  Any autonomous nonlinear dynamical system can be viewed as a superposition of
infinitely many linear processes, through the so-called Koopman mode
decomposition. Its data-driven approximation- Dynamic Mode Decomposition (DMD)-
has been extensively developed and deployed across a plethora of fields. In
this work, we study the effect of subtracting the temporal mean on the DMD
approximation, for observables possessing only a finite number of Koopman
modes.
  Pre-processing time-sequential training data by removing the temporal mean
has been a point of contention in the Companion matrix formulation of DMD. This
stems from the potential of said pre-processing to render DMD equivalent to a
temporal Discrete Fourier Transform (DFT). We prove that this equivalence is
impossible when the training data is linearly consistent and the order of the
DMD approximation exceeds the number of Koopman modes. Since model order and
training set size are synonymous in this variant of DMD, the parity of DMD and
DFT can, therefore, be indicative of inadequate training data.
