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

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Springer Science & Business Media, Oct 9, 2003 - Mathematics - 174 pages
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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
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