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 principal subject 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, distinctive capabilities, and the very fascinating 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 instructional creation is given to the required heritage fabric. the second one bankruptcy establishes a number of models of Hardy's theorem for the Fourier remodel at the Heisenberg workforce 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 here are legitimate within the normal context of solvable extensions of H-type groups.
The suggestions used to end up the most effects run the gamut of recent harmonic research comparable to illustration idea, round capabilities, Hecke-Bochner formulation and unique functions.
Graduate scholars and researchers in harmonic research will significantly reap the benefits of this book.
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Extra info for An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups
Suppose G is a Lie group and n is an irreducible unitary representation of G on a Hilbert space 11.. For u and v coming from 11. we consider the function
A )f/JO(k) = f (U- . ej). Here I on the right-hand side is the Euclidean Fourier transform of f on JRn. 5 includes Hardy's theorem for JRn. , a) can also be computed explicitly for functions of certain form. 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.
3) = 21x1 2 for j 1,2, ... , n, so that Jr;CC) -f). + "A H("A) is the (scaled) Hermite operator. The eigenfunctions of this operator H("A) are given by