# An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups

Springer Science & Business Media, 9 oct. 2003 - 174 pages
In 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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### Table des matières

 Euclidean Spaces 3 12 Hermite functions and L2 theory 9 13 Spherical harmonics and symmetry properties 13 14 Hardys theorem on R 20 15 Beurlings theorem and its consequences 31 16 Further results and open problems 40 Heisenberg Groups 47 22 Fourier transform on H 50
 29 Hardys theorem for the Heisenberg group 89 210 Further results and open problems 102 Symmetric Spaces of Rank 1 107 32 The algebra of radial functions on 5 113 33 Spherical Fourier transform 121 34 Helgason Fourier transform 128 35 HeckeBochner formula for the Helgason Fourier transform 138 36 Jacobi transforms 143

 23 Special Hermite functions 54 24 Fourier transform of radial functions 62 25 Unitary group and spherical harmonics 64 26 Spherical harmonics and the Weyl transform 71 27 Weyl correspondence of polynomials 79 28 Heat kernel for the sublaplacian 85
 37 Estimating the heat kernel 148 38 Hardys theorem for the Helgason Fourier transform 154 39 Further results and open problems 159 Bibliography 171 Index Droits d'auteur

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