In a paper about fractional Gaussian fields I found a problem, thats supposed to be really easily done with a Gram Schmidt procedure but I couldnt find the solution. Here for a multi-index $\alpha=(\alpha_1,...,\alpha_n)$ where $\alpha_i$ are all nonnegative integers, we consider the monom $x^{\alpha}:=\prod_{i=1}^nx_i^{\alpha_i}$ for $x=(x_1,...,x_n)\in\mathbb{R}^n$ and the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ i.e. all $f\in C^{\infty}_c(\mathbb{R}^n,\mathbb{R})$ such that for all multi-indices $\alpha,\beta$ we have $\sup_{x\in\mathbb{R}^n}x^{\alpha}D^{\beta}f(x)<\infty$ (see the standard definition in Wikipedia). The task is the following:
Find a family $f_{\alpha}$ of Schwartz functions such that for all multi-indices $\alpha,\beta$ we have $\displaystyle\int_{\mathbb{R}^n}x^{\alpha}f_{\beta}(x)dx=\delta_{\{\alpha=\beta\}}$, where $\delta_{\{\alpha=\beta\}}$ is $1$ if $\alpha=\beta$ and $0$ else.
Does anyone have an idea how to find that family of Schwartz functions? I would really be happy for any hints.