An Introduction to the Uncertainty Principle [electronic resource] : Hardy’s Theorem on Lie Groups / by Sundaram Thangavelu
- Thangavelu, Sundaram
- Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2004.
- Physical Description:
- XIII, 174 pages : online resource
- Additional Creators:
- SpringerLink (Online service)
- Progress in Mathematics ; 217
- 1 Euclidean Spaces -- 1.1 Fourier transform on L1(?n) -- 1.2 Hermite functions and L2 theory -- 1.3 Spherical harmonics and symmetry properties -- 1.4 Hardy’s theorem on ?n -- 1.5 Beurling’s theorem and its consequences -- 1.6 Further results and open problems -- 2 Heisenberg Groups -- 2.1 Heisenberg group and its representations -- 2.2 Fourier transform on Hn -- 2.3 Special Hermite functions -- 2.4 Fourier transform of radial functions -- 2.5 Unitary group and spherical harmonics -- 2.6 Spherical harmonics and the Weyl transform -- 2.7 Weyl correspondence of polynomials -- 2.8 Heat kernel for the sublaplacian -- 2.9 Hardy’s theorem for the Heisenberg group -- 2.10 Further results and open problems -- 3 Symmetric Spaces of Rank 1 -- 3.1 A Riemannian space associated to Hn -- 3.2 The algebra of radial functions on S -- 3.3 Spherical Fourier transform -- 3.4 Helgason Fourier transform -- 3.5 Hecke-Bochner formula for the Helgason Fourier transform -- 3.6 Jacobi transforms -- 3.7 Estimating the heat kernel -- 3.8 Hardy’s theorem for the Helgason Fourier transform -- 3.9 Further results and open problems.
- Motivating this interesting monograph is the development of a number of analogs of Hardy's theorem in settings arising from noncommutative harmonic analysis. This is the central theme of this work. Specifically, it is devoted to connections among various theories arising from abstract harmonic analysis, concrete hard analysis, Lie theory, special functions, and the very interesting interplay between the noncompact groups that underlie the geometric objects in question and the compact rotation groups that act as symmetries of these objects. A tutorial introduction is given to the necessary background material. The second chapter establishes several versions of Hardy's theorem for the Fourier transform on the Heisenberg group and characterizes the heat kernel for the sublaplacian. In Chapter Three, the Helgason Fourier transform on rank one symmetric spaces is treated. Most of the results presented here are valid in the general context of solvable extensions of H-type groups. The techniques used to prove the main results run the gamut of modern harmonic analysis such as representation theory, spherical functions, Hecke-Bochner formulas and special functions. Graduate students and researchers in harmonic analysis will greatly benefit from this book.
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