Let a real polynomial representing a boolean function be $P(x_1,\dots,x_n) = \sum_{a\in\{0,1\}^n}c_ax^a = \sum_{a\in\{0,1\}^n}p(a)\prod_{i\in 1_a}x_i\prod_{j\in \bar{1}_a}(1-x_j)$ where $1_a$ is the set of $i$ such that $a_i=1$ and $\bar{1}_a$ is the complement of $1_a$. How do you represent $c_a$ in terms of $p(a)$?
Is for all $b\in\{0,1\}^n$, $c_{b} = \sum_{a\in\{0,1\}^n}(-1)^{<b,a>}p(a)$?