Generalized Appell Systems

Operators on Spaces Generated by Infinite Dimensional Appell Polynomials

It is known that many constructions arising in the classical Gaussian infinite dimensional analysis can be extended to the case of more general measures. One such extension can be obtained through biorthogonal systems of Appell polynomials and generalized functions. In this paper, we consider linear continuous operators from a nuclear Frechet space of test functions to itself in this more general setting. We construct an isometric integral transform (biorthogonal CS-transform) of those operators into the space of germs of holomorphic functions on a locally convex infinite dimensional nuclear s…

Dual $n_1$-Appell-like Systems in Infinite-Dimensional Analysis

We introduce and study dual $n_1$-Appell-like systems which are the simple generalization of generalized dual Appell systems in Infinite-Dimensional Analysis (IDA). We study connected with these systems objects of IDA: the analogues of Kondratiev spaces, $S$-transform, characterization theorems etc. The results we obtained are useful to application in the theory of probability.

Fueter-Sce映射与Clifford-Appell多项式

The Fueter-Sce mapping and the Clifford-Appell polynomials

本文利用Fueter-Sce映射与广义Cauchy-Kovalevskaya延拓的联系，显式计算其作用并刻画值域与核，定义新乘积，研究相关空间，并推广得到高维多解析单演多项式。

Fractional Poisson Analysis in Dimension one

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 den…

无限维分数阶 Poisson 测度的双正交方法

A biorthogonal approach to the infinite dimensional fractional Poisson measure

本文提出双正交方法分析无限维分数阶 Poisson 测度，利用广义 Appell 多项式系统及其对偶系统刻画 Hilbert 空间，并通过积分变换将测试与广义函数空间完全刻画为整函数。

Mittag-Leffler Analysis I: Construction and characterization

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak inte…