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 $0<\alpha<1/2$?

Let $f(x)$ be some unknown continuous square-integrable 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 $-\alpha \leq b_i \leq \alpha$ for $0\leq\alpha<1/2$.

Additionally, assume that for at least two of the samples, $b_i$ equals $-\alpha$ and $\alpha$ respectively, i.e. the whole domain of $f(x)$ is sampled. Furthermore, assume $\int_0^1 f(x)\mathrm{d}x=0$ and $\int_0^1 (f (x))^2\mathrm{d}x=1$. Finally, since the sign of $f(x)$ is not retrievable, we care only about retrieving either $f(x)$ or $-f(x)$.

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