论文标题
上的秩序和球形衍射，iii：阴影变换和衍射公式
Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula
论文作者
论文摘要
我们定义了在交换空间中的广泛加权点集的球形衍射度量，即与Gelfand Pairs相关的适当同质空间。在双曲机平面的情况下，我们可以将球形衍射度量解释为自动相关分布的梅林变换。我们表明，通勤空间中均匀的常规模型集具有纯点球形衍射度量。该度量的原子位于基础晶格的球形自动型光谱上，衍射系数可以通过窗口特征函数的所谓阴影变换来抽象地表征。在海森伯格组的情况下，我们可以根据贝塞尔和拉瓜尔的功能给出这些衍射系数的明确公式。
We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.
