
author: Olav Njåstad
author: Haakon Waadeland
published in: Journal of Computational and Applied Mathematics
publication date: 1993-08-11T00:00:00Z
review text: The paper investigates the problem of determining unknown frequencies \(\omega_ 1,\dots,\omega_ I\), using an observed time signal \(x_ N\) arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well-defined frequencies \(\omega_ j\) \((j=1,\dots,I)\). For the solution of the problem, the authors use the Wiener-Levinson method formulated in terms of Szegö polynomials \(p_ n(\psi_ N;z)\) with respect to a distribution function \(\psi_ N\) defined by the \(N\) observed values of the signal. The Wiener-Levinson method consists in the computation of the Szegö polynomials \(p_ n(\psi_ N;z)\) using Levinson's algorithm and in the approximation of the critical points \(e^{i\omega_ j}\) by zeros \(z(j,n,N)\) of \(p_ n(\psi_ N;z)\) nearest to the unit circle. The usefulness of this method depends on the validity of a conjecture of the first and the third author and \textit{E. B. Saff} [J. Comput. Appl. Math. 32, No. 3, 387-406 (1990; Zbl 0718.94002)]. In this paper, the authors prove one of the main parts of the conjecture: If \(n_ 0\) denotes the number of critical points \(e^{i\omega_ j}\), then for every \(n\geq n_ 0\) and \(N\geq 1\), the zeros \(z(j,n,N)\) of \(p_ n(\psi_ N;z)\) can be arranged so that \(\lim_{N\to\infty} z(j,n,N)= e^{i\omega_ j}\) for each of the frequencies \(\omega_ j\). Further, they give a related result on the convergence of corresponding two-point Padé approximants.
Mathematics Subject Classification ID: 65T40
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 94A11
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 41A21
zbMATH DE Number: 219122
zbMATH Keywords: frequency analysis
zbMATH Keywords: Wiener-Levinson filters
zbMATH Keywords: discrete signals
zbMATH Keywords: Wiener- Levinson method
zbMATH Keywords: Szegö polynomials
zbMATH Keywords: Levinson's algorithm
zbMATH Keywords: convergence
zbMATH Keywords: Padé approximants
reviewed by: Gabriele Steidl
MaRDI profile type: Publication
cites work: Q3231529
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description: scientific article; zbMATH DE number 219122