  The Dyson-Ising ferromagnet is a one-dimensional Ising model with a power law
interaction. When the power is between -1 and -2, the model has a phase
transition. Van Enter and Le Ny proved that at sufficiently low temperatures
the decimation renormalization group transformation is not defined in the sense
that the renormalized measure is not a Gibbs measure. We consider a modified
model in which the nearest neighbor couplings are much larger than the other
couplings. For a family of Hamiltonians which includes critical cases, we prove
that the first step of the renormalization group transformation can be
rigorously defined for majority rule and decimation.
