The Wick Powers of a Paley-Wiener Functional

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the Cameron-Martin space $H_{c}$, the Cameron-Martin squares $|h|_{c}^{2}$ and the Paley-Wiener functionals $\ell_{h}$ of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation §cameron-martin, the coordinates $h_{k}=\langle h,e_{k}\rangle$, and the Hermite polynomials $H^{v}_{n}$.

(Wick powers)¶ For $h\in H_{c}$ and $n\in\mathbb{N}_{0}$, the $n$-th Wick power of $\ell_{h}$ is the function ${:}\ell_{h}^{n}{:}\,:X\to\mathbb{R}$,

$${:}\ell_{h}^{n}{:}(x)=H^{|h|_{c}^{2}}_{n}\bigl(\ell_{h}(x)\bigr)\qquad(x\in X),$$

the Hermite polynomial being defined since $0\le|h|_{c}^{2}$ by The Cameron-Martin Space of a Diagonal Gaussian Measure on a Hilbert Space §square.