Let H_1 be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space S_m(H_1) of Schwartz functions of type m and the space S(\Sigma_m) consisting of restrictions of Schwartz functions on R^2 to a subset \Sigma_m of the Heisenberg fan with |m| of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H_1.
Fourier transform of Schwartz functions on the Heisenberg group
ASTENGO, FRANCESCA;
2013-01-01
Abstract
Let H_1 be the 3-dimensional Heisenberg group. We prove that a modified version of the spherical transform is an isomorphism between the space S_m(H_1) of Schwartz functions of type m and the space S(\Sigma_m) consisting of restrictions of Schwartz functions on R^2 to a subset \Sigma_m of the Heisenberg fan with |m| of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H_1.
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