Kernel B-splines and interpolation

This paper applies difference operators to conditionally positive definite kernels in order to generate {\em 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.