The classical theory of the Fourier–Stieltjes transform operates primarily within the algebra of bounded complex Borel measures  and its dual . However, analyzing singular, non-compactly supported, or ultra-singular measures requires embedding this transform into broader topological vector spaces of generalized functions. In this paper, we develop a rigorous distributional analysis of the Fourier–Stieltjes transform across dual topological spaces, focusing on the tempered distributions , compactly supported distributions , and ultra-distributional Gel'fand–Shilov dual spaces . We establish precise characterizations of the range of  under the Fourier map within these dual structures. Furthermore, by introducing a regularized duality pairing based on functional-analytic multipliers, we prove unified Sobolev bounds, structural decomposition theorems, and topological embedding properties that govern the action of the Fourier–Stieltjes transform on singular distribution spaces.

Keywords: Fourier–Stieltjes transform, Schwartz distribution spaces, Tempered distributions, Gel'fand–Shilov spaces, Paley–Wiener–Schwartz theorem, Dual-space duality