Frames for source recovery from non-uniform dynamical samples
Ruchi, Lalit Kumar Vashisht
TL;DR
This work develops a rigorous framework for stable source-term recovery in non-uniform discrete dynamical systems indexed by spectral-pair derived sets in infinite-dimensional Hilbert spaces. By embedding measurements in structured operator spaces and leveraging non-uniform frames, it derives precise necessary and sufficient conditions for stable recovery across finite and infinite time horizons. Key results show that finite-time stability requires the sampling set to form a frame, while infinite-time stability hinges on the dual action $S^*$ forming a frame on the source subspace $W$; in the contractive case $\rho(A)<1$, a related condition involves $P_W(I-A^*)^{-1}$. The findings have implications for space-time sampling, environmental monitoring, and non-uniform sampling frameworks where spectral-pair indexed measurements arise, with concrete operator-based reconstruction guarantees.
Abstract
Motivated by the work of Aldroubi et al., we investigate the stability of the source term of the discrete dynamical system indexing over a non-uniform discrete set arising from spectral pairs in infinite-dimensional separable Hilbert spaces. Extending results due to Aldroubi et al., firstly, we give a necessary and sufficient condition for the recovery of the source term in finitely many iterations. Afterwards, we derive a necessary condition for the stability of the source term in finitely many iterations when it belongs to the closed subspace of an infinite-dimensional separable Hilbert space. Finally, we give a necessary and sufficient condition for the recovery of the source term in infinitely many iterations.