General Inversion Theorems
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, L1, L2, ad lib. Harish-Chandra dealt fundamentally with the Schwartz space, which he defined in the context of semisimple Lie groups, and L2. 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.
KeywordsPolar Decomposition Chapter VIII Schwartz Space Fourier Inversion Iwasawa Decomposition
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