## An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie GroupsThe central theme of this work is the development of a number of analogs of Hardy's theorem, which is one interpretation of the mathematical Uncertainty Principle, in settings arising from noncommutative harmonic analysis. A tutorial introduction gives the requisite background material. The techniques used to prove the main results run the gamut of modern harmonic analysis such as representation theory, spherical functions, Hecke Bochner formulas and special functions. Graduate students and researchers in harmonic analysis will greatly benefit from this unique book. |

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### Contents

Euclidean Spaces | 1 |

Heisenberg Groups | 45 |

Symmetric Spaces of Rank 1 | 105 |

Copyright | |

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### Common terms and phrases

Abel transform automorphisms Beurling's theorem bounded change of variables coefficients commutes completes the proof condition constant multiple convolution Corollary cosech2r defined eigenfunctions eigenvalue entire function Euclidean Fourier transform explicitly following result function f function on Hn functions satisfying fx(z given group Fourier transform Gx(P Haar measure Hardy's theorem harmonic analysis harmonics of degree heat kernel heat kernel associated Hecke-Bochner identity Heisenberg group Helgason Fourier transform hence Hermite functions Hermite operator hk(x Hp,q hypothesis inner product integral inversion formula Jacobi transforms Laguerre functions left invariant Lemma Let f e linear Mehler's formula Note obtain orthogonal orthonormal basis Plancherel theorem polynomial of degree properties Proposition prove the following px(x qs(z radial function Recalling the definition satisfies the estimate solid harmonic Sp,q special Hermite functions spherical Fourier transform spherical functions spherical harmonics subgroup sublaplacian symmetric spaces theory version of Hardy's Weyl transform