Tempered Generalized Functions Algebra, Hermite Expansions and Itô Formula

The space of tempered distributions $\mathcal{S}^{\prime}$ can be realized as a sequence spaces by means of the Hermite representation theorems (see \cite{Miku}). In this work we introduce and study a new tempered generalized functions algebra $\mathcal{H}$, in this algebra the tempered distributions are embedding via its Hermite expansion. We study the Fourier transform, point value of generalized tempered functions and the relation of the product of generalized tempered functions with the Hermite product of tempered distributions. Furthermore, we give a generalized It\^o formula for elements of $\mathcal{H}$ and finally we show some applications to stochastic analysis.