  The focus of this paper is to extend Fisher's linear discriminant analysis
(LDA) to both densely re-corded functional data and sparsely observed
longitudinal data for general $c$-category classification problems. We propose
an efficient approach to identify the optimal LDA projections in addition to
managing the noninvertibility issue of the covariance operator emerging from
this extension. A conditional expectation technique is employed to tackle the
challenge of projecting sparse data to the LDA directions. We study the
asymptotic properties of the proposed estimators and show that asymptotically
perfect classification can be achieved in certain circumstances. The
performance of this new approach is further demonstrated with numerical
examples.
