Convergence of PPC-continued fraction approximants in frequency analysis
type: Item
instance of: scholarly article
title: Convergence of PPC-continued fraction approximants in frequency analysis
zbMATH Open document ID: 1042.94005
author: W. B. Jones
author: Vigdis Brevik Petersen
author: Haakon Waadeland
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-03-17T00:00:00Z
review text: Let \(x(m):=G(m\Delta t)\) be observations of a signal \[ G(t)=\sum_{j=-I}^I\alpha_j\,e^{2\pi i f_j t},\quad f_{-j}=f_j,\quad \alpha_{-j}=\overline{\alpha_j},\quad 0=f_0<f_1<\dots <f_I, \] at regular time intervals \(\Delta t\). The problem considered is to recover an approximation of the signal from these observations. In the \(N\)-process this is done by starting with the \(N\) first observations \(x(m)\) to form the PPC-fraction \[ \delta_0^{(N)}-\frac{2\delta_0^{(N)}}1\,{\quad\atop +}\,\frac 1{\delta_1^{(N)}z}\,{\quad\atop +}\,\frac{(1-\delta_1^{(N)^2})z}{\delta_1^{(N)}}\,{\quad\atop +}\,\frac 1{\delta_2^{(N)}z}\,{\quad\atop +}\,\frac{(1-\delta_2^{(N)^2})z}{\delta_2^{(N)}}\,{\quad\atop +\dots} \tag{\(*\)} \] whose reflection coefficients \(\delta_n^{(N)}\) turn out to be real and less than \(1\) in absolute value. Let \(R_m(N;z)\) be the classical approximants of (\(*\)). It is known that as \(N\to\infty\), \(\{\frac 1N R_{2m}(N;z)\}\) converges for \(| z| <1\) and \(\{\frac 1N R_{2m+1}(N;z)\}\) converges for \(| z| >1\), uniformly on compact subsets, to \[ H(z)=\sum_{j=-I}\widehat I | \alpha_j| ^2\,\frac{e^{i\omega_j}+z}{e^{i\omega_j}-z},\quad \omega_j=2\pi f_j\Delta t. \] In the present paper the authors extend the regions for this convergence.
reviewed by: Lisa Lorentzen
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 40A15
Mathematics Subject Classification ID: 30B70
zbMATH DE Number: 2060091
zbMATH Keywords: continued fractions
zbMATH Keywords: convergence
zbMATH Keywords: frequency analysis
zbMATH Keywords: signal processing
zbMATH Keywords: PPC-fraction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069965
OpenAlex ID: W2035778315
cites work: Q4177796
cites work: Q5508534
cites work: Q3856819
cites work: Q3231529
cites work: Q3285751
cites work: Applications of Szegö polynomials to digital signal processing
cites work: Szegö polynomials associated with Wiener-Levinson filters
cites work: Continued fractions associated with trigonometric and other strong moment problems
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
cites work: Szegő polynomials applied to frequency analysis
cites work: Q4295466
cites work: Q4295467
cites work: Application of Szegö Polynomials to Frequency Analysis
cites work: Continued fractions and Szegö polynomials in frequency analysis and related topics
cites work: Q4016586
cites work: A constructive proof of convergence of the even approximants of positive PC-fractions
cites work: Asymptotics of columns in the table of orthogonal polynomials with varying measures
cites work: Linear Prediction of Speech
cites work: Generalized Szegö theory in frequency analysis
cites work: Asymptotics for zeros of Szegő polynomials associated with trigonometric polynomial signals
cites work: Anharmonic Frequency Analysis
cites work: Q4339505
cites work: Q4252002
cites work: On measures in frequency analysis
cites work: Q5817617
DOI: 10.1216/RMJM/1181069965
description: scientific article; zbMATH DE number 2060091