  The aim of this paper is to deduce a discrete version of the fractional
Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear
chain of finite length.We obtain explicit expressions for this fractional
Laplacianmatrix and deduce also its periodic continuum limit kernel. The
continuum limit kernel gives an exact expression for the fractional Laplacian
(Riesz fractional derivative) on the finite periodic string.In this approach we
introduce two material parameters, the particle mass $\mu$ anda frequency
$\Omega\_{\alpha}$. The requirement of finiteness of the the total mass and
total elastic energy in the continuum limit (lattice constant $h\rightarrow 0$)
leads to scaling relations for the two parameters, namely$\mu \sim h$ and
$\Omega\_{\alpha}^2\sim h^{-\alpha}$.The present approach can be generalized to
define lattice fractional calculus on periodic lattices in full analogy to the
usual `continuous' fractional calculus.
