Abstract
The aim of this chapter is to prove Hardy’s theorem for the Euclidean Fourier transform. We define the Fourier transform on ℝn 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.
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© 2004 Springer Science+Business Media New York
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Thangavelu, S. (2004). Euclidean Spaces. In: An Introduction to the Uncertainty Principle. Progress in Mathematics, vol 217. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8164-7_1
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DOI: https://doi.org/10.1007/978-0-8176-8164-7_1
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6468-2
Online ISBN: 978-0-8176-8164-7
eBook Packages: Springer Book Archive