Convergence of discrete and penalized least squares spherical splines (Q719351)

The authors consider the approximation of data \(\mathcal{V}\) on the unit sphere \(\mathbb{S}^2\) in \(\mathbb{R}^3\) by discrete least squares and penalized least squares spherical splines. More precisely, given a set of discrete values \(\{f(v) : v\in \mathcal{V}\}\) of a function \(f:\mathbb{S}^2\to\mathbb{R}\), find a penalized least squares spline \(s_{\lambda,f}\in S^r_d (\Delta)\), where \(d > r \geq 0\) and \(\Delta\) is a triangulation of \(\mathbb{S}^2\). It is assumed that the elements in \(\mathcal{V}\) are evenly distributed over \(\Delta\). Here \(\lambda \geq 0\) denotes the penalty parameter, which when set equal to zero yields the discrete least squares spherical splines. Error bounds for the approximation of sufficiently smooth functions by discrete and penalized least squares spherical splines are derived and it is shown that in the case of discrete least squares spherical splines this error bound depends explicitly on the mesh size of the triangulation \(\Delta\), whereas in the case of penalized least squares spherical splines it also depends on the penalty parameter \(\lambda\).