  Fourier multiplier analysis is developed for nonlocal peridynamic-type
Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The
Fourier multipliers are given through an integral representation. We show that
the integral representation of the Fourier multipliers is recognized explicitly
through a unified and general formula in terms of the hypergeometric function
$_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is
utilized to identify the asymptotic behavior of the Fourier multipliers
$m(\nu)$ as $\|\nu\|\rightarrow \infty$. We show that the multipliers are
bounded when the peridynamic Laplacian has an integrable kernel, and diverge
when the kernel is singular. The bounds and decay rates are presented
explicitly in terms of the dimension $n$, the integral kernel, and the
peridynamic Laplacian nonlocality. The asymptotic analysis is applied in the
periodic setting to prove a regularity result for the peridynamic Poisson
equation and, moreover, show that its solution converges to the solution of the
classical Poisson equation.
