  Existing works on functional data classification focus on the construction of
classifiers that achieve perfect classification in the sense that
classification risk converges to zero asymptotically. In practical
applications, perfect classification is often impossible since the optimal
Bayes classifier may have asymptotically nonzero risk. Such a phenomenon is
called as imperfect classification. In the case of Gaussian functional data, we
exploit classification problem in imperfect classification scenario. Sharp
convergence rates for minimax excess risk are derived when data functions are
either fully observed or discretely observed. Easily implementable classifiers
based on discriminant analysis are proposed which are proven to achieve minimax
optimality. In discretely observed case, we discover a critical sampling
frequency that governs the sharp convergence rates. The proposed classifiers
perform favorably in finite-sample applications, as we demonstrate through
comparisons with other functional classifiers in simulations and one real data
application.
