Zak transform

Remarks

For fixed $x$, $Zf(x, \cdot)$ is a Fourier series with coefficients $f(x + k)$. The map $Z$ extends to a unitary operator from $L^2(\R)$ onto $L^2([0, 1)^2)$. The values are quasi-periodic: $Zf(x, \omega + 1) = Zf(x, \omega)$ and $Zf(x + 1, \omega) = e^{2\pi i \omega} Zf(x, \omega)$. The second rule follows by putting $j = k + 1$ in the series. Some authors include a lattice parameter $a > 0$ and use $f(x + ak)$.

Example: let $f = 1_{[0, 1)}$. For $x \in [0, 1)$ only the term $k = 0$ is nonzero, so $Zf(x, \omega) = 1$ on $[0, 1)^2$. Then $\norm{Zf}^2_{L^2([0,1)^2)} = 1 = \norm{f}_2^2$, as unitarity requires.

Non-example: the map $f \mapsto \sum_{k \in \Z} f(x + k)$ without phase factors is the periodization of $f$. It loses information, for instance it sends $1_{[0, 1/2)} - 1_{[1, 3/2)}$ to $0$.