**2. Auxiliary Tools**

In order to make the paper self-contained and have a complete basis for the theoretical analysis in the later sections, we introduce some concepts and theorems related to radial functions in this section.

#### *2.1. Radial Basis Functions*

**Definition 1.** *A multivariate function* Φ : R*n* → R *is called radial if its value at each point depends only on the distance between that point and the origin, or equivalently provided there exists a univariate function ϕ* : [0, ∞) → R *such that* <sup>Φ</sup>(*x*) = *ϕ*(*r*) *with r* = *<sup>x</sup> . Here,* · *is usually the Euclidean norm. Then, the radial basis functions are defined by translation* <sup>Φ</sup>*k*(*x*) = *ϕ*( *x* − *<sup>x</sup>k* ) *for any fixed center xk* ∈ R*n.*

**Definition 2.** *A real-valued continuous function* Φ : R*n* → R *is called positive definite on* R*n if it is even and:*

$$\sum\_{j=1}^{N} \sum\_{k=1}^{N} c\_j c\_k \Phi(\mathbf{x}\_j - \mathbf{x}\_k) \ge 0 \tag{1}$$

*for any N pairwise different points x*1, ···, *xN* ∈ R*n, and c* = [*<sup>c</sup>*1, ···, *cN*]*<sup>T</sup>* ∈ R*N. It is the Fourier transform of a (positive) measure. The function* Φ *is strictly positive definite on* R*n if the quadratic (1) is zero only for c* ≡ **0***.*

The strictly positive definiteness of the radial function can be characterized by considering the Fourier transform of a univariate function. This is described in the following theorem. Its proof can be found in [2].

**Theorem 1.** *A continuous function ϕ* : [0, ∞) → R *such that r* → *rn*−<sup>1</sup>*ϕ*(*r*) ∈ *L*<sup>1</sup>[0, ∞) *is strictly positive definite and radial on* R*n if and only if the n-dimensional Fourier transform:*

$$\mathcal{F}\_n \varphi(r) = \frac{1}{\sqrt{r^{n-2}}} \int\_0^\infty \varphi(t) t^{\frac{n}{2}} I\_{(n-2)/2}(rt) dt \tag{2}$$

*is non-negative and not identically equal to zero. Here, J*(*<sup>n</sup>*−<sup>2</sup>)/2 *is the classical Bessel function of the first kind of order* (*n* − <sup>2</sup>)/2*.*
