  In his classical paper, L. Schwartz proved that on the real line, in every
linear translation invariant space of continuous complex valued functions,
which is closed under compact convergence the exponential monomials span a
dense subspace. He studied so-called local ideals in the space of Fourier
transforms, and his proof based on the observation that, on the one hand, these
local ideals are completely determined by the exponential monomials in the
space, and, on the other hand, these local ideals completely determine the
space itself. D.I.Gurevich used a similar idea of localization to give
counterexamples for Schwartz's theorem in higher dimension. This localization
process depends on differential operators and differentiability properties of
the Fourier transforms. We note that in his two papers R.J. Elliott used
somewhat similar ideas, but unfortunately, some of his proofs and results are
incorrect. In this paper we show that the ideas of localization can be extended
to general locally compact Abelian groups using abstract derivations on the
Fourier algebra of compactly supported measures. Based this method we present
necessary and sufficient conditions for spectral synthesis for varieties on
locally compact Abelian groups.
