  In this study, we investigate the availability of approaching to perfect
classification on functional data with finite samples. The seminal work
(Delaigle and Hall (2012)) showed that classification on functional data is
easier to define on a perfect classifier than on 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 (DH) 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 DH
condition holds because a convergence of misclassification errors of functional
data is significantly slow. Specifically, a minimax rate of the convergence of
errors with functional data has a logarithm order in the sample size. This
study solves this complication by proving that the DH condition also achieves
fast convergence of the misclassification error in sample size. Therefore, we
study a classifier with empirical risk minimization using reproducing kernel
Hilbert space (RKHS) and analyse its convergence rate under the DH condition.
The result shows that the convergence speed of the misclassification error by
the RKHS classifier has an exponential order in sample size. Technically, the
proof is based on the following points: (i) connecting the DH condition and a
margin of classifiers, and (ii) handling metric entropy of functional data.
Experimentally, we validate that the DH condition and the associated margin
condition have a certain impact on the convergence rate of the RKHS classifier.
We also find that some of the other classifiers for functional data have a
similar property.
