Let $f(x)$ be some unknown continuous function defined on the interval [0,1].

Suppose we have $i=1,...,n$ samples $f_i$ of $f$ of the following form:
$$f_i(x)=a_i*f(x+b_i)+c_i$$
where each sample is defined on the interval $[\alpha,1-\alpha]$ and $-\alpha \leq b_i \leq \alpha$.

What would be an efficient way to estimate $f(x)$, $a_i$, $b_i$ and $c_i$? For $\alpha=b_i=0$, a PCA-like analysis would work, but how to treat the case $0<\alpha<1/2$?