An Introduction to the Uncertainty Principle: Hardy’s by Sundaram Thangavelu

By Sundaram Thangavelu

Motivating this fascinating monograph is the improvement of a few analogs of Hardy's theorem in settings coming up from noncommutative harmonic research. this is often the principal subject of this work.
Specifically, it really is dedicated to connections between a variety of theories coming up from summary harmonic research, concrete demanding research, Lie idea, distinctive capabilities, and the very fascinating interaction among the noncompact teams that underlie the geometric items in query and the compact rotation teams that act as symmetries of those objects.
A instructional creation is given to the required heritage fabric. the second one bankruptcy establishes a number of models of Hardy's theorem for the Fourier remodel at the Heisenberg workforce and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier remodel on rank one symmetric areas is handled. many of the effects provided listed here are legitimate within the normal context of solvable extensions of H-type groups.
The suggestions used to end up the most effects run the gamut of recent harmonic research comparable to illustration idea, round capabilities, Hecke-Bochner formulation and unique functions.
Graduate scholars and researchers in harmonic research will significantly reap the benefits of this book.

Show description

Read Online or Download An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups PDF

Best linear books

The Symmetric Eigenvalue Problem (Classics in Applied Mathematics)

A droll explication of ideas that may be utilized to appreciate a few of an important engineering difficulties: these facing vibrations, buckling, and earthquake resistance. whereas containing sizeable idea, this is often an utilized arithmetic textual content that reads as though you're eavesdropping at the writer conversing out loud to himself.

Signal Enhancement with Variable Span Linear Filters

This publication introduces readers to the radical inspiration of variable span speech enhancement filters, and demonstrates the way it can be utilized for potent noise relief in a number of methods. extra, the ebook presents the accompanying Matlab code, permitting readers to simply enforce the most rules mentioned. Variable span filters mix the information of optimum linear filters with these of subspace tools, as they contain the joint diagonalization of the correlation matrices of the specified sign and the noise.

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

Sample text

Suppose G is a Lie group and n is an irreducible unitary representation of G on a Hilbert space 11.. For u and v coming from 11. we consider the function (g) = (zr(g)u, v) which is called an entry function (or matrix coefficient) of Jr . As n is unitary, it is clear that (g) is a bounded continuous function on G. It need not be square integrable in general. Let Ps be the left regular representation of G realised on L 2 (G ). We can naturally extend P to the space Cb(G) bounded continuous functions.

A )f/JO(k) = f (U- . ej). Here I on the right-hand side is the Euclidean Fourier transform of f on JRn. 5 includes Hardy's theorem for JRn. , a) can also be computed explicitly for functions of certain form. 6 Let f E L1(M(n)) be of the form f(x, u) = g(lxl) P(x) h(u) where P is a solid harmonic of degree m. Then for every '). > 0, a E tV! and qJ E H(k , a) we have io.. ,a) qJ (k ) = '). ) is the (n + 2m )-dimensional Fourier transform of g(lxl). A proof of this lemma uses several results from Euclidean Fourier analysis.

3) = 21x1 2 for j 1,2, ... , n, so that Jr;CC) -f). + "A H("A) is the (scaled) Hermite operator. The eigenfunctions of this operator H("A) are given by ~(x) = I"AI£ a('J'iXTx), introduced earlier. 4) Thus the entry functions (JrA(z, t)~, ~) as a , f3 range over N n give a family of eigenfunctions for the sublaplacian. Since 54 2. 5) it is clear that these eigenfunctions are not in L 2(Hn ) . However, they are in L 2(Cn ) considered as functions of z alone. c(e iAt f(z» = eiAt L Af(z).

Download PDF sample

Rated 4.45 of 5 – based on 33 votes