Fractional Poisson Analysis in Dimension one

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 ${\pi }_{\lambda ,\beta }$, 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 $((N{)}_{{\pi }_{\lambda ,\beta }}^{\kappa }{)}^{\prime }=(N{)}_{{\pi }_{\lambda ,\beta }}^{-\kappa }$. 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.

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