  Extracting the latent underlying structures of complex nonlinear local and
nonlocal flows is essential for their analysis and modeling. In this work, we
attempt to provide a consistent framework through Koopman theory and its
related popular discrete approximation -- dynamic mode decomposition (DMD). We
investigate the conditions to perform appropriate linearization, dimensionality
reduction, and representation of flows in a highly general setting. The
essential elements of this framework are Koopman Eigenfunction (KEF), for which
existence conditions are formulated. This is done by viewing the dynamic as a
curve in state-space. These conditions lay the foundations for system
reconstruction, global controllability, and observability for nonlinear
dynamics. We examine the limitations of DMD through the analysis of Koopman
theory and propose a new mode decomposition technique based on the typical time
profile of the dynamics. An overcomplete dictionary of decay profiles is used
to sparsely approximate the flow. This analysis is also valid in the full
continuous setting of Koopman theory, which is based on variational calculus.
We demonstrate applications of this analysis, such as finding KEFs and their
multiplicities, dynamics reconstruction, and global linearization.
