The frame set of the first Hermite function

Abstract

We determine the frame set of the first Hermite function, proving a conjecture of Lyubarskii and Nes. We use a characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler to transform the problem into a question about the existence of a nonzero Gaussian shift-invariant entire function $F$, with bounded coefficients, whose derivative vanishes on a lattice with spacing $δ=ab$. By taking the Wronskian of $N$ translates of $F$, we amplify its critical points to zeros of multiplicity at least $N-1$. After rescaling, this Wronskian is again a Gaussian shift-invariant function. A Gaussian zero-density theorem then gives $(N-1)/(Nδ)\leq1$. Varying $N$ over the range allowed by the construction forces every subcritical lattice product $δ=ab$ for which the system is not a frame to equal $(q-1)/q$ for some integer $q\geq2$.