| Abstract: | If  is a foliation of an open set  by smooth  -dimensional surfaces, we define a class of functions  , supported in  , that are, roughly speaking, smooth along  and of bounded variation transverse to  . We investigate geometrical conditions on  that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable. |
