In *Science and Hypothesis*, chapter XI, *The calculus of probabilities*, Henri PoincarĂ© deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/items/scienceandhypoth00poinuoft/scienceandhypoth00poinuoft.pdf, page 206):

*Why, then, do I draw a curve without sinuosities? Because I consider a priori a law represented by a continuous function (or function the derivatives of which to a higher order are small), as more probable than a law not satisfying those conditions. But for this conviction the problem would have no meaning; interpolation would be impossible; no law could be deduced from a finite number of observations; science would cease to exist.*

To my knowledge, PoincarĂ© interpolation was finally addressed in 1992 by Bretthorst using Jaynes' Principle of Maximum Entropy. Please refer to *Bayesian interpolation and deconvolution*, http://bayes.wustl.edu/glb/deconvolution.pdf.
The principle is quite simple:

- By Bayes' rule, we just need to assign the prior probability distribution for the smooth theoretical signal $u(t)$ to be estimated (eq. 10).
- Approximate for instance $\displaystyle \frac{\textrm{d}^{2}u}{\textrm{d}t^{2}}$ numerically by finite differences (eq. 16). We seek for our prior distribution among all continuous distributions having same Euclidean norm $\displaystyle\left\|\frac{\textrm{d}^{2}u}{\textrm{d}t^{2}}\right\| $ (eq. 17).
- Call the Principle of Maximum Entropy: finally, you get a multivariate Gaussian prior distribution (eq. 28) involving the fractional variance regularizer $\epsilon^{2}$.
- Assign a prior distribution for $\epsilon$ and compute its marginal posterior distribution (eq. 61). Then you get for instance its MAP estimate by maximization (fig. 3);
- Estimate $u_{j}$ by maximizing the marginal posterior $p(u_{j}|\epsilon,D,I)$ (eq. 52, fig. 4).

Bretthorst considers that $u(t)$ is to be estimated at $\nu+2$ points where $\nu=\beta(N-1)+1$ (eq. 5), $N$ is the number of observations and $\beta$ is a positive integer. We can think for instance about a regular grid with step $\Delta t/\beta$. In order to make the finite differences (eq. 16) and consequently our estimates more accurate, we should certainly increase $\beta$ even if we want to estimate $u(t)$ at, say, only the $N$ measurement points. In practice, we actually observe that the interpolation converges quickly as $\beta$ increases. Therefore the following problems may arise:

**Problem 1:** Can we pass to the limit $\beta\to+\infty$ in the interpolation problem?

$p(u_{j}|\epsilon,D,I)$ (eq. 52) involves the eigenvalues and the eigenvectors of the $j^{th}$ cofactor of the square matrix $g_{ik}=\epsilon^{2}R_{ik} + S_{ik}$ (eq. 43-52) of dimension $\nu+2$. Therefore, there are two possibilities:

- Either we can get those eigenvalues and eigenvectors analytically for any $\beta$ and $\epsilon$ or we can approximate them. Then, we would just need to take the limit in the series $h_{l}(u_{j})$ (eq. 43).
- Or we can get those eigenvalues and eigenvectors only for the infinite matrix $g_{ik}$. Then, we may need to justify that we can interchange the limits.

**Problem 2:** Can we pass to the limit $\beta\to+\infty$ in the deconvolution problem?

Now the matrix $S_{ik}$ depends on the impulse response function $r(t)$ (eq. 112) so that we can't use the nice structure of the interpolation matrix (eq. 35).