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

Springer Science & Business Media, Oct 9, 2003 - Mathematics - 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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### 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 Bibliography 169 Index 173 Copyright