I'm having a difficult time finding any theory on an inverse problem I've come up against. Let's say I have an unknown function $f:[0,1] \rightarrow \mathbb{R}$, and I know $\int\_{a}^{b} f$ for some collection $A$ of pairs $(a,b)\in[0,1]^2$. I'm looking for pointers to any material that discusses condtions on $f$ and $A$ that are sufficient recover (all of? some of?) the values of $f$. Google searches just keep turning up elementary-calculus-help-type pages. I'm a beginning graduate student, if it matters. Thanks.


**Edit:** I'm actually looking for something broader than I asked for. I know that recovering the values of $f$ is a lot to ask and is very unlikely unless $A=[0,1]^2$; I'm also looking for approximations to $f$, anything that can be said about its properties/behaviour, etc. when $A\subsetneqq [0,1]^2$.



Reference Request - Recovering a function from its definite integrals (inverse problem)
Here is how you make an inverse problem of this problem: Choose a space $X$ for the function $f$ you are looking for (e.g. $L^2(0,1)$ to work in Hilbert spaces, but other spaces may be more suitable, depending on your needs).


I assume that you only have finitely many definite integrals (since I assume that this is a practical problem where the definite integrals come from measurements). 
Now let us denote your tuples as $(a\_1,b\_1),\dots (a\_N,b\_N)$. You forward operator is
$$\newcommand{\RR}{\mathbb{R}}
K:X\to\RR^N
$$
mapping $f$ to the $N$-vector with components $\int\_{a\_i}^{b\_i}f(x)\,dx$. So you are given a vector $g\in\RR^N$ and want some solution to 
$$
Kf = g.
$$
Now you are in business with the standard theory for linear inverse problems.


You have some of the usual problems coming with an inverse problem: Non-uniqueness (the operator is not injective) and probably instability in some sense (depending on you data and values $(a\_i,b\_i)$). (As far as I see, non-solvability should not be an issue as $K$ should be surjective for meaningful tupels $(a\_i,b\_i)$).


To deal with non-uniqueness: You may view this as an advantage as you can choose among all solutions of $Kf=g$. To pick one, you can choose regularization functional $R:X\to [0,\infty]$ and define a minimum-$R$-solution as solution of
$$
\min\{R(f)\mid f\in X,\ Kf=g\}.
$$
From a computational point of view, convex functional $R$ are beneficial and you can choose $R$ to impose some structure on your solution, e.g. $R(f) = \int\_0^1 |f'(x)|^2\, dx$ imposes some smoothness (effectively this means that you constrain your solutions to the Sobolev space $H^1$). The most straight-forward choice would be $R(f) = \int\_0^1 |f(x)|^2\, dx$ which should produce a linear equality as optimality condition (and you are effectively computing the Moore-Penrose pseudo-inverse). I could say more about regularizing functionals if needed.


If your data vector $g$ is also uncertain, i.e. it may be given by measurement data with an error, you may want to relax your problem and look for solutions of
$$
\min\{R(f)\mid d(Kf,g)\leq\delta\}
$$
for some discrepancy functional $d$ and some value $\delta>0$. Both should be related to the error in your data. Note that this is in some way equivalent to (generalized) Tikhonov regularization which would be solving
$$
\min\_f d(Kf,g) + \lambda R(f) 
$$
for some regularization parameter $\lambda>0$. The most simple case of this would be standard Tikhonov regularization in Hilbert spaces:
$$
\min\_f \|Kf-g\|\_{2}^2 + \lambda\|f\|\_{L^2(0,1)}^2
$$
leading to the linear optimality condition
$$
K^\*(Kf-g) + \lambda f = 0.
$$
The adjoint operator $K^\*:\RR^N\to L^2(0,1)$ is given by
$$
K^\*g = \sum\_{i=1}^N g\_i\chi\_{[a\_i,b\_i]}
$$
(where $\chi\_{[a\_i,b\_i]}$ is the characteristic function of $[a\_i,b\_i]$).
So the optimality condition is actually
$$
\sum\_i \left[\langle f,\chi\_{[a\_i,b\_i]}\rangle - g\_i\right]\chi\_{[a\_i,b\_i]} + \lambda f = 0.
$$
This shows that the regularized solution is also a linear combination of the characteristic functions $\chi\_{[a\_i,b\_i]}$ and thus, we still get a finite dimensional linear problem for the coefficients.


If you want some smoothness, try $R(f) = \int\_0^1 |f'(x)|^2\, dx$. This would give an optimality conditions like
$$
\sum\_i \left[\langle f,\chi\_{[a\_i,b\_i]}\rangle - g\_i\right]\chi\_{[a\_i,b\_i]} - \lambda f'' = 0
$$
and thus the solution is piecewise quadratic.


