The aim of this chapter is to prove Hardy’s theorem for the Euclidean Fourier transform. We define the Fourier transform on ℝ n and study some of its important properties. We introduce spherical harmonics via representation theory of the orthogonal group and give a group theoretic proof of the Hecke-Bochner identity. We also introduce Hermite functions and prove some of their properties that are needed in the study of the Heisenberg group. We establish Hardy’s theorem and its refinements. We also prove a higher dimensional version of Beurling’s theorem and deduce some of its consequences.
KeywordsEuclidean Space Entire Function Spherical Harmonic Heat Kernel Heisenberg Group
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