  We investigate the availability of approaching perfect classification on
functional data with finite samples. The seminal work (Delaigle and Hall
(2012)) showed that perfect classification for functional data is easier to
achieve than for finite-dimensional data. This result is based on their finding
that a sufficient condition for the existence of a perfect classifier, named a
Delaigle--Hall condition, is only available for functional data. However, there
is a danger that a large sample size is required to achieve the perfect
classification even though the Delaigle--Hall condition holds, because a
minimax convergence rate of errors with functional data has a logarithm order
in sample size. This study solves this complication by proving that the
Delaigle--Hall condition also achieves fast convergence of the
misclassification error in sample size, under the bounded entropy condition on
functional data. We study a reproducing kernel Hilbert space-based classifier
under the Delaigle--Hall condition, and show that a convergence rate of its
misclassification error has an exponential order in sample size. Technically,
our proof is based on (i) connecting the Delaigle--Hall condition and a margin
of classifiers, and (ii) handling metric entropy of functional data. Our
experiments support our result, and also illustrate that some other classifiers
for functional data have a similar property.
