  This manuscript introduces a space of functions, termed occupation kernel
Hilbert space (OKHS), that operate on collections of signals rather than real
or complex functions. 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.
