Heisenberg群中球面测度的Fourier变换

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

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，Ｈλ）；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Ｈλ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．

In this paper,the unit sphere S(H n) in the Heisenberg group H n is defined by the homogeneous gauge.The surface measure dσ on S(H n) is induced from the Haar Measure of H n.Let {(π λ,H[WT5 BX] λ);λ∈R 1 }} be the equivalent class of infinite dimensional irreducible unitary representations of H n[WT5 BZ].The author shows that there exist orthonormal basis on H[WT5 BX] λ, so that the Fourier transformation π λ(dσ) of dσ is an infinite diaognal matrix.And the asymptotic values of the diaognal elements are given.Moreover,a generalization of Pompieu's problem on H n is studied.