Higher Fourier interpolation on the plane

Let $l\geq 6$ 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 $x\in \R^2.$ For every integer $k\geq 0$ and $x\in \R^2,$ we express $f(x)$ in terms of the values of $\frac{d^k f}{du^k}$ and $\frac{d^k \mathcal{F}(f)}{du^k}$ 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 $\frac{d^k f}{du^k}$ and $\frac{d^k \mathcal{F}(f)}{du^k}$ 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.