Occupation Kernel Hilbert Spaces for Fractional Order Liouville Operators and Dynamic Mode Decomposition
This provides a theoretical framework for analyzing fractional order systems, which is incremental as it extends existing DMD methods to nonlocal operators.
The paper introduces occupation kernel Hilbert spaces (OKHS) to handle collections of signals, enabling the definition of fractional order Liouville operators and a fractional order dynamic mode decomposition (DMD) routine for fractional order dynamical systems, with computations that differ only slightly from integer-order methods.
This manuscript gives a theoretical framework for a new Hilbert space of functions, the so called occupation kernel Hilbert space (OKHS), that operate on collections of signals rather than real or complex numbers. To support this new definition, an explicit class of OKHSs is given through the consideration of a reproducing kernel Hilbert space (RKHS). This space enables the definition of nonlocal operators, such as fractional order Liouville operators, as well as spectral decomposition methods for corresponding fractional order dynamical systems. In this manuscript, a fractional order DMD routine is presented, and the details of the finite rank representations are given. Significantly, despite the added theoretical content through the OKHS formulation, the resultant computations only differ slightly from that of occupation kernel DMD methods for integer order systems posed over RKHSs.