(1) Suppose I have a (bounded, measurable) function $f$ on a compact subset $K$ of $R^n$, and for every open interval $U\subset K$ I know only the mean value
$$ m_U := \frac{1}{|U|} \int_U f dx. $$
What additional constraints are required to uniquely recover $f$ almost everywhere? Or: what is the largest space of functions you can think of which allows this?
(2) Same situation, but I know the mean not for every $U$, but only nested sequences of intervals, in the sense of a finer and finer discretization of $K$. For simplicity, let's assume $K=[0,1]$ and $U$ is of the form $( k/2^n, (k+1) / 2^n)$ with $n \geq 0$ and $0 \leq k \leq 2^n-1$.
If you can point me to theorems/sections of books which might yield an answer to the questions or sketch an idea for a possible answer and proof, I would be most happy.