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Spectral convergence of multiquadric interpolation

Published online by Cambridge University Press:  20 January 2009

Martin Buhmann  and
Nira Dyn
Martin Buhmann
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England
Nira Dyn
Affiliation:
Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
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Abstract

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In this paper, we consider interpolants on h·ℤn from the closure of the space spanned by translates of the function (‖·‖2 + 1)β/2 (β>−n and not an even nonnegative integer) along h·ℤn. We show that these interpolants approximate a function, whose Fourier transform satisfies certain asymptotic conditions, up to an error of order hp, on any compact domain in ℝn, where p is only restricted by the smoothness of the function.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 319 - 333
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

REFERENCES

1.Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions (Dover Publications, 1970).Google Scholar
2.Buhmann, M. D., Multivariable Interpolation using Radial Basis Functions (Ph.D. Dissertation, University of Cambridge, 1989).Google Scholar
3.Buhmann, M. D., Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics, Constr. Approx. 6 (1990), 21–34.CrossRefGoogle Scholar
4.Buhmann, M. D., Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6 (1990), 225–255.CrossRefGoogle Scholar
5.Buhmann, M. D. and Dyn, N., Error estimates for multiquadric interpolation, in Curves and Surfaces (Laurent, P.-J., LeMéhauté, A., Schumaker, L. L., eds., Academic Press, New York, 1991), 51–58.CrossRefGoogle Scholar
6.Dyn, N., Interpolation and approximation by radial and related functions, in Approximation Theory VI (Chui, C. K., Schumaker, L. L., Ward, J. D., eds., Academic Press, 1989), 211–234.Google Scholar
7.Dyn, N. and Ron, A., Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems, Trans. Amer. Math. Soc. 319 (1990), 381–403.CrossRefGoogle Scholar
8.Ismail, M. E. H., Complete monotonicity of modified Bessel functions (ICM-Report 88–013, University of South Florida, 1988).Google Scholar
9.Jones, D. S., The Theory of Generalized Functions (Cambridge University Press, Cambridge, 1982).CrossRefGoogle Scholar
10.Light, W. A. and Cheney, E. W., Quasi-interpolation with translates of a function having non-compact support, Constr. Approx. 8 (1992), 35–48.CrossRefGoogle Scholar
11.Madych, W. R. and Nelson, S. A., Error bounds for multiquadric interpolation, in Approximation Theory VI (Chui, C. K., Schumaker, L. L., Ward, J. D., eds., Academic Press, 1989), 413–416.Google Scholar
12.Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive definite functions II, Math. Comp. 54 (1990), 211–230.CrossRefGoogle Scholar
13.Powell, M. J. D., The theory of radial basis function approximation in 1990, in Advances in Numerical Analysis II (Light, W. A., ed., Clarendon Press, Oxford, 1992), 105–210.CrossRefGoogle Scholar
14.Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1971).Google Scholar
15.Wu, Z. and Schaback, R., Local error estimates for radial basis function interpolation of scattered data (Report University of Göttingen, 1990).Google Scholar