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 . For any irreducible unitary representation of *G* and vector *v* in the representation space of define to be those functions in which transform (under the action *R*) according to the vector *v*. The Fourier transform preserves the class . A Bochner identity asserts that for different choices of *G, R*, the Fourier transform is the same (up to a constant multiple). It is proved here that for *G, R*, and and a map which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity for all holds if and only if for all . 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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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1977-0433147-6

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