Convergent Dynamic Mode Decomposition

This manuscript addresses convergence of dynamic mode decomposition (DMD) algorithms and the existence of associated Koopman modes. Convergence relies on reformulation of dynamic mode decomposition in terms of newly defined compact operators defined with pairs of Hilbert spaces selected separately as the domain and range of the operator. With the Hilbert spaces selected so that the domain is embedded in the range, an eigenfunction approach to DMD is developed by leveraging a finite rank representation. The finite rank representation is proven to converge, in norm, to the original operator with increasing rank. The manuscript concludes with the description of a DMD algorithm that converges when a dense collection of occupation kernels, arising from the data, are leveraged in the analysis.