## Relatório de pesquisa 14/09

Tempered Generalized Functions Algebra, Hermite Expansions and Itô Formula, Pedro Catuogno and Christian Olivera, submitted May 07, 2009.

Abstract

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.

Mathematics Subject Classifications
(2000):

Keywords: Hermite functions, Generalized functions Algebras, It\^o formula, Stochastic Integration.

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May 07, 2009

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