Zak transform

Summary

The Zak transform turns a function on $\R$ into a function of two variables by periodizing it with phase factors. It is a unitary map from $L^2(\R)$ onto $L^2([0, 1)^2)$, and its values are quasi-periodic. It diagonalizes Gabor systems at critical density, which makes it the main tool behind the Balian–Low theorem. It has been discovered several times; Joshua Zak introduced it in solid-state physics.