Orthogonal rational functions and frequency analysis
Given a trigonometric signal \(x(m)=\sum_{j=-I}^I A_j \exp(i\omega_jm)\) with \(A_0\geq 0\), \(A_{-j}=\bar{A}_j\) complex and \(\omega_{-j}=-\omega_j\) real, one has to extract the frequencies \(\omega_j\) from the signal. Traditionally, this is done using orthogonal Szegő polynomials. These polynomials are orthogonal with respect to the spectral measures of the signal \(x(m)\), \(m=1,\ldots,N\). Als \(n\to\infty\), the zeros of the orthogonal polynomials approach the points on the circle corresponding to the desired frequencies. In this paper the analog in the case where the polynomials are replaced by orthogonal rational functions with prescribed poles is considered [see \textit{O.~Njåstad} and \textit{H.~Waadeland}, J. Math. Anal. Appl. 206, No. 1, 280-307 (1997; Zbl 0872.42006); J. Comput. Appl. Math. 77, No. 1-2, 255-275 (1997; Zbl 0864.42010)]. This paper reviews the method and its problems and gives explicit expressions for the moment integrals and the orthogonal rational functions, which is elaborated for the simple example \(x(m)=\exp(im\omega)+\exp(-im\omega)\).
scientific article; zbMATH DE number 1546754