The Radchenko–Viazovska question on Fourier interpolation

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VibeMathed record: The Radchenko–Viazovska question on Fourier interpolation evidence for this event

The Radchenko–Viazovska question on Fourier interpolation evidence for this event

This event attributed to Kristian Seip

This event attributed to ChatGPT (OpenAI, model version unstated)

The Radchenko–Viazovska question on Fourier interpolation parent of this event

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. parent of this event

This event attributed to Andriy Bondarenko