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

By Sundaram Thangavelu

Motivating this attention-grabbing monograph is the improvement of a few analogs of Hardy's theorem in settings bobbing up from noncommutative harmonic research. this is often the relevant topic of this work.
Specifically, it really is dedicated to connections between a variety of theories coming up from summary harmonic research, concrete tough research, Lie concept, certain features, 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 educational advent is given to the mandatory heritage fabric. the second one bankruptcy establishes a number of types 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 remodel on rank one symmetric areas is handled. many of the effects provided listed below are legitimate within the common context of solvable extensions of H-type groups.
The strategies used to end up the most effects run the gamut of contemporary harmonic research corresponding to illustration concept, round services, Hecke-Bochner formulation and specific functions.
Graduate scholars and researchers in harmonic research will vastly make the most of this book.

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

Example text

A}Cz) which is the jth derivative of Pz(t) at t = 0 is holomorphic. 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.

2, sn - I whe re the fun ctions P and Q are such that . log P (r ) . log Q ().. ) hm sup 2 = hm sup 2 = O. ->- 00 i, and when ab = i, ).. 40 1. Euclidean Spaces where p and q are such that their L 2 -norms on and Q(r) respectively. Ixl = r do not grow faster than per) This theorem is proved in [61] where the authors have also proved analogues of Hardy's theorem for the Dunk! transform. 5 and uses the Heeke-Bochner formula and Poisson integral representation of Bessel functions. 1 to complete the proof.

00 Thus 1(1;) is an entire function that satisfies the estimates and Hardy's theorem will follow once we prove the following result concerning entire functions. 2 Suppose F(I;) is an entire function of one complex variable that satisfies the following estimates: W(I;)I IF(~)I :s C (1 + 1S-1 2 )me blIm<012, :s C (l + l~e)me-b~2 for S- E C, ~ E R Then F(I;) = P(l;)e- bI;2 where P(I;) is a polynomial of degree at most 2m. As we have said earlier we prove this theorem by appealing to the PhragmenLindelof maximum principle, which we establish now.

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