  We prove that functions with compact support in non-quasianalytic classes of
Roumieu-type and of Beurling-type defined by a weight matrix with some mild
regularity conditions can be characterized by the decay properties of their
Fourier transform. For this we introduce the abstract technique of constructing
from the original matrix multi-index matrices and associated function spaces.
We study the behaviour of this construction in detail and characterize its
stability. Moreover non-quasianalyticity of the classes is characterized.
