局部紧阿贝尔群上的概率迹与Poisson求和公式
Probabilistic Trace and Poisson Summation Formulae on Locally Compact Abelian Groups
摘要：
研究了局部紧阿贝尔群上具有连续密度的概率测度的卷积半群，其中有一个离散子群使得因子群是紧的。商结构的两个有趣的例子是$d$ -维环面和阿德利克圆。我们的主要结果是表明，泊松求和公式的密度可以解释为一个概率的迹公式，链接的值的密度的因素组上的相关半群的迹$L^{2}$ -空间。高斯是一个非常重要的例子。对于旋转不变的$α$ -稳定密度，迹公式是有效的，但我们不能验证Poisson求和公式.为了准备研究半稳定的法律上的adèles，我们首先调查这些$p$-adics，在那里我们表明，他们有连续的密度，可以表示为一系列的扩展。我们利用这些规律构造了一个密度不满足概率迹公式的卷积半群。
英文摘要：
We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the $d$--dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on $L^{2}$-space. The Gaussian is a very important example. For rotationally invariant $α$-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the $p$--adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
