Often, the Boolean ___domain is taken as <math>\{-1, 1\}</math>, with false ("0") mapping to 1 and true ("1") to -1−1 (see [[Analysis of Boolean functions]]). The polynomial corresponding to <math>g(x): \{-1,1\}^n \rightarrow \{-1,1\}</math> is then given by:<math display="block">g^*(x) = \sum_{a \in {\{-1,1\}}^n} g(a) \prod_{i:a_i=-1} \frac{1-x_i}{2} \prod_{i:a_i=1} \frac{1+x_i}{2}</math>Using the symmetric Boolean ___domain simplifies certain aspects of the [[Analysis of Boolean functions|analysis]], since negation corresponds to multiplying by -1−1 and [[Parity function|linear functions]] are [[monomial]]s (XOR is multiplication). This polynomial form thus corresponds to the ''Walsh transform'' (in this context also known as ''Fourier transform'') of the function (see above). The polynomial also has the same statistical interpretation as the one in the standard Boolean ___domain, except that it now deals with the expected values <math>E(X) = P(X=1) - P(X=-1) \in [-1, 1]</math> (see [[piling-up lemma]] for an example).
