Convergence of PPC-continued fraction approximants in frequency analysis (Q1425663)

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.