Abstract: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 [Publ. Math. Inst. Hautes Études Sci. 129 (2019), pp. 51–81] 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 Poincaré 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 [Fourier interpolation with zeros of zeta and L-functions, arXiv:2005.02996, 2020]

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中文翻译：

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来自球体的傅立叶插值

摘要：在每个维度 $d \geq 2$ 中，我们给出了一个明确的公式，该公式仅根据其限制和傅立叶变换的限制来表达 $\mathbb {R}^d$ 上任何施瓦茨函数的值，到所有以原点为中心的球体，其半径为整数的平方根。因此，我们概括了 Radchenko 和 Viazovska [Publ. 数学。研究所 Hautes Études Sci. 129 (2019), pp. 51–81] 到更高维度。我们开发了一种通用工具，可以将更高维度中径向函数的傅立叶唯一性和插值结果转换为固定维度中非径向函数的相应结果。在大于或等于 $5$ 的维度中，我们使用与经典庞加莱级数密切相关的构造来解决径向问题。在剩余的小维度中，zeta 和 L 函数为零的傅立叶插值，arXiv:2005.02996, 2020]

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