  We study the problem of embedding arbitrary $\mathbb{Z}^k$-actions into the
shift action on the infinite dimensional cube
$\left([0,1]^D\right)^{\mathbb{Z}^k}$. We prove that if a $\mathbb{Z}^k$-action
satisfies the marker property (in particular if it is a minimal system without
periodic points) and if its mean dimension is smaller than $D/2$ then we can
embed it in the shift on $\left([0,1]^D\right)^{\mathbb{Z}^k}$. The value $D/2$
here is optimal. The proof goes through signal analysis. We develop the theory
of encoding $\mathbb{Z}^k$-actions into band-limited signals and apply it to
proving the above statement. Main technical difficulties come from higher
dimensional phenomena in signal analysis. We overcome them by exploring
analytic techniques tailored to our dynamical settings. The most important new
idea is to encode the information of a tiling of the Euclidean space into a
band-limited function which is constructed from another tiling.
