*Bounty: 100*

*Bounty: 100*

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!