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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!

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