Tempered generalized functions and Hermite expansions

In this work we introduce an 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. Furthermore, we give a generalized Ito formula in this context and some applications to stochastic analysis. (C) 2010 Elsevier Ltd. All rights reserved.