Vectorial interpolation using radial-basis-like functions

This paper deals with vector field interpolation, i.e., the data are 3 values located in scattered 3 points, while the interpolating function is a function from 3 into 3. In order to take into account possible connections between the components of the interpolant, we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant, and depending on a parameter ρ significative of the balance of the rotational part and of the divergence part, and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm. We so obtain interpolants whose expression is σ(x) = Σni=1 Φ(x − xi)ai + pm−1(x), where Φ is some 3 × 3 matrix function, pm−1 is a degree m − 1 vectorial polynomial, and where the ai are 3-vectors. Besides, the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m − 1. For ρ = 1, we find the usual m-harmonic splines in each component of σ. Numerical examples show the interest of the method, and we compare the so-obtained functions with the ones obtained by Matlabʹs procedures.