Fourier interpolation from spheres

富克斯群的上同调与傅里叶插值

Cohomology of Fuchsian groups and Fourier interpolation

本文从富克斯群在韦伊表示系数下第一上同调的消失结果出发，为Radchenko-Viazovska首次证明的傅里叶插值结果提供了一种新的证明方法。

扰动插值公式及其应用

Perturbed interpolation formulae and applications

该研究运用泛函分析技术证明傅里叶分析中部分经典及新近插值结果可被扰动，推广了Paley-Wiener空间的Kadec 1/4定理，得到多个高维傅里叶插值相关结果的扰动结论，并给出若干相关应用。

从球堆积到傅里叶插值

From sphere packing to Fourier interpolation

本文探讨维亚佐夫斯卡用模形式构造特殊函数解决八维球堆积问题，进而引出傅里叶分析中拉琴科与维亚佐夫斯卡的插值定理。

Fourier non-uniqueness sets from totally real number fields

Let $K$ be a totally real number field of degree $n \geq 2$. The inverse different of $K$ gives rise to a lattice in $\mathbb{R}^n$. We prove that the space of Schwartz Fourier eigenfunctions on $\mathbb{R}^n$ which vanish on the "component-wise square root" of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres $\sqrt{m}S^{n-1}$ for integers $m \geq 0$ and, as $m \rightarrow \infty$, there are $\sim c_{K} m^{n-1}$ many points on the $m$-th sphere for some explicit constant $c_{K}$, proportional to the square root…

Polynomial interpolation of modular forms for Hecke groups

Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F = \{f_m\}_{m = 3, 4, ...}$ of modular forms for Hecke groups $G(λ_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient in the Fourier expansion of $f_m$. We express the $P_{n,F}$ in terms of the Fourier expansions of well-known Hauptmoduln, or in terms of certain divisor-sums. By studying the complex roots of the $P_n$, we relate them to Lehmer's question about Ramanujan's tau function. We review the theory of triangle functions and Hecke's theory of modular forms in order to esta…

Fourier uniqueness in $\mathbb{R}^4$

We show an interrelation between the uniqueness aspect of the recent Fourier interpolation formula of Radchenko and Viazovska and the Heisenberg uniqueness study for the Klein-Gordon equation and the lattice-cross of critical density, studied by Hedenmalm and Montes-Rodriguez. This has been known since 2017.

Higher Fourier interpolation on the plane

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…

A Note on Fourier Eigenfunctions in Four Dimensions

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proo…