  We consider renormalization group (RG) transformations for classical
Ising-type lattice spin systems in the infinite volume limit. Formally, the RG
maps a Hamiltonian H into a renormalized Hamiltonian H':
exp(-H'(\sigma'))=\sum_\sigma T(\sigma, \sigma')exp(-H(\sigma)), where
T(\sigma, \sigma') denotes a specific RG probability kernel, \sum_\sigma'
T(\sigma, \sigma')=1, for every configuration \sigma. With the help of the
Dobrushin uniqueness condition and standard results on the polymer expansion,
Haller and Kennedy gave a sufficient condition for the existence of the
renormalized Hamiltonian in a neighborhood of the critical point. By a more
complicated but reasonably straightforward application of the cluster expansion
machinery, the present investigation shows that their condition would further
imply a band structure on the matrix of partial derivatives of the renormalized
interaction with respect to the original interaction. This in turn gives an
upper bound for the RG linearization.
