MR3428804 Reviewed Benci, Vieri(I-PISA); Luperi Baglini, Lorenzo(A-WIENM) Generalized functions beyond distributions. (English, Arabic summary) Arab. J. Math. (Springer) 4 (2015), no. 4, 231–253. 26E30 (46F30)

Generalized functions are here intended to mean a particular class V˜(Ω) of ultrafunctions, i.e. functions defined on a non-Archimedean field that extends a class V(Ω) of L2(Ω)-integrable continuous functions. The particular class of ultrafunctions is selected by some desiderata to obtain the properties sufficient for applications to PDE. One of these is to maintain the locality property of a local operator defined on V(Ω) when it is extended to V˜(Ω). This fact is related to the possibility of defining a sort of orthogonality between "Delta ultrafunctions''. The results are applied to the definition of the derivative operator, the definite integral and to associate an ultrafunction to every C−∞(R) distribution.