  A central topic in functional data analysis is how to design an
optimaldecision rule, based on training samples, to classify a data function.
We exploit the optimal classification problem when data functions are Gaussian
processes. Sharp nonasymptotic convergence rates for minimax excess
mis-classification risk are derived in both settings that data functions are
fully observed and discretely observed. We explore two easily implementable
classifiers based on discriminant analysis and deep neural network,
respectively, which are both proven to achieve optimality in Gaussian setting.
Our deepneural network classifier is new in literature which demonstrates
outstanding performance even when data functions are non-Gaussian. In case of
discretely observed data, we discover a novel critical sampling frequency
thatgoverns the sharp convergence rates. The proposed classifiers perform
favorably in finite-sample applications, as we demonstrate through
comparisonswith other functional classifiers in simulations and one real data
application.
