# Reference Request - Recovering a function from its definite integrals (inverse problem)

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 $A\subsetneqq [0,1]^2$.

• Well, if $f$ is integrable, then the integral is jointly continuous in $a$ and $b$, so dense sets suffice. So for instance $\{0\} \times \mathbb{Q}$ would work. – Nate Eldredge Feb 5 '18 at 0:00
• Recovery of function from its integral is called differentiation (Newton-Leibniz formula). – Alexandre Eremenko Feb 5 '18 at 0:27
• If $A=\{0\}\times[0,1]$, you’re in business also. Otherwise, maybe you could set this up as an optimization problem: find the “simplest “ function satisfying certain constraints, whatever your definition of simple is. Finally, why not work with the anti derivative, $F$ instead of with $f$? Your conditions immediately give you information about $F$. – Anthony Quas Feb 5 '18 at 1:00
• Is $f$ assumed continuous? – LSpice Feb 5 '18 at 16:14
• Could you add a little more detail on the background of the problem? Is this a theoretical question or a practical one? Do you assume "exact data" (i.e. the values $\int_a^b f$ are exact)? Any more information about $A$? If you aim at approximations of $f$, do you aim at error bounds or just convergence? What do you assume about $f$? (Barely integrable, continuity, smoothness, bounds on derivatives,…?) Up to now, the answers seem to poke around not knowing for what to look… – Dirk Feb 5 '18 at 16:57

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.

• It's not clear whether the set $A$ of sample pairs is meant to be finite. – LSpice Feb 5 '18 at 16:15
• @lspice Sure. Somehow my gut tells me that this is an applied math question where many things are open and open to modelling (whick would imply a finite number of data points) but I may well be wrong... – Dirk Feb 5 '18 at 16:27

In general, it appears that hardly anything interesting can be said. E.g., let $A=\{(1/5,3/5),(2/5,4/5)\}$; here, it will be convenient to think of $A$ as a set of (say) open intervals, rather than a set of pairs of endpoints of intervals.

However, note first that, without loss of generality, for each open interval in $A$, all the intervals (closed, left-open, right-open) with the same endpoints may be assumed to belong to $A$. Next, let us assume that $\int_0^1|f|<\infty$ and that $A$ is a semi-ring (see measures on semi-rings) and $[0,1]\in A$. Then the formula $\mu(I):=\int_I f$ for $I\in A$ defines a finite signed countably-additive measure $\mu$ on $A$, which can be uniquely extended to a signed measure $\bar\mu$ on the sigma-algebra $\Sigma$ generated by $A$.

The measure $\bar\mu$ determines, and is determined by, the conditional expectation $E(f|\Sigma)$ (of $f$ given $\Sigma$), equal the Radon--Nikodym derivative $\dfrac{d\bar\mu}{d\lambda|_\Sigma}$ (with respect to the underlying Lebesgue measure $\lambda$ over $[0,1]$), and this conditional expectation is then precisely all that we can get from the knowledge of the map $A\ni I\mapsto \int_I f$. (One might note that the appearance of the Radon--Nikodym derivative here is in broad agreement with the comment "Recovery of function from its integral is called differentiation" by Alexandre Eremenko.)

E.g., if $A$ consists of all intervals with endpoints in the set $\{j/n\colon j=0,\dots,n\}$, then all that we will know is, in essence, the "histogram" of the average values of $f$ over the intervals $[0,1/n],\dots,[1-1/n,1]$, and this "histogram" is the best approximation to $f$ that we can get in this case.

Extended comment: Dirk suggested an inverse-problem approach. One may note that such an approach will work perfectly well (and, generally, even better) within the above framework of the conditional expectation. Indeed, for a space $X$ of (say) real-valued integrable functions on $[0,1]$ we have the map $X\overset K\to\mathbb R^A$ defined by the formula $Kf:=(\int_I f)_{I\in A}$ for $f\in X$. This map can be factored as follows: \begin{equation} X\overset{E(\cdot|\Sigma)}\longrightarrow X_\Sigma\overset{K_\Sigma}\longrightarrow\mathbb R^A, \end{equation} where $X_\Sigma$ is the set of all integrable $\Sigma$-measurable functions in $X$ and $K_\Sigma$ is the restriction of $K$ to $X_\Sigma$; indeed, by the definition of the conditional expectation/Radon--Nikodym derivative, we have
$K_\Sigma E(f|\Sigma)=Kf$ for all $f\in X$. Thus, instead of $K$, one can deal with its restriction $K_\Sigma$, with the same (or greater) degree of success. In particular, if $A$ is finite, then we have to deal with the finite-dimensional space $X_\Sigma$ instead of the possibly infinite-dimensional space $X$.

This comment may be viewed as an illustration of what was said previously: that the conditional expectation $E(f|\Sigma)$ is precisely all that we can get from the knowledge of the map $A\ni I\mapsto \int_I f$.

• I have added an extended comment about an inverse-problem approach. – Iosif Pinelis Feb 5 '18 at 13:17
• Nice! Do I see correctly that $X_\Sigma$ contains functions that are piecewise constant in the case where $A$ is finite? In this case, a regularization may be in order if one aims at a different structure. – Dirk Feb 5 '18 at 13:22
• @Dirk : Right, $X_\Sigma$ will consist of piecewise constant functions if $A$ is finite, and then one may indeed want to regularize/smooth those functions. – Iosif Pinelis Feb 5 '18 at 13:32

If $A$ is infinite and dense then you can find the function exactly by taking limits. I'll look at the finite case. In the event that the function is a polynomial of degree $n$ then knowing the integral over $n+1$ internally disjoint intervals is enough for recovery. Without knowing that the function is a polynomial , but given $n+1$ integrals one could find the unique polynomial of degree $n$ that matches the data. Or the polynomial of degree $n-m$ that best fits.