Year: 2022
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G8442367376
OpenAlex ID: W4225387472