This article aims to transform the infinite-order Lagrangian density for ghost-free infinite-derivative linearized gravity into non-local form. To achieve it, we use the theory of generalized functions and the Fourier transform in the space of tempered distributions S′ . We show that the non-local operator domain is not defined on the whole functional space but on a subset of it. Moreover, we prove that these functions and their derivatives are bounded in all R3 and, consequently, the Riemann tensor is regular and the scalar curvature invariants do not present any spacetime singularity. Finally, we explore what conditions we need to satisfy so that the solutions of the linearized equations of motion exist in S′ .

中文翻译：

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卷积形式的无限导数线性化引力

本文旨在将无重影无限导数线性引力的无限阶拉格朗日密度转换为非局部形式。为了实现它，我们使用广义函数理论和回火分布空间中的傅里叶变换 小号‘ . 我们表明非本地运算符域不是在整个功能空间上定义的，而是在它的一个子集上定义的。此外，我们证明这些函数及其导数在所有方面都是有界的 R3个 因此，黎曼张量是规则的，标量曲率不变量不存在任何时空奇点。最后，我们探索需要满足哪些条件才能使线性化运动方程的解存在于 小号‘ .