a polynomial representation of boolean functions I came up with this linear transformation to map boolean functions to polynomials and it seems to have some nice properties. I was wondering if there is any reference describing this (and/or similar) mappings and their application. 

For three variables, the transformation matrix is given below: 



I have posted this with some details and a description for reducing 3-SAT to checking the existence of specific term in an ordinary polynomial product on math.stackexchange (here). I have not got much feedback there. I was wondering if someone here could help. 

If this is not the right forum to ask this sort of questions, or more details are needed, please let me know in comments.

EDIT: I found similarities to Zhegalkin polynomials and the $f:\left \{ -1,1 \right \}^n \rightarrow \left \{ -1,1 \right \}$ representation described here with the difference that this is a $f:\left \{ -1,1 \right \}^n \rightarrow \left \{ 0,1 \right \}$ representation.
Well done on your independent discovery.

This is a Hadamard matrix of Sylvester type, written in a different order. There is a massive literature on this topic. It is used in coding theory, cryptography, is directly related to Reed-Muller codes of degree 1, it can be used to obtain best affine approximations of functions, etc.

Ryan O'Donnell's notes on Analysis of Boolean functions available here which are now published as a book, Claude Carlet's chapters on Boolean 
functions here are easily accessible online.

Just use the order $[1,X,Y,XY,Z,ZX,ZY,ZXY]$ which corresponds to lexicographic order over the integers, and apply the corresponding permutation to the columns as well. 

In that order your matrix is simply the 3-fold Kronecker product $$H_2 \otimes H_2\otimes H_2$$of the basic 2 by 2 Hadamard matrix.

$$
H_2=\left(\begin{array}{cc} 1 &amp; 1 \\ 1 &amp; -1 \end{array}
\right),
$$

This paper considers efficient evaluation of Hadamard representation coefficients.