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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. © Springer Science 2006.

We present an adaptive method to extract shape-preserving information from a univariate data sample. The behavior of the signal is obtained by interpolating at adaptively selected few data points by a linear combination of multiquadrics with variable scaling parameters. On the theoretical side, we give a sufficient condition for existence of the scaled multiquadric interpolant. On the practical side, we give various examples to show the applicability of the method