Bochner identities for Fourier transforms

Abstract

Let G be a compact Lie group and R an orthogonal representation of G acting on R n {{\mathbf {R}}^n} . For any irreducible unitary representation π \pi of G and vector v in the representation space of π \pi define S ( π , v ) \mathcal {S}(\pi ,v) to be those functions in S ( R n ) \mathcal {S}({{\mathbf {R}}^n}) which transform (under the action R ) according to the vector v . The Fourier transform F \mathcal {F} preserves the class S ( π , v ) \mathcal {S}(\pi ,v) . A Bochner identity asserts that for different choices of G, R , π , v \pi ,v the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R , π , v \pi ,v and G ′ , R ′ , π ′ , v ′ G’,R’,\pi ’,v’ and a map T : S ( π , v ) → S ( π ′ , v ′ ) T:\mathcal {S}(\pi ,v) \to \mathcal {S}(\pi ’,v’) which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity F ′ T f = c T F f \mathcal {F}’Tf = cT\mathcal {F}f for all f ∈ S ( π , v ) f \in \mathcal {S}(\pi ,v) holds if and only if Δ ′ T f = c 1 T Δ f \Delta ’Tf = {c_1}T\Delta f for all f ∈ S ( π , v ) f \in \mathcal {S}(\pi ,v) . From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.