Function-on-function linear model

A function-on-function linear model has $Y(t)=\int \beta (t,s)X(s)ds+\epsilon (t)$. If the centered error is independent of $X$, then $\mathbb{E}[Y\mid X]$ is the regression operator applied to $X$.

For centered Hilbert-space random variables $X$ and $Y$, the cross-covariance operator is $C_{YX}h=\mathbb{E}[⟨X,h⟩Y]$. In the functional linear model $Y=BX+\epsilon$ with independent centered error, $C_{YX}=B{C}_X$.