Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

中文翻译：

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具有指定零的傅里叶变换的特征函数

具有规定零的傅里叶变换的特征函数在证明乙8Cohn、Kumar、Miller、Radchenko 和 Viazovska 在各自的尺寸 8 和 24 中给出了最好的球体填充和 Leech 晶格。用于线性规划参数的函数被构造为某些模和准模形式的拉普拉斯变换。Cohn 和 Gonçalves 使用类似的构造来找到满足 12 维中最优不确定性原理的函数。本文对这些构造给出了统一的看法，并开发了一种机制来找到所有维度中可被 4 整除的基本形式。此外，正性讨论了在这种情况下出现的准模形式的傅立叶系数。