Spherical Fourier transform - Imported Knowlpedia Content

Definition

Let $G$ be a connected semisimple Lie group with finite center, let $K$ be a maximal compact subgroup, and fix a left Haar measure on $G$ . For an integrable $K$ -bi-invariant function $f$ , its spherical Fourier transform is

\mathcal Sf(\lambda)=\int_G f(g)\varphi_{-\lambda}(g)\,dg,

where $\varphi_\lambda$ is the normalized elementary spherical function with spectral parameter $\lambda$ in the complexified dual of a maximal abelian subspace of the noncompact Cartan component. The transform is Weyl-invariant in $\lambda$ . The sign on $\lambda$ is conventional and is chosen to match the stated inversion convention.

Diagonalizing spherical convolution

For $K$ -bi-invariant integrable functions $f$ and $h$ ,

\mathcal S(f*h)(\lambda) =\mathcal Sf(\lambda)\mathcal Sh(\lambda).

Thus the spherical transform is the scalar Fourier transform of the commutative convolution algebra associated with the Gelfand pair $(G,K)$ . The multiplicative property follows from the product formula for spherical functions Helgason, Chapter IV, §§2–3.

Inversion and Plancherel measure

On suitable test functions, $f$ is recovered by integrating $\mathcal Sf(\lambda)\varphi_\lambda$ against the spherical Plancherel measure. For the standard real parameter space, that measure has density proportional to $\lvert c(\lambda)\rvert^{-2}$ , where $c$ is the Harish-Chandra $c$ -function. This is the radial, commutative part of the nonabelian Plancherel formula.

Schwartz-space theorem

On the $K$ -bi-invariant Harish-Chandra Schwartz space, the transform is a topological algebra isomorphism onto an explicitly described Weyl-invariant Schwartz space. The Trombi–Varadarajan theorem gives the corresponding $L^p$ -Schwartz versions and their holomorphic spectral domains Trombi–Varadarajan, main theorem.

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, 2000. AMS record. Relevant: Chapter IV on spherical functions, spherical transforms, inversion, and Plancherel theory.
P. C. Trombi and V. S. Varadarajan, “Spherical Transforms on Semisimple Lie Groups,” Annals of Mathematics 94 (1971), 246–303. DOI record. Relevant: the main Schwartz-space isomorphism theorem.