The Fourier–Stieltjes transform extends classical Fourier analysis to bounded complex Borel measures on . While inversion formulas for standard  functions are well established via pointwise or mean convergence, reconstructing a general measure  from its Fourier–Stieltjes transform  presents fundamental measure-theoretic challenges, particularly at points or hypersurfaces of discontinuity. In this paper, we establish a generalized multidimensional inversion formula for the Fourier–Stieltjes transform applicable to arbitrary bounded Borel measures on . By introducing a parameterized family of regularizing kernels based on Gaussian and spherical summation operators, we construct a limit process that recovers the measure value on -dimensional intervals (hyper-rectangles) whose boundary has zero -measure. Furthermore, we extend this framework to tempered distributions, showing that our formula unifies the singular, absolutely continuous, and discrete components of  under a singular integral limit. The structural properties of the inversion operator are rigorously analyzed, yielding explicit bounds and convergence rates in the weak-* topology.

Keywords: Fourier–Stieltjes transform, Multidimensional inversion formula, Bounded variation measures, Schwartz distributions, Harmonic analysis, Bochner's theorem