麻豆AV

Title:聽Higher Fourier interpolation on the plane

Abstract:聽聽Let $lgeq 6$ be any integer, where $lequiv 2$ mod $4$. Let $f(x)=int e^{ipi au |x|^2}dmu( au)$ and $mathcal{F}(f)$ be the Fourier transform of $f$, where $xin R^2$ and $mu$ is a measure with bounded variation and supported on a compact subset of $ au inCC$, where $Im( au),Im(-rac{1}{ au})>sin(rac{pi}{l}).$ For every integer $kgeq 0$ and $xin R^2,$

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

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