Cylindrical Hermite Polynomials: Growth and Integrability, Partial Derivatives, Orthogonality, and Monomials as Hermite Combinations

In the setting of Gaussian Analysis Relative to a Diagonal Gaussian Reference Measure with Noise Weights: Standing Notation, with the multi-indices $\alpha\in\mathcal{A}$, their length bounds, orders $|\alpha|$, factorials $\alpha!$ and powers $c^{\alpha}$, the multi-indices $\varepsilon_{k}$ and $\alpha-\varepsilon_{k}$ of Finitely Supported Multi-Indices: Order, Factorial, Powers and Weighted Order, and the cylindrical Hermite polynomials $H_{\alpha}$ of The Cylindrical Hermite Polynomials of a Diagonal Gaussian Measure on a Hilbert Space. Differentiability on $X$ is that of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space, applied with $E=U=X$. Let $\alpha,\beta\in\mathcal{A}$ and let $n\in\mathbb{N}$ be a length bound for $\alpha$.

1. (Cylindrical form, growth and integrability)¶ Let $h_{\alpha}:\mathbb{R}^{n}\to\mathbb{R}$, $h_{\alpha}(y)=\prod_{k=1}^{n}H^{c_{k}}_{\alpha_{k}}(y_{k})$. Then $h_{\alpha}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ and $H_{\alpha}=h_{\alpha}\circ p_{n}$; there is $M\in\mathbb{R}$ with $|h_{\alpha}(y)|\le M(1+\lVert y\rVert^{|\alpha|})$ for every $y\in\mathbb{R}^{n}$ and $|H_{\alpha}(x)|\le M(1+|x|^{|\alpha|})$ for every $x\in X$; and $H_{\alpha}$ is Borel with $\int_{X}|H_{\alpha}|^{p}\,d\gamma_{c}<\infty$ for every real $p\ge1$, so that $H_{\alpha}\in L^{p}(\gamma_{c})$.

2. (Partial derivatives)¶ $H_{\alpha}$ is differentiable on $X$, and for every $k\in\mathbb{N}$ and $x\in X$: $\partial_{k}H_{\alpha}(x)=\alpha_{k}\,H_{\alpha-\varepsilon_{k}}(x)$ if $\alpha_{k}\ge1$, and $\partial_{k}H_{\alpha}(x)=0$ if $\alpha_{k}=0$.

3. (Orthogonality)¶

$$\langle H_{\alpha},H_{\beta}\rangle_{L^{2}(\gamma_{c})}=\begin{cases}\alpha!\,c^{\alpha}&\text{if }\alpha=\beta,\\0&\text{if }\alpha\ne\beta.\end{cases}$$

4. (Monomials)¶ The function $x\mapsto\prod_{k=1}^{n}x_{k}^{\alpha_{k}}$ on $X$ is a finite linear combination of functions $H_{\beta}$ with $\beta\in\mathcal{A}$, $\beta_{k}\le\alpha_{k}$ for every $k\in\mathbb{N}$, and $|\beta|\le|\alpha|$.