Spectral Estimation Problem in Infinite Dimensional Spaces
S. A. Avdonin, V. S. Mikhaylov
TL;DR
This work addresses the generalized spectral estimation problem in infinite-dimensional spaces by employing a boundary control method from inverse theory. It develops an operator-based framework to recover spectral data from boundary observations, representing the target signal as $S(t)=\sum_{k=1}^{\infty} a_k(t) e^{\lambda_k t}$ with $a_k(t)=\sum_{i=0}^{L_k} a_k^i t^i$, and reduces recovery to generalized eigenvalue problems for $\lambda_k$ and $L_k$, followed by reconstruction of the polynomial coefficients $a_k^i$. The approach is then applied to the continuation of inverse data for a 1D first-order hyperbolic system, showing how boundary measurements on $(0,T)$ can be extended to all times by recovering spectral data and modal coefficients via a moment method, under exact controllability and a Riesz-basis structure. The results provide a rigorous link between boundary control, infinite-dimensional spectral estimation, and PDE identification problems (TY5), enabling principled data extension and parameter recovery from boundary observations.
Abstract
We consider the generalized spectral estimation problem in infinite dimensional spaces. We solve this problem using the boundary control approach to inverse theory and provide an application to the initial boundary value problem for a hyperbolic system.