Heisenberg 群中球面测度的 Fourier 变换

本文按 Heisenberg 群 Hn 上齐次度规定义,其中单位球面 S(Hn).Hn 上 Haar 测度诱导出 S(Hn)上 面积元 dσ. 设{ (πλ ,H λ ); λ∈ R1 { 0} }是 Hn 的无穷维不可约酉表示等价类. 作者证明可选取 H λ的正交基使 dσ的群 Fourier 变换 πλ( dσ)对角化,且给出了对角线元素的渐近估计. 此 外,作者还考虑了 Hn 上 Pompeiu 问题的一个推广.

Fourier Transformation of the Surface Measure on the Sphere in the Heisenberg Group

In this paper, the unit sphere S(Hn) in the Heisenberg group Hn is defined by the homogeneous gauge. The surface measure dσon S(Hn) is induced from the Haar Measure of Hn . Let{ (πλ ,H λ ); λ∈ R1 { 0}}be the equivalent class of infinite dimensional irreducible unitary representations of Hn . The author show s that there exist orthonormal basis on H λ , so that the Fourier transformation πλ( dσ)of dσis an infinite diaognal matrix. And the asymptotic values of the diaognal elements are given. Moreover, a generalization of Pompieu' s problem on Hn is studied.