Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces
We establish sharp sampling and interpolation criteria for the radial--angular Gaussian mixed-norm Fock spaces $\mathcal F_α^{p,q}$, where $α>0$ and $0 \fracαπ$, together with relative separation when $p<\infty$; no relative-separation condition is required when $p=\infty$. Here $D_{\mathrm{sep}}^{-}$ is the supremum of the lower Beurling densities over separated subsets of $Λ$. Interpolation is equivalent to separation and $D^{+}(Λ)<\fracαπ$. The same criterion characterizes interpolation for the little endpoint $f_α^{p,\infty}$, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on $\ell^2$. The proof first passes to $\ell^\infty$ by translated polynomial cutoffs and a commutator estimate, and then uses the $p$-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.