Fourier transforms of spherical distributions on compact symmetric spaces
Gestur Olafsson, Henrik Schlichtkrull
备注：
New version. Our previous results were for symmetric spaces of the compact type. The new version including an appendix sketching the extension of the Paley-Wiener theorem for K-invariant functions and distributions to compact symmetric spaces
在我们以前的文章“A local Paley-Wiener theorem for compact symmetric spaces”，Adv. Math. 218（2008），202- 215和“傅里叶级数on compact symmetric spaces”（已提交）中，我们研究了紧致对称空间M=U/K上的傅里叶级数。特别地，我们证明了M上的光滑函数的Paley-Wiener型定理，这些光滑函数具有足够小的支撑，并且是K-不变的，分别是K-有限的。在本文中，我们将这些结果推广到M上的K不变分布。我们证明了一个分布的傅里叶变换，这是支持在一个足够小的球周围的基点，扩展到一个全纯函数的指数型。我们在全纯函数空间中描述了傅里叶变换的象。我们用分布的傅里叶变换刻画了分布的奇异支撑。最后，我们利用Paley-Wiener定理刻画了在给定不变微分算子范围内的小支撑的分布。
英文摘要：
In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that the Fourier transform of a distribution, which is supported in a sufficiently small ball around the base point, extends to a holomorphic function of exponential type. We describe the image of the Fourier transform in the space of holomorphic functions. We characterize the singular support of a distribution in terms of its Fourier transform. Finally, we use the Paley-Wiener theorem to characterize the distributions of small support, which are in the range of a given invariant differential operator.
Integrals of spherical harmonics with Fourier exponents in multidimensions
考虑球面$S^n ,\, n \geq 1$上具有Fourier指数的球谐函数的积分。这种变换产生于加权Radon变换理论和电磁场理论中的矢量衍射理论的框架内。我们给出了这些积分的解析公式，这些积分直到乘法常数都是精确的。这些常数取决于球上基的选择。此外，我们发现这些常数明确的谐波产生的加权Radon变换的理论框架中的类。我们还建议公式找到这些常数的一般情况下。
