On the approximation of $C^M$-functions by means of boundary values

In this note we consider the problem of the approximation of a real function, sufficiently regular in a bounded closed domain, in a generic point of the interior by means of opportune boundary values. We will give a solution for this problem in the case of a $N$-cell of $\Omega $, i.e. \[ \Omega =\left\{ x\in \mathbb{R}^{N}:a_{i}\leq x_{i}\leq b_{i},a_{i}<b_{i}\text{ for }i=1,\dots ,N\right\} \] for $N=1,2$. This new method of approximation has many applications in numerical analysis.