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.
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Bozzini, M., Lenarduzzi, L. & Schaback, R. Kernel B-splines and interpolation. Numer Algor 41, 1–16 (2006). https://doi.org/10.1007/s11075-005-9000-8
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DOI: https://doi.org/10.1007/s11075-005-9000-8