Kernel B-splines and interpolation

This paper applies difference operators to conditionally positive definite kernels in order to generate kernel B-splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel B-spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel B-spline is constructed adaptively on the data knot set X, or we use a fixed difference scheme and shift its associated kernel B-spline around. In the latter case, the kernel B-spline so obtained is strictly positive in general. Furthermore, special kernel B-splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.