By Sundaram Thangavelu
Motivating this attention-grabbing monograph is the advance of a few analogs of Hardy's theorem in settings coming up from noncommutative harmonic research. this can be the significant subject of this work.
Specifically, it truly is dedicated to connections between numerous theories bobbing up from summary harmonic research, concrete demanding research, Lie conception, particular services, 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 heritage fabric. the second one bankruptcy establishes a number of types 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 handled. lots of the effects provided listed here are legitimate within the normal context of solvable extensions of H-type groups.
The strategies used to end up the most effects run the gamut of contemporary harmonic research resembling illustration thought, round features, Hecke-Bochner formulation and distinctive functions.
Graduate scholars and researchers in harmonic research will significantly make the most of this book.
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Additional resources for An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups
Sample text
6 Let f E L1(M(n)) be of the form f(x, u) = g(lxl) P(x) h(u) where P is a solid harmonic of degree m. Then for every '). > 0, a E tV! and qJ E H(k , a) we have io.. ,a) qJ (k ) = '). ) is the (n + 2m )-dimensional Fourier transform of g(lxl). A proof of this lemma uses several results from Euclidean Fourier analysis. First of all it requires the Heeke-Bochner formula which says that if f(x) = g(lxl) P(x) with P, a solid harmonic of degree m, then I(n = c-tv" P(~) C(I~I) where C( I~I) is the (n + 2m)-dimensional Fourier transform of g(lxl).
In fact, the theorem amounts to saying that if F(z) is an entire function of one complex variable of order 2 and type b that decays like e-b~2 when restricted to the real line, then F(z) = c e- bz2. 1 due to Pfannschmidt [49] which uses several properties of entire functions including the notion of proximate orders. 1 and a more refined version in the higher-dimensional case. 1. With S = ~ + i 1] E C consider the function 1, 20 1. rr)-! f(x)e-ix(HiTJ)dx . -00 This integral converges absolutely and uniformly in every strip 11J1 f :s A, since 00 11(1;)1 sc e-ax2+XTJdx :s C e bTJ2.
Hardy's theorem on IR n Let Qm j ().. , w) be the projection of Sm,j(w) , I S j S d m. W)Sm,j(w)dW)Sm ,j(W) . 4 which is a genuine result in harmonic analysis! 5 Suppose f ELI (Iftn ) satisfies the following estimates: for almost every r > 0 and for every m for every A > O. Then f = 0 whenever ab > ! and when ab f (x) = P (x )e-al xI2 where P is a polynomial ofdegree S 2l. 4. We also note that the conditions can be written in terms of individual spherical harmonic coefficients of f(rw) and jp..