General Inversion Theorems

Abstract

The Fourier inversion formula is a standard fact of elementary analysis. Harish-Chandra developed an inversion for K-bi-invariant functions on a semisimple Lie group G, in other words he developed the theory of a spherical transform [Har 58a], [Har 58b], which is an integral transform, with a kernel called the spherical kernel. There are variations to the setting of this transform, involving various factors. Harish-Chandra’s general Plancherel inversion in the non-K-bi-invariant case is a lot more complicated, and highly non-abelian. The K-bi-invariant case turns out to be abelian. However, in this case, one can look at the inversion on various function spaces, ranging over C ^∞ _c , C _c , Schwartz space, L ¹ , L ², ad lib. Harish-Chandra dealt fundamentally with the Schwartz space, which he defined in the context of semisimple Lie groups, and L ². Helgason pointed to the correspondence between C ^∞ _c and the Paley-Wiener space [Hel 66], complemented by Gangolli [Gan 71]. Then Rosenberg [Ros 77] gave a much simpler version of some parts of Harish inversion, using some lemmas from Helgason and Gangolli. Here we give this Rosenberg material, suitably axiomatized in a way which gives rise to a setting for pure inversion theorems in ordinary euclidean space, essentially along the lines of the similar elementary and standard theory of Fourier inversion. Roughly speaking, the situation is as follows.