Fourier-invariant functions with dense zero sets

For every $0\leqβ\leq1/2$, we construct a nonzero real-valued continuous function $f_β$ in $L^1(\mathbb R)\cap L^2(\mathbb R)$ such that $\widehat {f}_β=f_β$ and $f_β(\sqrt{n}/[\log(e+n)]^β)=0$ for all $n\geq 0$. The case $β=0$ settles in the negative a question raised by Radchenko and Viazovska regarding their Fourier interpolation formula. The construction uses a scale of reproducing kernel Hilbert spaces generated by the Fourier-invariant Hermite functions. Applying the Mehler formula, we identify the reproducing kernels of these spaces. By suitable estimates of these kernels, we show that $(\sqrt{n}/[\log(e+n)]^β)$, with one auxiliary point added to it, is a universal interpolating sequence for at least one of the Hilbert spaces under consideration. However, this result fails when $β>1/2$.