Linear Theory of Colombeau Generalized Functions
Results from the now-classical distribution theory involving convolution and Fourier transformation are extended to cater for Colombeau's generalized functions. Indications are given how these particular generalized functions can be used to investigate linear equations and pseudo differential operators. Furthermore, applications are also given to problems with nonregular data.
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algebras amplitudes analysis applications assertion associated sense bounded calculus Chipot coefficients Colombeau generalized functions compact set compact support construction continuous function convolution corresponding defined Definition Denote differential operator Dirichlet problem distribution sense element elliptic equal exists TV finite Fourier transformation given holds hypoelliptic ออ implies integration Lemma Let G microlocally neighbourhood Nonlinear notation Note NQ there exists null obtain open set partial differential equations partition of unity polynomials problems properly supported properties pseudodifferential operator quotient space resp Section seminorms sharp topology sheaf simplified model singular singular support smoothing special unit subset supp suppG symbol tempered generalized functions Theorem 2.4 unit nets vector spaces wave front set zero АО ст