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

Sundaram Thangavelu

ISBN 10: 1461264685 / ISBN 13: 9781461264682

Editore: Birkhäuser Boston, Birkhäuser Boston, 2012

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Druck auf Anfrage Neuware - Printed after ordering - 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.

Codice articolo 9781461264682

Dati bibliografici

• Titolo: An Introduction to the Uncertainty Principle : Hardy's Theorem on Lie Groups
• Editore: Birkhäuser Boston, Birkhäuser Boston
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 1461264685
• ISBN 13: 9781461264682
• Rilegatura: Taschenbuch
• Condizione: Neu
• Dimensioni: 235 L x 155 A x 11 P (millimetri) 235 millimetri di larghezza per 155 millimetri di altezza per 11 millimetri di profondità
• Peso: 306 grammi

Descrizione articolo

Riassunto

The central theme of this work is the development of a number of analogs of Hardy's theorem in settings arising from noncommutative harmonic analysis. A tutorial introduction gives the requisite background material. 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.

Dalla quarta di copertina

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.