By Sundaram Thangavelu

The Heisenberg staff performs an immense position in different branches of arithmetic, reminiscent of illustration idea, partial differential equations, quantity idea, a number of complicated variables and quantum mechanics. This monograph offers with numerous facets of harmonic research at the Heisenberg crew, that is the main commutative one of the non-commutative Lie teams, and for that reason supplies the best chance for generalizing the striking result of Euclidean harmonic research. the purpose of this article is to illustrate how the normal result of abelian harmonic research take form within the non-abelian setup of the Heisenberg group.

Several ends up in this monograph look for the 1st time in ebook shape, and a few theorems haven't seemed in different places. The certain dialogue of the illustration thought of the Heisenberg workforce is going way past the elemental Stone-von Neumann idea, and its family to classical distinct capabilities is worthy for any reader drawn to this crew. subject lined comprise the Plancherel and Paley—Wiener theorems, spectral concept of the sublaplacian, Wiener-Tauberian theorems, Bochner—Riesz potential and multipliers for the Fourier transform.

Thangavelu’s exposition is apparent and good constructed, and results in a number of difficulties useful of additional attention. Any reader who's attracted to pursuing examine at the Heisenberg crew will locate this special and self-contained textual content invaluable.

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For example {An (Tn is rapidly decreasing if and only if T is unitarily equivalent to an integral operator whose kernel is in the Schwartz class. We are interested in operators whose singular numbers are exponentially decreasing. We state our conditions on the operator in terms of the unbounded operator eaH , a > O. These operators are densely defined; their domain consists of all finite linear combinations of Hermite functions. If f = E~o CkPk! then we define eaH f = L Ck e(2k+n)aPkf. m k=O We consider operators T for which Te aH is Hilbert-Schmidt for some positive a.

When ab = ~, (Rf)(w,s) = ](ws) = g(w)e- bS2 where 9 is a function on sn-l. As f is integrable, ] is continuous and so by taking limit as s ~ 0, we get g(w) = ](0). Hence ](0 = ](0)e-bleI2 so that f(x) = Ce-alxI2. Finally when ab < ~, the n dimensional Hermite functions

Consequently the above series giving IILm /1l2 converges uniformly and gives the estimate 00 IILm III~ ::; C E(2k + n)2m e -2(2k+n)a. k=O This series can be estimated by the integral Jo 00 t2me-2at dt which gives the estimate IILm III~ ::; M 2m +l(2m)! which is more than what we need. Hence by the regularity theorem I is real analytic. We also have the following version of the above theorem for the group Fourier transform. 4 Let I E L2(Hn) be compactly supported. Suppose there is a compact set B 01 JR* such that lor every A not in B there exists a(A) > 0 lor which j(A)ea()')H E S2.