# Rediscovering a function from samples

A friend shared this nice problem with me. Suppose you have a fixed function, \(f\), and a family of probability distributions, defined by, say, prob. density functions, \(P_t\), \(t \in A\). If we know \(E_t=E_{P_t}[f]\) for every \(t\in A\), can we recover \(f\)?

Clearly the answer depends on both how rigid \(f\) is and the family \(P_t\). We can cast this problem as a functional analysis problem by defining \(k(x,t)=P_t(x)\) to be a kernel and the expectation to be an integral transform. Then the question becomes: is there an inverse kernel, say, \(k^{-1}(x,t)\), such that

\[\int k^{-1}(x,t)E(t)dt=f(x)?\] When does that exist and when is it unique? Hints can be taken from the Laplace transform, i.e., \(P_t(x)\propto e^{-tx}\) - up to a normalizing constant this is the just the exponential distribution. In general, this can be a hard problem though.

## Fredholm equations

If we know \(E(t)=E_{P_t}[f]\) for every \(t\in A\), can we recover \(f\)? Formally, we have the equation:

\[E(t)=\int k(x,t)f(x)dx,\] with appropriate limits for the integral. This equation is called a “Fredholm Equation of the first kind” and is closely studied in functional analysis and signal processing.

## Practical stuff

If we assume the existence of an inverse kernel, how can we approximate it? One idea — which is also kind of a standard approach — is to fix a set of orthonormal basis functions, describe everything in terms of them, and then resolve them to arrive to a linear algebra problem.