Szegő polynomials applied to frequency analysis
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.
scientific article; zbMATH DE number 219122
scientific article