Working on a physics branch that make use of (multidimensional) Hermite polynomials, I struggle to find an accurate insight about expansions on this basis. Let me explain:
Consider a function $f : \mathbb{R}^{D} \rightarrow \mathbb{R}$ and multidimensional extension of Hermite polynomials in $D$ dimension:
$$ H_0(\boldsymbol{x}) = 1, \ H_1(\boldsymbol{x}) = \boldsymbol{x}, [H_2(\boldsymbol{x})](i,j) = x_i x_j - \delta_{ij}, \ [H_3(\boldsymbol{x})](i,j,k) = x_i x_j x_k - x_i\delta_{jk}-x_j \delta_{ik} - x_k \delta_{ij} $$ with $\delta_{ij}$ being the Kronecker symbol. A lot of paper I find claims without further explanation $$ f(\boldsymbol{x}) = \sum_{n=0}^{+\infty}\frac{\boldsymbol{a}_n:H_n(\boldsymbol{x})}{n!} $$ where $\boldsymbol{a}_n = <f,H_n>_{\omega} = \frac{1}{\left(2\pi\right)^{D/2}}\int_{\mathbb{R}^{D}} f \left(\boldsymbol{x}\right)H_n\left(\boldsymbol{x}\right)\text{exp}\left(-\left\|\boldsymbol{x}\right\|^2 / 2\right)\text{d}\boldsymbol{x}$. The $:$ symbol denotes the sum of all same-indexes products of the $n^{D}$ coefficients of $\boldsymbol{a}_n$ and $H_n$
For a 1D case, I'd say that $$f(x) = \sum_{n=0}^{+\infty}\frac{\langle f, H_n\rangle_{\omega}}{\langle H_n, H_n \rangle_{\omega}} H_n(x) = \sum_{n=0}^{+\infty}\frac{a_n}{n!}H_n\left(x\right)$$
So I feel like it is a multidimensional generalization of the projection formula on hermitian basis, but why is it $\boldsymbol{a}_n : H_n\left(\boldsymbol{x}\right)$ and where does this come from ? Thanks in advance