  We consider the problem of classification of functional data into two groups
by linear classifiers based on one-dimensional projections of functions. We
reformulate the task to find the best classifier as an optimization problem and
solve it by regularization techniques, namely the conjugate gradient method
with early stopping, the principal component method and the ridge method. We
study the empirical version with finite training samples consisting of
incomplete functions observed on different subsets of the domain and show that
the optimal, possibly zero, misclassification probability can be achieved in
the limit along a possibly non-convergent empirical regularization path. Being
able to work with fragmentary training data we propose a domain extension and
selection procedure that finds the best domain beyond the common observation
domain of all curves. In a simulation study we compare the different
regularization methods and investigate the performance of domain selection. Our
methodology is illustrated on a medical data set, where we observe a
substantial improvement of classification accuracy due to domain extension.
