  In this paper we characterize the discrete H\"older spaces by means of the
heat and Poisson semigroups associated to the discrete Laplacian. These
characterizations allow us to get regularity properties of fractional powers of
the discrete Laplacian and the Bessel potentials along these spaces and also in
the discrete Zygmund spaces in a more direct way than using the pointwise
definition of the spaces.
  To obtain our results, it has been crucial to get boundedness properties of
the heat and Poisson kernels and their derivatives in both space and time
variables. We believe that these estimates are also of independent interest.
