  We consider SU(2) gauge potentials over a space with a compactified
dimension. A non-Abelian Fourier transform of the gauge potential in the
compactified dimension is defined in such a way that the Fourier coefficients
are (almost) gauge invariant. The functional measure and the gauge field
strengths are expressed in terms of these Fourier coefficients. The emerging
formulation of the non-Abelian gauge theory turns out to be an Abelian gauge
theory of a set of fields defined over the initial space with the compactified
dimension excluded. The Abelian theory contains an Abelian gauge field, a
scalar field, and an infinite tower of vector matter fields, some of which
carry Abelian charges. Possible applications of this formalism are discussed
briefly.
