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

By Sundaram Thangavelu

Motivating this fascinating monograph is the improvement of a couple of analogs of Hardy's theorem in settings bobbing up from noncommutative harmonic research. this is often the relevant topic of this work.
Specifically, it truly is dedicated to connections between quite a few theories bobbing up from summary harmonic research, concrete challenging research, Lie conception, designated capabilities, 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 educational creation is given to the required heritage fabric. the second one bankruptcy establishes a number of types of Hardy's theorem for the Fourier remodel at the Heisenberg staff and characterizes the warmth kernel for the sublaplacian. In bankruptcy 3, the Helgason Fourier remodel on rank one symmetric areas is taken care of. many of the effects awarded listed here 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 contemporary harmonic research akin to illustration idea, round services, Hecke-Bochner formulation and distinct functions.
Graduate scholars and researchers in harmonic research will significantly reap the benefits of this book.

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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). are irreducible . For the rest of the theorem we refer to Folland [24].

Assume that IF (z) I is bounded by C (l + [z i) N on the boundary ofthe angular sector {z = re'" : f3 :s e :s f3 + a} . Then the same bound is valid inside the angular sector. 3 and we leave the details to the reader. In order to apply this let us fix g E JRn and consider G~ :

42 1. Euclidean Spaces The coefficients Cmj are given in term s of ultras pherica1polynomials. More precisely, if Gi - ' (t) are ultraspherical polynomials of type f(m + I)f(n - 2) Cmj = wn - 2 f(m + n - 2) f (I - 1 1- 1 1), then 2 n- 3 a(t) G m (t) (1 - t ) 2" dt. - I Let us write A, for the Laplacian on jRn and p~ for the associated heat kernel. n)a f) 1\(~) = I ~ 120: f (~ ). With this notation we now state the most general form of Hardy's theorem on M (n ). 7 Suppose f and / satisfy (i) If(x , k)1 s c(l + Ixl) N Ps(x) , (x , k ) (ii) II/(A, 0')11 S c(I + 2 A) N e- o .

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