  We consider a one-dimensional lattice system of unbounded, real-valued spins
with arbitrary strong, quadratic, finite-range interaction. We show the
equivalence of the grand canonical ensemble (gce) and the canonical ensemble
(ce), in the sense of observables and correlations. A direct consequence is
that the correlations of the ce decay exponentially plus a volume correction
term. The volume correction term is uniform in the external field, the mean
spin and scales optimally in the system size. This extends prior results of
Cancrini & Martinelli for bounded discrete spins to unbounded continuous spins.
The result is obtained by adapting Cancrini & Martinelli's method combined with
authors' recent approach on continuous real-valued spin systems.
