transform becomes more regular (in terms of growth at infinity) when f is truncated with the smooth cutoff . The most difficult "problem", from a theoretical point of view, is finding the adequate sheaf O characterizing functions belonging to a given subsheaf E of the space G of generalized functions. Example If we take G = D′ the space of Schwartz distributions and want to characterize distributions which are locally functions, we must take for O(Ω) the classical function spaces called O′M(Ω) in the literature. Then the projection on the first component of a distribution's wave front set is nothing else than its classical singular support, i.e. the complement of the set on which its restriction would be a smooth function. Applications The wave front set is useful, among others, when studying propagation of singularities by pseudodifferential operators. See also FBI transform Singular spectrum Essential support References Lars Hörmander, Fourier integral operators I, Acta Math. 127 (1971), pp. 79–183. Chapter VIII, Spectral Analysis of Singularities Mathematical analysis Generalized functions