  We propose an alternative to $k$-nearest neighbors for functional data
whereby the approximating neighboring curves are piecewise functions built from
a functional sample. Using a locally defined distance function that satisfies
stabilization criteria, we establish pointwise and global approximation results
in function spaces when the number of data curves is large enough. We exploit
this feature to develop the asymptotic theory when a finite number of curves is
observed at time-points given by an i.i.d. sample whose cardinality increases
up to infinity. We use these results to investigate the problem of estimating
unobserved segments of a partially observed functional data sample as well as
to study the problem of functional classification and outlier detection. For
such problems, our methods are competitive with and sometimes superior to
benchmark predictions in the field.
