Fourier-invariant functions with dense zero sets

Abstract

For every $0\le \beta \le 1/2$, we construct a nonzero real-valued continuous function $f_{\beta }$ in $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ such that ${\hat{f}}_{\beta }=f_{\beta }$ and $f_{\beta }(\sqrt{n}/[\log (e+n){]}^{\beta })=0$ for all $n\ge 0$. The case $\beta =0$ 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 $(\sqrt{n}/[\log (e+n){]}^{\beta })$, 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 $\beta >1/2$.