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

ISBN-13: 9781461264682 / Angielski / Miękka / 2012 / 174 str.

Sundaram Thangavelu

Opis

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.

Recenzje

"This nicely written book by Thangavelu is concerned with extensions of Hardy's theorem to settings that arise from noncommutative harmonic analysis.... Each chapter contains several applications to the heat equation in various settings and ends with an extensive presentation of related topics, current research, detailed references to the literature, and lists of open problems. This makes the book an invaluable resource for graduate students and researchers in harmonic analysis and applied mathematics."

-SIAM Review

"...Each chapter ends with useful notes and open problems. Everything is written in sufficient detail to benefit specialized interested readers..."

-MATHEMATICAL REVIEWS

"The authoer discusses inthe present book the original theorem of Hardy and some of its generaliztions and its connections to noncommunitave analysis, harmonic analysis and special functions. First Hardy's theorem for the Euclidian Fourier transform is treated, and a theorem of Beurling and Hömander Subsequently Hardy's theorem is dicussed for the Fourier transfom on the Heisenberg group. finally the author discusses generaliztions of Hardy's theorem involving the Helgason Fourier transform for rank one symmetric spaces and for H-type groups. This unique book will be of great value for readers interested in this branch of analysis."

---Monatshefte für Mathematik

Spis treści

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.

O autorze

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.

Blurb

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.