Bivariate mean value interpolation on circles of the same radius (Q1954016)

The problem is to find a bivariate interpolating polynomial of degree \(n\), i.e., \(p(x,y)\in\Pi_n^2\), given the values for its mean over \(N=\mathrm{dim}(\Pi_2)=(n+2)(n+1)/2\) shifted disks \(D_k=D+\mathbf{m}_k\), \(k=1,\dots,N\), where \(D\) is a disk with radius \(r\) centered at the origin and the \(\mathbf{m}_k\) are the centers of the shifted disks. It has been proved by the author in another paper [East J. Approx. 17, No. 2, 151--157 (2011; Zbl 1234.41004)] that there is a unique solution if and only if the ordinary Lagrange interpolation problem with function values given in the centers \(\{\mathbf{m}_k\}\) has a unique solution. In this short note, it is proved that there does not exist a unique solution if \(n+2\geq 3\) of the centers are on a straight line.