Let be a function on -Biased Cube with and .
Define by where and .
The averaging projection is defined by
where .
Note depends only on coordinates in .
Lemma
Each is self-Adjoint.
Proof
so we are done by symetry.
Lemma
Each is an orthogonal projection and if then
Proof
Since we have .
To prove orthogonality we need
But this is clear as is self-Adjoint.
Now let and let .
We will take , and and .