Multidimensional natural splines of odd degree (Q2640810)

Let us denote by \({\mathcal P}_{r-1}\) the set of polynomials in \(X=(x_ 1,...,x_ n)\) of degree \(\leq r-1\), and let \(S(X)=Q_ 0(X)+\sum^{N}_{k=1}d_ k\| X-X_ k\|^{2r-1},\) where \(Q_ 0\in {\mathcal P}_{r-1}\), \(\| X\| =\sqrt{(X,Y)}\), \((X,Y)=x_ 1y_ 1...x_ ny_ n\), and the coefficients \(d_ k\) satisfy \(\sum^{N}_{k=1}d_ kQ(X_ k)=0,\forall Q\in {\mathcal P}_{r-1}\). \(X_ k\) are fixed points of \({\mathbb{R}}^ n\), S(X) is called natural spline. The interpolation problem for natural splines \(S(X_ j)=Q_ 0(X_ j)+\sum^{N}_{k=1}d_ k\| X_ j-X_ k\|^{2r-1}=y_ j,\) \(j\in 1,...,N\), is considered in this paper.