Higher Fourier interpolation on the plane

Abstract

Let $l\ge 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $\mu (\tau )d\tau$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where $\Im (\tau ),\Im (-\frac{1}{\tau })>\sin \left(\frac{\pi }{l}\right).$ Let $f(x)=\int e^{i\pi \tau |x{|}^2}d\mu (\tau )$ and $\mathcal{F}(f)$ be its Fourier transform, where $x\in {\mathbb{R}}^2.$ For every integer $k\ge 0$ and $x\in {\mathbb{R}}^2,$ we express $f(x)$ in terms of the values of $\frac{d^{k}f}{d{u}^k}$ and $\frac{d^k\mathcal{F}(f)}{d{u}^k}$ at $u=\frac{2n}{\lambda },$ where $n$ is a non-negative integer, $u=|x{|}^2$ and $\lambda =2\cos \left(\frac{\pi }{l}\right).$ We show that the condition $\Im (\tau ),\Im (-\frac{1}{\tau })>\sin \left(\frac{\pi }{l}\right)$ is optimal. We also identify the summation formulas among the values of $\frac{d^{k}f}{d{u}^k}$ and $\frac{d^k\mathcal{F}(f)}{d{u}^k}$ at $u=\frac{2n}{\lambda },$ with the space of holomorphic modular forms of weight $2k+1$ of the Hecke triangle group $(2,l,\infty )$. Using our formulas for $l=6$ and developing new methods, we prove a conjecture of Cohn, Kumar, Miller, Radchenko and Viazovska~\cite[Conjecture 7.5]{Maryna3}. This conjecture was motivated by the universal optimality of the hexagonal lattice.

We corrected a few typos. Comments are welcome!