  We prove the existence as well as regularity of a finite range decomposition
for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} +
m^{2})^{-1} (x-y) $, for $0<\alpha<2$ and all real $m$, in the lattice
${\mathbf Z}^{d}$ as well as in the continuum ${\mathbf R}^{d}$ for dimension
$d\ge 2$. This resolvent occurs as the covariance of the Gaussian measure
underlying weakly self- avoiding walks with long range jumps (stable L\'evy
walks) as well as continuous spin ferromagnets with long range interactions in
the long wavelength or field theoretic approximation. The finite range
decomposition should be useful for the rigorous analysis of both critical and
off-critical renormalisation group trajectories. The decomposition for the
special case $m=0$ was known and used earlier in the renormalisation group
analysis of critical trajectories for the above models below the critical
dimension $d_c =2\alpha$. This revised version makes some changes, adds new
material, and also corrects some errors in the previous version. It refers to
the author's published article with the same title in J Stat Phys (2016) 163:
1235-1246, as well as to an erratum to be published in J Stat Phys.
