Higher Fourier interpolation on the plane

Abstract: Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Let $f(x)=\int e^{i\pi \tau |x|^2}d\mu(\tau)$ and $\mathcal{F}(f)$ be the Fourier transform of $f$, where $x\in \R^2$ and $\mu$ is a measure with bounded variation and supported on a compact subset of $\tau \in\CC$, where $\Im(\tau),\Im(-\frac{1}{\tau})>\sin(\frac{\pi}{l}).$ For every integer $k\geq 0$ and $x\in \R^2,$

We express $f(x)$ by the values of $\frac{d^k f}{du^k}$ and $\frac{d^k \mathcal{F}f}{du^k}$  at $u=\frac{2n}{\lambda},$ where $u=|x|^2$ and $\lambda=2\cos(\frac{\pi}{l}).$ We show that the condition $\Im(\tau),\Im(-\frac{1}{\tau})>\sin(\frac{\pi}{l})$ is optimal.

We also identify the cokernel to these values with a specific space of holomorphic modular forms of weight $2k+1$ associated to the Hecke triangle group $(2,l,\infty)$. Using our explicit formulas for $l=6$ and developing new methods, we prove a conjecture of Cohn, Kumar, Miller, Radchenko and Viazovska~\cite[Conjecture 7.5]{Maryna3} motivated by the universal optimality of the hexagonal lattice.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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