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.