# Functional Linear Regression without Orthonormal Basis for Prediction

What do you do in functional linear regression when for whatever reason you don't want to use an orthonormal basis expansion? In functional linear regression for scalar on function regression, one may do regression on the centered process $$X$$ as follows \begin{align} Y&=\alpha+\int_\mathcal{T} X(t)\beta(t)dt+\epsilon \end{align} One generally expands $$X(t)=\sum_{i=1}^\infty x_i\phi_i(t)$$ and $$\beta(t)=\sum_{i=1}^\infty \beta_i\phi_i(t)$$ into the same orthonormal basis of $$L^2(\mathcal{T})$$, obtaining \begin{align} Y&=\alpha+\int_\mathcal{T}(\sum_{i=1}^\infty x_i\phi_i(t))(\sum_{k=1}^\infty \beta_k\phi_k(t))dt+\epsilon\\ &=\alpha+\sum_{i=1}^\infty x_i\beta_i+\epsilon \end{align} However if the basis is not orthonormal, could we set $$\gamma_i=\sum_{k=1}^\infty \beta_k\int_\mathcal{T} \phi_i(t)\phi_k(t)dt$$ and then use the model \begin{align} Y&=\alpha+\sum_{i=1}^\infty x_i\gamma_i \end{align} and fit $$\gamma_i$$ instead of $$\beta_i$$? Is this correct/reasonable, and if so, what are the tradeoffs of doing this? For inference the $$\beta_i$$ are important because they tell you about the importance of specific basis functions. However if we simply want good predictions this seems like it should be fine. Any other issues?

So I talked to someone who knows more FDA, and he said that usually you want to learn $$\beta(t)$$ for its interpretation, in which case you need to compute the inner products between basis functions. However, if you only care about prediction, then the model above is fine.