I have function values at each of the vertices of the hyper cube. What would be a natural interpolation of the function to each point on and inside the cube that can be written as a positive linear combination of polynomial (in the number of dimensions) number of vertices ? This is in context of lovasz extension of submodular functions.
Edit :
The vertices of the hypercube do not exhibit hyper-cubical symmetry. Instead the vertices correspond to the subsets of a set. For example the vertices (0,0,...0) and (1,1,1...1) are the null set and complete set respectively over a ground set $V$. What i want to understand is how to interpret the following proposed extension :
Let $c = (c(1), c(2), ... c(n))$ in $[0,1]^n$ and let $p_1> p_2 > ... > p_k$ be the distinct values in $\{c(1), c(2), ... c(n)\}$. Define $q_k = p_k$ and $q_j = p_j - p_{j+1}$ for $j = 1, 2, ... k-1$.
For $1 \le j \le k$ we let $U_j = \{ i | c(i) \ge p_j \}$. Define f' as follows :
f'(c) = (1- p_1)f(\phi) + \sum_{j=1}^{k} q_jf(U_j)
Think of each vertex of the hyper cube as being an indicator vector of a subset $U \subseteq V$, where $\left|{V}\right| = n$
The interpolation is as explained here