In white noise analysis, one starts with a *real* Gelfand triple $\mathcal{N}\subset \mathcal{H} \subset \mathcal{N}^{*}$ and produces out of it, using complexifications along the way, the *complex* spaces of white noise test functions $(\mathcal{N})$ and white noise distributions $(\mathcal{N})^{*}$. The elements of $(\mathcal{N})$ are functions on $\mathcal{N}^{*}$ which are characterized by the analyticity and growth properties of their extensions to the complexification $\mathcal{N}_{\mathbb{C}}^{*}$.

My question is: Is there a smooth (or real analytic) version of white noise analysis developed in which the white noise test functions are characterized by their smoothness and growth on $\mathcal{N}^{*}$, rather than on $\mathcal{N}_{\mathbb{C}}^{*}$?

My motivation is as follows. I would like to work with a larger space of white noise test functions which contains all smooth (or real analytic) *bounded* real valued functions on $\mathcal{N}^{*}$. This is certainly not the case for the standard construction of $(\mathcal{N})$ because there are analytic functions on $\mathcal{N}_{\mathbb{C}}^{*}$ growing rapidly so that they do not belong to $(\mathcal{N})$ but their restrictions to $\mathcal{N}^{*}$ are bounded. Take, for example, $F(\cdot)=e^{-\langle \cdot, f \rangle^n}$, where $f \in \mathcal{N}$ and $n$ is even.