  This paper concerns the modeling of multi-way functional data where double or
multiple indices are involved. We introduce a concept of weak separability. The
weakly separable structure supports the use of factorization methods that
decompose the signal into its spatial and temporal components. The analysis
reveals interesting connections to the usual strongly separable covariance
structure, and provides insights into tensor methods for multi-way functional
data. We propose a formal test for the weak separability hypothesis, where the
asymptotic null distribution of the test statistic is a chi-square type
mixture. The method is applied to study brain functional connectivity derived
from source localized magnetoencephalography signals during motor tasks.
