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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Bochner identities for Fourier transforms

Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 228 (1977), 307-327
MSC: Primary 43A30; Secondary 22E45
MathSciNet review: 0433147
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Abstract: Let G be a compact Lie group and R an orthogonal representation of G acting on $ {{\mathbf{R}}^n}$. For any irreducible unitary representation $ \pi $ of G and vector v in the representation space of $ \pi $ define $ \mathcal{S}(\pi ,v)$ to be those functions in $ \mathcal{S}({{\mathbf{R}}^n})$ which transform (under the action R) according to the vector v. The Fourier transform $ \mathcal{F}$ preserves the class $ \mathcal{S}(\pi ,v)$. A Bochner identity asserts that for different choices of G, R, $ \pi ,v$ the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R, $ \pi ,v$ and $ G',R',\pi ',v'$ and a map $ 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 $ \mathcal{F}'Tf = cT\mathcal{F}f$ for all $ f \in \mathcal{S}(\pi ,v)$ holds if and only if $ \Delta 'Tf = {c_1}T\Delta f$ for all $ 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.

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Keywords: Bochner identity, Fourier transform, representations of compact Lie groups, spherical harmonics, adjoint representation, Stiefel harmonics, classical groups
Article copyright: © Copyright 1977 American Mathematical Society

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