  The application of Fourier analysis in combination with the Proper Orthogonal
Decomposition (POD) is investigated. In this approach to turbulence
decomposition, which has recently been termed Spectral POD (SPOD), Fourier
modes are considered as solutions to the corresponding Fredholm integral
equation of the second kind along homogeneous-periodic or homogeneous
coordinates. In the present work, the notion that the POD modes formally
converge to Fourier modes for increasing domain length is challenged. Numerical
results indicate that the discrepancy between POD and Fourier modes along
\textit{locally} translationally invariant coordinates is coupled to the Taylor
macro/micro scale ratio (MMSR) of the kernel in question. Increasing
discrepancies are observed for smaller MMSRs, which are characteristic of low
Reynolds number flows. It is observed that the asymptotic convergence rate of
the eigenspectrum matches the corresponding convergence rate of the exact
analytical Fourier spectrum of the kernel in question - even for extremely
small domains and small MMSRs where the corresponding DFT spectra suffer
heavily from windowing effects. These results indicate that the accumulated
discrepancies between POD and Fourier modes play a role in producing the
spectral convergence rates expected from Fourier transforms of translationally
invariant kernels on infinite domains.
