  We develop a test of normality for spatially indexed functions. The
assumption of normality is common in spatial statistics, yet no significance
tests, or other means of assessment, have been available for functional data.
This paper aims at filling this gap in the case of functional observations on a
spatial grid. Our test compares the moments of the spatial (frequency domain)
principal component scores to those of a suitable Gaussian distribution.
Critical values can be readily obtained from a chi-squared distribution. We
provide rigorous theoretical justification for a broad class of weakly
stationary functional random fields. We perform simulation studies to assess
the the power of the test against various alternatives. An application to
Surface Incoming Shortwave Radiation illustrates the practical value of this
procedure.
