## An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie GroupsIn 1932 Norbert Wiener gave a series of lectures on Fourier analysis at the Univer sity of Cambridge. One result of Wiener's visit to Cambridge was his well-known text The Fourier Integral and Certain of its Applications; another was a paper by G. H. Hardy in the 1933 Journalofthe London Mathematical Society. As Hardy says in the introduction to this paper, This note originates from a remark of Prof. N. Wiener, to the effect that "a f and g [= j] cannot both be very small". ... The theo pair of transforms rems which follow give the most precise interpretation possible ofWiener's remark. Hardy's own statement of his results, lightly paraphrased, is as follows, in which f is an integrable function on the real line and f is its Fourier transform: x 2 m If f and j are both 0 (Ix1e- /2) for large x and some m, then each is a finite linear combination ofHermite functions. In particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant; and if one x 2 2 is0(e- / ), then both are null. |

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### Contents

Euclidean Spaces | 1 |

12 Hermite functions and L2 theory | 7 |

13 Spherical harmonics and symmetry properties | 11 |

14 Hardys theorem on R | 18 |

15 Beurlings theorem and its consequences | 29 |

16 Further results and open problems | 38 |

Heisenberg Groups | 45 |

22 Fourier transform on H | 48 |

29 Hardys theorem for the Heisenberg group | 87 |

210 Further results and open problems | 100 |

Symmetric Spaces of Rank 1 | 105 |

32 The algebra of radial functions on 5 | 111 |

33 Spherical Fourier transform | 119 |

34 Helgason Fourier transform | 126 |

35 HeckeBochner formula for the Helgason Fourier transform | 136 |

36 Jacobi transforms | 141 |

23 Special Hermite functions | 52 |

24 Fourier transform of radial functions | 60 |

25 Unitary group and spherical harmonics | 62 |

26 Spherical harmonics and the Weyl transform | 69 |

27 Weyl correspondence of polynomials | 77 |

28 Heat kernel for the sublaplacian | 83 |

37 Estimating the heat kernel | 146 |

38 Hardys theorem for the Helgason Fourier transform | 152 |

39 Further results and open problems | 157 |

169 | |

173 | |

### Other editions - View all

An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups Sundaram Thangavelu No preview available - 2012 |

### 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 f and g following result function f function on H functions satisfying Further assume given group Fourier transform Haar measure Hardy's theorem harmonic analysis harmonics of degree heat kernel heat kernel associated Hecke-Bochner formula Hecke-Bochner identity Heisenberg group Helgason Fourier transform hence Hermite functions Hermite operator hk(x hypothesis inner product integral inversion formula Jacobi transforms Laguerre functions left invariant Lemma Let f e Lie algebra matrix Mehler's formula Note obtain orthogonal orthonormal basis Plancherel theorem polynomial of degree properties Proposition prove the following qs(z radial function Recalling the definition Riemannian satisfies the estimate solid harmonic Sp.q special Hermite functions spherical Fourier transform spherical functions spherical harmonics subgroup sublaplacian symmetric spaces version of Hardy's Weyl transform