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

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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
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