  In this paper we make an attempt to extend L. Schwartz's classical result on
spectral synthesis to several dimensions. Due to counterexamples of D. I.
Gurevich this is impossible for translation invariant varieties. Our idea is to
replace translations by proper euclidean motions in higher dimensions. For this
purpose we introduce the basic concepts of spectral analysis and synthesis in
the non-commutative setting based on Gelfand pairs, where "translation
invariance" will be replaced by invariance with respect to a compact group of
automorphisms. The role of exponential functions will be played by spherical
functions. As an application we obtain the extension of L. Schwartz's
fundamental result.
