# Asymptotic properties of functional models

When working in Functional Data Analysis, a classical "preprocessing" step is to represent the "observations" using a B-spline expansion:

$$X_i(t) \approx \sum_{j=1}^J \lambda_{ij} f_j(t) \qquad i=1, \ldots, n$$

where $$J$$ is the number of elements in the basis and $$f_1, \ldots, f_J$$ are suitably defined B-spline functions. Then, statistical methods are performed by working directly on the coefficients $$\{\lambda_{ij}\}$$.

My question is if there are some asymptotic guarantees that as the number of data $$n$$ and the truncation level $$J$$ increase to $$+\infty$$ the statistical methods converge to a "true" idealized solution.

In particular, I'm interested in functional on functional regression an functional PCA. I know the literature is huge, but it would be great to have some papers to start from!