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Distance Matrices and Ridge Function Interpolation

Published online by Cambridge University Press:  20 November 2018

Les Reid  and
Xingping Sun
Les Reid
Affiliation:
Department of Mathematics Southwest Missouri State University Springfield, Missouri 65804 USA.
Xingping Sun
Affiliation:
Department of Mathematics Southwest Missouri State University Springfield, Missouri 65804 USA.
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Abstract

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A geometric characterization is given for a collection of points in ℝd to produce a singular l1 -distance matrix. Some quantitative results are established in terms of "characteristic matrices". The results in this paper generalize those of Dyn, Light and Cheney and have application to ridge function interpolation.

Keywords

41A0541A63
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 6 , 01 December 1993 , pp. 1313 - 1323
Copyright
Copyright © Canadian Mathematical Society 1993

References

[B] Bellman, R., Introduction to Matrix Analysis, McGraw-Hill Book Company, 1960.Google Scholar
[BP] Braess, D. and Pinkus, A., Interpolation by ridge functions , J. Approx. Theory, to appear.Google Scholar
[DS] Diliberto, S. P. and Straus, E.G., On the approximation of a function of several variable by the sum of functions of fewer variables, Pacific J. Math. 1(1951), 195–210.Google Scholar
[DLC] Dyn, N., Light, W.A. and Cheney, E.W., Interpolation by piecewise-linear radial basis functions I, J. Approx. Theory 59(1989), 202–223.Google Scholar
[H] Herz, C.S., Une ébauche d'une théorie générale des fonctions définie s-négative , Séminaire Brelot- Choquet-Deny (Théorie du Potentiel), 7eannée, Inst. Henri Poincaré, Paris.Google Scholar
[L] Light, W.A., The singularity of distance matrices. In: Multivariate Approximation Theory, (eds. Schempp, W., Chui, C. and Zeller, K.), Birkhauser-Verlag (1989), 233–240.Google Scholar
[SI] Schoenberg, I.J., On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space, Ann. Math. 38(1937), 787–793.Google Scholar
[S2] Schoenberg, I.J., Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44(1938), 522–536.Google Scholar
[S3] Schoenberg, I.J., Metric spaces and completely monotone functions, Ann. Math. 39(1938), 811–841.Google Scholar
[SX] Sun, X., Multivariate interpolation by a general class of functions , J. Approx. Theory, to appear.Google Scholar