The Group Fourier Transform
In this chapter we introduce the Heisenberg group and study the group Fourier transform. The Heisenberg group is constructed as a group of unitary operators acting on L2(ℝn). All its irreducible, unitary representations are identified using a theorem of Stone and von Neumann. Then the group Fourier transform is defined and basic results such as the Plancherel theorem and the inversion formula are proved. To further study the properties of the Fourier transform, we introduce the Hermite and special Hermite functions. We prove versions of the Paley-Wiener theorem and Hardy’s theorem for the Fourier transform on the Heisenberg group.
KeywordsHeisenberg Group Laguerre Function Plancherel Theorem Singular Number Twisted Convolution
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