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 important subject of this work.

Specifically, it really is dedicated to connections between a variety of theories bobbing up from summary harmonic research, concrete difficult research, Lie thought, particular services, and the very attention-grabbing interaction among the noncompact teams that underlie the geometric gadgets in query and the compact rotation teams that act as symmetries of those objects.

A instructional advent is given to the required heritage fabric. the second one bankruptcy establishes numerous models of Hardy's theorem for the Fourier remodel at the Heisenberg team and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier rework on rank one symmetric areas is taken care of. many of the effects provided listed here are legitimate within the basic context of solvable extensions of H-type groups.

The strategies used to end up the most effects run the gamut of recent harmonic research reminiscent of illustration conception, round capabilities, Hecke-Bochner formulation and precise functions.

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

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**Example text**

Interchanging the roles of f and j we also get j ELI (R"). The function j though bounded need not have any exponential decay. Now we will define another function g in terms of f so that g will have exponential decay. 2. l xI Then it is clear that ~ 1/ /2 ~ Ig(y)1 = If(y)le- i Y I I 12 :::: Ilflhe- i Y 2 and also that g (y)e! Iy 1 is integrable. Y)ldydx :::: C(1 + R)N. 10) is equal to ff If(x)/Ij(y)lh(x , y)el(x,Y)ldxdy ~n ~n where the function h (x, y) is defined by h(x , y) = f (1 + Ix - r] + Iyl)-N e-!

Is a strongly continuous unitary representation of H" , We will shortly show that each 7f). is irreducible . A celebrated theorem of Stone and von Neumann says that up to unitary equivalence these are all the irreducible unitary representations of H " that are nontrivial at the centre. , A =1= are irreducible. If p is any other irreducible unitary representation of H " on a Hilbert space H. / I for some A =1= 0, then p is unitarily equivalent to 7f).. We will not prove this theorem completely. We will only show that 7f).

1). This is known as Hardy's theorem for the Fourier transform pairs on R 1. 1) hold for a measurable function f on R Then f = 0 whenever ab > and when ab = f is a constant multiple of the ax2 Gaussian e- . 1 1, Hardy's theorem is an example of the uncertainty principle for the Fourier transform on R It says that both f and f cannot have arbitrary Gaussian decay. In [31] Hardy points out the origin of this result with this remark : "This note originates from a remark of N. Wiener to the effect that 'a pair f and g' cannot be very small.