  We study the decimation to a sublattice of half the sites, of the
one-dimensional Dyson-Ising ferromagnet with slowly decaying long-range pair
interactions of the form $\frac{1}{{|i-j|}^{\alpha}}$, in the phase transition
region (1< $\alpha \leq$ 2, and low temperature). We prove non-Gibbsianness of
the decimated measure at low enough temperatures by exhibiting a point of
essential discontinuity for the finite-volume conditional probabilities of
decimated Gibbs measures. Thus result complements previous work proving
conservation of Gibbsianness for fastly decaying potentials ($\alpha$ > 2) and
provides an example of a "standard" non-Gibbsian result in one dimension, in
the vein of similar resuts in higher dimensions for short-range models. We also
discuss how these measures could fit within a generalized (almost vs. weak)
Gibbsian framework. Moreover we comment on the possibility of similar results
for some other transformations.
