Wavelet-Based Observables Enhance Koopman Analysis Framework

A new mathematical approach has been introduced that combines wavelet transforms with Koopman semigroup analysis. This approach identifies wavelet-based observables as eigenfunctions of the Koopman semigroup in the Banach space of continuous functions, specifically on a compact forward-invariant set defined by the supremum norm. The researchers have also derived closed-form representations for how the Koopman semigroup and its resolvent operate using these observables. To enable numerical approximations, they have merged Extended Dynamic Mode Decomposition (EDMD) with these wavelet observables, resulting in the cWDMD algorithm (Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform). The theoretical results are validated through two numerical examples, providing a fresh perspective for analyzing nonlinear dynamical systems with a new spectral decomposition method.