On a representation of the inverse Fq transform

A recent generalization of the Central Limit Theorem consistent with nonextensive statistical mechanics has been recently achieved through a generalized Fourier transform, noted $q$-Fourier transform. A representation formula for the inverse $q$-Fourier transform is here obtained in the class of functions $\mathcal{G}=\bigcup_{1\le q<3}\mathcal{G}_q,$ where $\mathcal{G}_{q}=\{f = a e_{q}^{-\beta x2}, \, a>0, \, \beta>0 \}$. This constitutes a first step towards a general representation of the inverse $q$-Fourier operation, which would enable interesting physical and other applications.Comment: 4 page

Fractional generalizations of filtering problems and their associated fractional Zakai equations

In this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process