Polynomial ridge functions are dense on compact sets

Fix $k\ge 1$ and linear forms ${\mathrm{\ell }}_j(x)=v_j\cdot x$ for $1\le j\le k$. Expanding the powers and summing signs gives the polarization identity ${\prod }_{j=1}^k{\mathrm{\ell }}_j(x)=\frac{1}{2^{k}k!}{\sum }_{ϵ\in \{-1,1{\}}^k}({\prod }_{j=1}^k{ϵ}_j)({\sum }_{j=1}^k{ϵ}_j{\mathrm{\ell }}_j(x){)}^k$. Indeed, in the multinomial expansion, a monomial ${\prod }_j{\mathrm{\ell }}_j^{r_j}$ has sign sum ${\prod }_j{\sum }_{{ϵ}_j=\pm 1}{ϵ}_j^{r_j+1}$, which is zero unless every $r_j$ is odd. Since ${\sum }_j{r}_j=k$, the only surviving tuple is $r_1=\cdots =r_k=1$, with coefficient $2^{k}k!$. Thus every product of $k$ linear forms lies in ${\mathcal{R}}_d$.

Every coordinate monomial $x_1^{{\alpha }_1}\cdots x_d^{{\alpha }_d}$ of positive total degree $k$ is a product of $k$ coordinate linear forms, with $x_i$ repeated ${\alpha }_i$ times. Step 1.1 puts it in ${\mathcal{R}}_d$. Constants lie in ${\mathcal{R}}_d$ by $k=0$, so every real polynomial restricts to an element of ${\mathcal{R}}_d{\mathrm{\mid }}_K$.

The restrictions to $K$ of real polynomials form a unital real function algebra: sums, scalar multiples, and products are polynomials, and the constant-one polynomial is a unit. If $x,y\in K$ are distinct, some coordinate differs, so the corresponding coordinate polynomial separates them. By [F1] these polynomial restrictions are uniformly dense in $C(K,\mathbb{R})$. Since step 2.1 places them inside ${\mathcal{R}}_d{\mathrm{\mid }}_K$, the latter is dense as well.