The Radchenko–Viazovska question on Fourier interpolation

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.

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.