For $U\subseteq I$, average a function $F$ over the coordinates outside $U$, retaining the input on $U$. The resulting function has Fourier coefficient $\widehat F(S)$ when $S\subseteq U$, and zero otherwise. This is Lemma 4.18 of Håstad's paper: take $I$ to be the set of possible encoded words, and $U$ to be the words satisfying the side condition.

The definition averages over a full independent cube; the unused coordinates give equal multiplicities, hence the same uniform average as over just the coordinates outside $U$. Averaging preserves the bound $|F|\leq1$. At an input whose entire averaging fiber has one value, that value is preserved.