Fractional Poisson Analysis in Dimension one
| Autor: | Bendong, Jerome B., Menchavez, Sheila M., da Silva, José Luís |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\pi_{\lambda,\beta}$, that is, a probability measure in the set of natural (or real) numbers. The Hilbert space $L^{2}(\pi_{\lambda,\beta})$ of complex-valued functions plays a central role in the construction, namely, the test function spaces $(N)_{\pi_{\lambda,\beta}}^{\kappa}$, $\kappa\in[0,1]$ is densely embedded in $L^{2}(\pi_{\lambda,\beta})$. Moreover, $L^{2}(\pi_{\lambda,\beta})$ is also dense in the dual $((N)_{\pi_{\lambda,\beta}}^{\kappa})'=(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. Hence, we obtain a chain of densely embeddings $(N)_{\pi_{\lambda,\beta}}^{\kappa}\subset L^{2}(\pi_{\lambda,\beta})\subset(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. 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. Comment: 32 pages, 0 figures |
| Databáze: | arXiv |
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