The Bochner Identity in Harmonic Analysis
Zachary J. Smith · DigitalCommons (California Polytechnic State University) · 2007
In a general sense, harmonic analysis is the study of decomposition and synthesis formulas for functions in terms of eigenfunctions to differential operators. Here the differential operator of interest is the Laplace-Beltrami operator. This differential operator is characterized by its invariance under isometries. In the settings of Euclidean space and real hyperbolic space, the rotation group is an important subgroup of isometries; its natural representation on the L2 Hilbert space of functions will decompose the latter into irreducible invariant subspaces. Bochner's identity arises from studying how the Fourier transform acts on these subspaces. The utility of the Bochner identity will then be demonstrated by considering versions of Hardy's theorem. Hardy's theorem comes from looking at special cases to the Uncertainty principle from quantum physics.