平面上的高阶Fourier插值
Higher Fourier interpolation on the plane
摘要：
令$l\geq 6$为任意整数，其中$l\equiv 2$ mod $4$。假设$μ(τ)dτ$是具有有界变差的测度并且在复平面的紧致子集上得到支持，其中$\Im(τ),\Im(-\frac{1}τ)>\sin\left(\fracπ{l}\right).$令$f(x)=\int e^{iπτ|x|^2}dμ(τ)$和$\mathcal{F}(f)$是其傅立叶变换，其中$x\in \R^2.$对于每个整数$k\geq 0$和$x\in \R^2,$，我们用$\frac{d^k f}{du^k}$和$\frac{d^k \mathcal{F}(f)}{du^k}$在$u=\frac{2n}λ,$的值来表示$f(x)$，其中$n$是非负整数，我们证明了条件$λ=2\cos\left(\fracπ{l}\right).$是最优的。利用Hecke三角形群$(2,l,\infty)$的权$2k+1$的全纯模形式空间，我们还确定了$\frac{d^k f}{du^k}$和$\frac{d^k \mathcal{F}(f)}{du^k}$在$u=\frac{2n}λ,$的值之间的求和公式.利用我们的$l=6$公式和发展新的方法，我们证明了Cohn，Kumar，Miller，Radchenko和Viazovska的一个猜想~ \cite [猜想7.5]{Maryna 3}.这个猜想是由六角晶格的普适最优性激发的。
英文摘要：
Let $l\geq 6$ be any integer, where $l\equiv 2$ mod $4$. Suppose that $μ(τ)dτ$ is a measure with bounded variation and is supported on a compact subset of the complex plane, where $\Im(τ),\Im(-\frac{1}τ)>\sin\left(\fracπ{l}\right).$ Let $f(x)=\int e^{iπτ|x|^2}dμ(τ)$ 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 $u=\frac{2n}λ,$ where $n$ is a non-negative integer, $u=|x|^2$ and $λ=2\cos\left(\fracπ{l}\right).$ We show that the condition $\Im(τ),\Im(-\frac{1}τ)>\sin\left(\fracπ{l}\right)$ 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 $u=\frac{2n}λ,$ 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.
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