  This article aims to transform the infinite-order Lagrangian density for
ghost-free infinite-derivative linearized gravity into non-local. To achieve
it, we use the theory of generalized functions and the Fourier transform in the
space of tempered distributions $\mathcal{S}^\prime$. 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 $\mathbb{R}^3$ 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 $\mathcal{S}^\prime$.
