  In this work we introduce a new algebra of tempered generalized functions.
The tempered distributions are embedded in this algebra via their Hermite
expansions. The Fourier transform is naturally extended to this algebra in such
a way that the usual relations involving multiplication, convolution and
differentiation are valid. We study the elementary properties of the
association, embedding, point values and Fourier transform. Furthermore, we
give a generalized Ito formula in this context and some applications to
stochastic analysis.
