From the introduction: We propose kernel-based principal components for functional data, and study their asymptotic properties. There are two common ways of performing smooth principal component analysis. The first is to smooth the functional data and then perform PCA. The second is to directly define smoothed the principal components. This can be achieved, for example, by adding a penalty term to the sample variance and maximizing the penalizid sample variance. If a kernel-based smoothing method is used, it will be shown that both approaches are the same.
On the other hand, the kernel-based approach allows to derive the asymptotic distribution of the smooth principal components, which is unknown for penalized methods as in other nonparametric settings. It is also shown that the degree of regularity of kernel-based principal components is given by that of the kernel function used. Strong consistency and the asymptotic distribution are derived under mild conditions.