Interpolation with radially symmetric thin plate splines (Q1815886)

The author studies the problem of determining a radially symmetric function defined by \(f(x,y)=g(\sqrt{x^2+y^2})\) such that, for given data \((x_i,y_i,z_i)_{i\in I}\), \(I\subset\mathbb{N}\), and for all \(i\in I\), \(f(x_i,y_i)=z_i\). In the paper \(g\) is defined in such a way that the function \(f\) minimizes the seminorm \[ \left\{\int_{R^2_1\leq x^2+y^2\leq R^2_2}(\partial^2 f/\partial x^2)^2 + 2(\partial^2 f/\partial x\partial y)^2 +(\partial^2f/\partial y^2)^2\right\}^{1/2} \] over all radially symmetric functions which interpolate the data. Existence and uniqueness of the functions \(f\) and \(g\) are settled by using general theorems on variational splines, and general expressions for \(f\) and \(g\) are given by using the reproducing kernel of the associated Hilbert space. Furthermore, it is shown how to determine \(f\) and \(g\) such that \(f\) interpolates the data.