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For spatially dependent functional data, a generalized Karhunen-Lo\`{e}veexpansion is commonly used to decompose data into an additive form of temporalcomponents and spatially correlated coefficients. This structure provides aconvenient model to investigate the space-time interactions, but may not holdfor complex spatio-temporal processes. In this work, we introduce the conceptof weak separability, and propose a formal test to examine its validity fornon-replicated spatially stationary functional field. The asymptoticdistribution of the test statistic that adapts to potentially diverging ranksis derived by constructing lag covariance estimation, which is easy to computefor practical implementation. We demonstrate the efficacy of the proposed testvia simulations and illustrate its usefulness in two real examples: ChinaPM$_{2.5}$ data and Harvard Forest data.

Test of Weak Separability for Spatially Stationary Functional Field