Error analysis beyond exponential kernels for Prony-series approximations

Develop rigorous error analysis for generalized Langevin equations when the memory kernel is approximated by a Prony series, beyond the simple one-dimensional harmonic oscillator case with exponentially decaying kernels; in particular, derive trajectory-level error bounds for more general kernels and settings.

Background

In prior work on kernel identification and approximation, Prony-series representations have been analyzed for a one-dimensional harmonic oscillator under exponentially decaying kernels, yielding error bounds in that restricted setting.

The present paper highlights that assuming exponential decay is a strong limitation and explicitly states that extending error analysis to more general cases remains open, motivating broader theoretical frameworks for non-exponential (e.g., subexponential) kernels and higher-dimensional dynamics.