I'm interested in whether exists a smooth function such that it passes through given $N$ points.
Particularly, consider a compact subset $\Omega \in \mathbb{R}^N$ and a family $\mathbb{C}^2(\Omega)$ of twice continuously differentiable functions from $\Omega$ to $\mathbb{R}$. Additionally, two sets are given: $A = \{ X_i \}_{i = 1}^{N}$, $B = \{ Y_i \}_{i = 1}^{N}$, where $X_i \in \Omega$, $Y_i \in \mathbb{R}$, $i \neq j \rightarrow X_i \neq X_j$, and $N$ is finite. Moreover, assume that the distance (e.g. Euclidean, not sure if it matters) between every pair $X_i$ and $X_j$ is greater than some constant $\epsilon > 0$.
For given $A$ and $B$, when exist $f \in \mathbb{C}^2(\Omega)$ s.t. $f(X_i) = Y_i$? Does it always exist? Or its existence depends on some properties of $A$ and $B$? How existence of $f$ can be proved?
Thanks a lot