Functions on the plane as combinations of powers of distances to points (Q624489)

The author claims that he proves the following theorem: Let \(m\) and \(n\) be two natural numbers. There are \(N\) numbers \(a_1,\dots,a_N\) and \(N\) points \(p_1,\dots,p_N \in {\mathbb{R}}^2\) such that \(r^{n+2m} \cos (n\theta )=\sum_{i=1}^{N} a_i |p-p_i|^{2(m+n)}\) for \(p=(r\cos\theta ,r\sin\theta )\in {\mathbb{R}}^2\), where \(|\cdot |\) is the Euclidean distance on the plane.