Narcowich, Francis J.; Ward, Joseph D. Norms of inverses and condition numbers for matrices associated with scattered data. (English) Zbl 0724.41004 J. Approximation Theory 64, No. 1, 69-94 (1991). 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 \(\log (1+\| x\|^ 2_ 2)\). Reviewer: H.Mühlig (Dresden) Cited in 3 ReviewsCited in 45 Documents MSC: 41A05 Interpolation in approximation theory 41A63 Multidimensional problems 65F35 Numerical computation of matrix norms, conditioning, scaling Keywords:interpolation matrix PDFBibTeX XMLCite \textit{F. J. Narcowich} and \textit{J. D. Ward}, J. Approx. Theory 64, No. 1, 69--94 (1991; Zbl 0724.41004) Full Text: DOI References: [1] K. Ball; K. Ball [2] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Tables of Integral Transforms (1954), McGraw-Hill: McGraw-Hill New York · Zbl 0055.36401 [3] Duchon, J., Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces, RAIRO Anal. Numér., 10, 5-12 (1976) [4] Duchon, J., Splines minimizing rotation invariant semi-norms in Sobolov spaces, (Schempp, W.; Zeller, K., Constructive Theory of Functions of Several Variables (1977), Springer-Verlag: Springer-Verlag Berlin), 85-100, Oberwolfach 1976 [5] N. Dynin; N. Dynin · Zbl 0705.41006 [6] N. Dyn, W. A. Light, and E. W. CheneyJ. Approx. Theory; N. Dyn, W. A. Light, and E. W. CheneyJ. Approx. Theory · Zbl 0678.41001 [7] Franke, R., Scattered data interpolation: Tests of some methods, Math. Comp., 38, 181-199 (1982) · Zbl 0476.65005 [8] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 1905-1915 (1971) [9] Lighthill, M. J., Fourier Analysis and Generalised Functions (1958), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0078.11203 [10] Madych, W. R.; Nelson, S. A., Multivariate interpolation: A variational theory (1983), manuscript · Zbl 0703.41008 [11] Madych, W. R.; Nelson, S. A., Multivariate interpolation and conditionally positive definite functions, Approx. Theory Appl., 4, 77-79 (1988) · Zbl 0703.41008 [12] Micchelli, C. A., Interpolation of scattered data: Distances, matrices, and conditionally positive definite functions, Constr. Approx., 2, 11-22 (1986) · Zbl 0625.41005 [13] Schoenberg, I. J., Metric spaces and completely monotone functions, Ann. of Math., 39, 811-841 (1938) · Zbl 0019.41503 [14] Treves, F., Topological Vector Spaces, Distributions and Kernels (1967), Academic Press: Academic Press New York · Zbl 0171.10402 [15] Watson, G. N., Theory of Bessel Functions (1966), Cambridge Univ. Press: Cambridge Univ. Press London · Zbl 0174.36202 [16] Wells, J. H.; Williams, R. L., Embeddings and Extensions in Analysis, (Ergebnisse, Vol. 84 (1975), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0324.46034 [17] Zygmund, A., (Trigonometric Series, Vols. I and II (1959), Cambridge Univ. Press: Cambridge Univ. Press London) · Zbl 0628.42001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.