#StackBounty: #self-study #pca #linear-model #functional-data-analysis Asymptotic properties of functional models

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!

Get this bounty!!!

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