the space of polynomials on ${\mathbb{R}}^d$ of degree no greater than $\lfloor {H}\rfloor$. This means that $h$ is defined as a linear functional on the subspace of test functions $\phi$ satisfying $\int_{{\mathbb{R}}^d} \phi(x) L(x)dx = 0$ for all polynomials $L$ of degree $\lfloor {H}\rfloor$. Alternatively, at the cost of breaking translation invariance, we may define $\operatorname{FGF}_s({\mathbb{R}}^d)$ as a random element of $C^{\lceil {H}\rceil -1}({\mathbb{R}}^d)$ by fixing the derivatives of $h$ at 0 up to order $\lceil {H}\rceil - 1$. The FGF covariance structure is described by the Hurst parameter $H$. When ${H}$ is a positive non-integer, we have $$\operatorname{Cov}[(h, \phi_1), (h, \phi_2)] = C(s,d)\int_{{\mathbb{R}}^d} \int_{{\mathbb{R}}^d}
|x-y|^{2{H}} \phi_1(x) \phi_2(y) dxdy,$$ for some constant $C(s,d)$. A variant of this statement applies for negative and integer values of ${H}$ (see Theorem \[KernelComp\]).

Note that ${H}$ is an affine function of $s$ and can be used instead of $s$ to parameterize the family of FGFs. We use the parameter $s$ in part to highlight the connection to the fractional Laplacian and white noise. With our convention, white noise is $\operatorname{FGF}_0({\mathbb{R}}^d)$ and the Gaussian free field is $\operatorname{FGF}_1({\mathbb{R}}^d)$. However, in many of our formulas and theorems ${H}$ will be the more natural parameter to use; 