Test of Weak Separability for Spatially Stationary Functional Field

For spatially dependent functional data, a generalized Karhunen-Lo\`{e}ve expansion is commonly used to decompose data into an additive form of temporal components and correlated coefficients. This structure provides convenient model investigate the space-time interactions, but may not hold for complex spatio-temporal processes. In this work, we introduce concept weak separability, propose formal test examine its validity non-replicated stationary field. The asymptotic distribution statistic that adapts potentially diverging ranks derived by constructing lag covariance estimation, which easy compute practical implementation. We demonstrate efficacy proposed via simulations illustrate usefulness in two real examples: China PM$_{2.5}$ Harvard Forest data.