论文标题
尺寸中的分数泊松分析
Fractional Poisson Analysis in Dimension one
论文作者
论文摘要
在本文中，我们使用生物三相方法（Appell系统）来构建和表征与分数泊松测量$π_{λ，β} $相关的测试和广义功能的空间，也就是说，在天然（或真实）数字集中的概率度量。复合价值功能的Hilbert Space $ l^{2}（π_{λ，β}）$在结构中起着核心作用，即，测试功能空间$（n）_ {π_{π_{λ，β}}}^κ$，$κ$，$κ\ in [0,1,1] $ l^{2}（π_{λ，β}）$。此外，$ l^{2}（π_{λ，β}）$在双$（（n）_ {π_{π_{λ，β}}}^κ）'=（n）_ {π_{π_{λ{λ，β}}}}^{ - κ} $中也很密集。因此，我们获得了一系列密集的嵌入$（n）_ {π_{λ，β}}^κ\ subset l^{2}（π_{λ，β}）\ subset（subset（subset）_ {π_{π_{π_{π_{λ，β}}}}^ - κ} $。这些空间的表征是通过不同类型和生长顺序的整个功能的整体变换和链来实现的。 Wick Cyculus以直接的方式从高斯分析到目前的非高斯框架。最后，在附录B中，我们在（广义）appell多项式和贝尔多项式之间给出了明确的关系。
In this paper, we use a biorthogonal approach (Appell system) to construct and characterize the spaces of test and generalized functions associated to the fractional Poisson measure $π_{λ,β}$, that is, a probability measure in the set of natural (or real) numbers. The Hilbert space $L^{2}(π_{λ,β})$ of complex-valued functions plays a central role in the construction, namely, the test function spaces $(N)_{π_{λ,β}}^κ$, $κ\in[0,1]$ is densely embedded in $L^{2}(π_{λ,β})$. Moreover, $L^{2}(π_{λ,β})$ is also dense in the dual $((N)_{π_{λ,β}}^κ)'=(N)_{π_{λ,β}}^{-κ}$. Hence, we obtain a chain of densely embeddings $(N)_{π_{λ,β}}^κ\subset L^{2}(π_{λ,β})\subset(N)_{π_{λ,β}}^{-κ}$. The characterization of these spaces is realized via integral transforms and chain of spaces of entire functions of different types and order of growth. Wick calculus extends in a straightforward manner from Gaussian analysis to the present non-Gaussian framework. Finally, in Appendix B we give an explicit relation between (generalized) Appell polynomials and Bell polynomials.
