A nonlocal Laplacian in one dimension

Abstract

An overview on nonlocal Laplace operators acting on real-valued one-dimensional functions is presented. We provide a definition for nonlocal Laplace operators and present some basic examples. In addition, we show that, for vanishing nonlocality, the nonlocal Laplacian of a sufficiently differentiable one-dimensional function approaches the second derivative of the function. Moreover, we compute the Fourier multipliers of the nonlocal Laplacian and show that these multipliers converge to the multipliers of the Laplacian in the limit of vanishing nonlocality. Furthermore, we consider a nonlocal diffusion equation and provide an integral representation for its solution in terms of the Fourier multipliers of the nonlocal Laplacian.