By Sundaram Thangavelu

Motivating this fascinating monograph is the advance of a couple of analogs of Hardy's theorem in settings bobbing up from noncommutative harmonic research. this is often the significant subject matter 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, targeted features, and the very fascinating 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 mandatory history fabric. the second one bankruptcy establishes numerous models of Hardy's theorem for the Fourier rework 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. many of the effects offered listed below are legitimate within the common context of solvable extensions of H-type groups.

The strategies used to turn out the most effects run the gamut of recent harmonic research akin to illustration idea, round features, Hecke-Bochner formulation and unique functions.

Graduate scholars and researchers in harmonic research will tremendously take advantage of this book.

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

**Example text**

Y )leabl(x,Y)1 + Ix l) N (1 + ly l)N d xdy (l is finite as it is dominated by 38 1. -' f(x) = P(x)e- ! x) for some positive definite matrix A and polynomial P with degree P < N - n. Now if p > 2, the integral of f with the exponential factors cannot be finite a; P as f cannot decay faster than e Ixl • Similarly, if q > 2 the corresponding integral of f will not be finite, which forces p = q = 2. 10 (Cowling-Price type). -' then f = O. -' then = f(x) s p,q s 00 . Assume that P(x)e-alxI2for some polynomial P.

11), choose 0 > 1 and consider { Ig( x)1 J'xl:,: R (1IYI~28R Ig (y )l el(x,Y)ld y + 1 lyl:,:28 R Ig (Y) lel(X,Y)ld Y) dx. g(Y)le l(x,Y)1 and therefore { 1 J1xl:,:R l yl ~28R s Ce- ~ (l- ! e I(x,Y)ld xdy::: C. On the other hand the second integral is bounded by 1 lyl:,:28R Ig(x)ll g (Y)l el(x,Y)ldxdy < ( 1 + R)N - ff Ig (x )lI g( Y)1 el(x,Y)ld xd < C (l (1+l xl+l yI) N Y- + R)N. 11). We now claim that the function g admits a holomorphic extension to the whole of and the extension is of order 2. Moreover, for z E g(z)g(iz) = R(z) where R(z) is a polynomial.

We already know that it is homogeneous and hence a polynomial of degree j . Therefore, P (z) is a polynomial of degree ~ N. This completes the proof of the lemma. Beurling's theorem has several interesting consequences. 2. 7 Let IE L 2(lRn ) . (Y)1 + [x] + IYI)NexP JR" JR" dxdy < 00, J=I then I(x) = P(x)exp(On the other hand, (~ ) LJ IXjYjl if I n L f3jX;) where P is a polynomial and f3 j > Olor all j. (Y)1 elxllYldxd < + [x] + lyl) N Y 00 , JR" JR" then I(x) = P(x)e-,Blx I2 where P is a polynomial and f3 > O.