论文标题
球体的傅立叶插值
Fourier interpolation from spheres
论文作者
论文摘要
在每个尺寸$ d \ geq 2 $中，我们给出一个明确的公式，该公式仅在$ \ mathbb {r}^d $上表达任何schwartz函数的值，仅在其限制方面以及其傅立叶变换的限制，以及所有以原点为中心的球的限制，它们的radius是integer的正方形根。因此，我们将radchenko和Viazovska的插值定理推广到更高的维度。我们开发了一种通用工具，用于在较高维度中转化径向函数的傅立叶唯一性和插值结果，以在固定维度中的非radial功能的相应结果。在更大或等于5的维度中，我们使用与经典庞康科（Poincare）系列密切相关的构造解决了径向问题。在其余的小维度中，我们将该技术与radchenko-viazovska公式的直接概括结合到了较高的径向函数，我们从Bondarenko，Radchenko和Seip的一般结果中推断出这种技术。
In every dimension $d \geq 2$, we give an explicit formula that expresses the values of any Schwartz function on $\mathbb{R}^d$ only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square root of an integer. We thus generalize an interpolation theorem by Radchenko and Viazovska to higher dimensions. We develop a general tool to translate Fourier uniqueness- and interpolation results for radial functions in higher dimensions, to corresponding results for non-radial functions in a fixed dimension. In dimensions greater or equal to 5, we solve the radial problem using a construction closely related to classical Poincare series. In the remaining small dimensions, we combine this technique with a direct generalization of the Radchenko--Viazovska formula to higher-dimensional radial functions, which we deduce from general results by Bondarenko, Radchenko and Seip.
