# An Introduction to the Uncertainty Principle: Hardy’s - download pdf or read online

By Sundaram Thangavelu

ISBN-10: 0817681647

ISBN-13: 9780817681647

ISBN-10: 1461264685

ISBN-13: 9781461264682

Motivating this attention-grabbing monograph is the advance of a few analogs of Hardy's theorem in settings bobbing up from noncommutative harmonic research. this is often the critical subject matter of this work.

Specifically, it truly is dedicated to connections between a number of theories coming up from summary harmonic research, concrete demanding research, Lie conception, particular services, and the very attention-grabbing 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 mandatory history fabric. the second one bankruptcy establishes a number of models of Hardy's theorem for the Fourier remodel at the Heisenberg crew and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier rework on rank one symmetric areas is taken care of. lots of the effects awarded listed here are legitimate within the basic context of solvable extensions of H-type groups.

The options used to turn out the most effects run the gamut of contemporary harmonic research equivalent to illustration idea, round capabilities, Hecke-Bochner formulation and targeted functions.

Graduate scholars and researchers in harmonic research will tremendously reap the benefits of this book.

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

**Sample text**

F * ). (G) *1 G. 21) for F and G in L 1 (C") . For some interesting properties of the twisted convolution see the monograph [24]. We show that they are expressible in terms of Laguerre functions . 3. Special Hermite functions 53 us with important tools required in order to establish analogues of Hardy's theorem for the Fourier transform on H" , Using special Hermite functions we can write down another version of inversion formula for the Fourier transform on H", We begin with some abstract considerations which motivate us to introduce the special Hermite functions .

F, g ; z ) = gives IIfll211g112 = which is a contradiction as f and g are nontrivial. Hence M has to be the whole space. (f, g) is called the Fourier-Wigner transform of f and g which is very useful in the study of Fourier transforms on the Heisenberg group. 4) by polarisation. 1. 2 Let tp; 1ft, I, g be in L 2(lR n ) . Then fen V).. (cp , 1ft; z) V ).. (f, g ; z)dz = (cp , f) (g, 1ft ) where the inner products on the right-hand side are taken in L 2(lRn ). The Fourier-Wigner transform and its close relative the Wigner transform have many interesting properties.

The convolution f * Pt makes sense whenever f is a tempered distribution. 5 to say something about the initial condition. 8 for the Hermite operator H. Let K, (x , y) be the heat kernel associated to H which is explicitly given by L e-(2I al+n )t*a (y) . 15) The following theorem characterises this heat kernel. 10 Let f (x, y) be a function on jRn X jRn that satisfies the following three conditions: (i) If(x , y)1 :s cs,», y), (ii) I f(x, y) a(x)dx I :s Cl (y)e-2Ialt, (iii) I J f(x, y )a (y )d y l s Cl (x)e-2Ial t, where Cl and Cs are independent ofa, Then f(x, y ) is a constant multiple of Kitx ; y). *

### An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups by Sundaram Thangavelu

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