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Norms of inverses and condition numbers for matrices associated with scattered data. (English) Zbl 0724.41004

For matrices arising in connection with interpolating scattered, multivariate data, a general method is given for obtaining bounds both on the norm of the inverse of the interpolation matrix and on the condition number of that matrix. Interpolation functions are translates of a so called conditionally negative definite, radially symmetric function h(x) of order 1. As application of the method quantitative estimates are obtained in several cases, including those where h(x) is one of the functions \(\sqrt{1+\| x\|^ 2_ 2}\) and \(\log (1+\| x\|^ 2_ 2)\).

MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems
65F35 Numerical computation of matrix norms, conditioning, scaling
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