Spherical splines and interpolation on a sphere (Q1911007)

The author considers certain functions on the unit sphere \(\Omega\) in \(\mathbb{R}^3\), denoted as spherical splines. Here, a spherical spline, associated with the ``knots'' \(\{x_j\}\), is a function \(s\) of the form \[ s(x) = Q(x) + \sum^N_{j =1} d_j G(x,x_j), \quad x \in \Omega, \] where \(Q\) is a spherical harmonic and \(G\) is a function of the form \[ G(x,y) = \sum^\infty_{k = 1} g_k P_k (xy), \] \(P_k\) being the \(k\)th Legendre polynomial. Moreover, the coefficients \(d_j\) are to be taken from a specific point set described in the paper. The author proves the unique solvability of a certain interpolation problem for these spherical splines, and shows that they have a minimum norm property w.r.t. all solutions of this interpolation problem.