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**An introduction to the uncertainty principle. Hardy’s theorem on Lie groups. With a foreword by Gerald B. Folland.**
*(English)*
Zbl 1188.43010

Progress in Mathematics (Boston, Mass.) 217. Boston, MA: Birkhäuser (ISBN 0-8176-4330-3). xiv, 174 p. (2004).

Publisher’s description: 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 sub-Laplacian. 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.

Contents: Foreword (G. Folland).- Preface.- Euclidean Spaces.- Heisenberg Groups.- Symmetric Spaces of Rank one.- Bibliography.- Index.

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 sub-Laplacian. 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.

Contents: Foreword (G. Folland).- Preface.- Euclidean Spaces.- Heisenberg Groups.- Symmetric Spaces of Rank one.- Bibliography.- Index.

### MSC:

43A80 | Analysis on other specific Lie groups |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

33C55 | Spherical harmonics |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

43A20 | \(L^1\)-algebras on groups, semigroups, etc. |