The Radchenko–Viazovska question on Fourier interpolation

What was actually shown

For every $0\le \beta\le \tfrac12$, Bondarenko and Seip construct a nonzero real-valued continuous even function$$f_\beta\in L^1(\mathbb R)\cap L^2(\mathbb R)$$such that$$\widehat f_\beta=f_\beta$$and$$f_\beta\!\left(\frac{\sqrt n}{[\log(e+n)]^\beta}\right)=0 \qquad(n\ge0).$$They normalize the construction by requiring $f_\beta(1/2)=1$, so the function is genuinely nontrivial.

For $\beta=0$, this gives a nonzero Fourier-invariant function vanishing at every $\sqrt n$, which answers the question negatively: their interpolation formula for even Schwartz functions does not extend merely under the assumption that the interpolation series is well-defined and absolutely convergent.

More strongly, for every $0<\beta\le1/2$ the zero set can be logarithmically denser than the square-root sequence. These sampling points, together with $1/2$, form a universal interpolating sequence for a suitable reproducing-kernel Hilbert space of Fourier-invariant Hermite expansions.