## Multiquadric prewavelets on nonequally spaced knots in one dimension

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## Abstract:

In this paper, we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with*nonequally spaced*centers (or "knots"). Although the multiquadric function and its relations are our prime examples, the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well.

## References

- Carl de Boor,
*Odd-degree spline interpolation at a biinfinite knot sequence*, Approximation theory (Proc. Internat. Colloq., Inst. Angew. Math., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1976, pp. 30–53. MR**0613677** - Carl de Boor, Ronald A. DeVore, and Amos Ron,
*On the construction of multivariate (pre)wavelets*, Constr. Approx.**9**(1993), no. 2-3, 123–166. MR**1215767**, DOI 10.1007/BF01198001 - Martin D. Buhmann,
*New developments in the theory of radial basis function interpolation*, Multivariate approximation: from CAGD to wavelets (Santiago, 1992) Ser. Approx. Decompos., vol. 3, World Sci. Publ., River Edge, NJ, 1993, pp. 35–75. MR**1359544** - M. D. Buhmann,
*Discrete least squares approximation and prewavelets from radial function spaces*, Math. Proc. Cambridge Philos. Soc.**114**(1993), no. 3, 533–558. MR**1236000**, DOI 10.1017/S0305004100071814 - M. D. Buhmann,
*On quasi-interpolation with radial basis functions*, J. Approx. Theory**72**(1993), no. 1, 103–130. MR**1198376**, DOI 10.1006/jath.1993.1009 - M. D. Buhmann,
*Pre-wavelets on scattered knots and from radial function spaces: a review*, The mathematics of surfaces, VI (Uxbridge, 1994) Inst. Math. Appl. Conf. Ser. New Ser., vol. 58, Oxford Univ. Press, New York, 1996, pp. 309–324. MR**1430590** - Martin D. Buhmann and Charles A. Micchelli,
*Spline prewavelets for nonuniform knots*, Numer. Math.**61**(1992), no. 4, 455–474. MR**1155333**, DOI 10.1007/BF01385520 - M. D. Buhmann and N. Dyn,
*Error estimates for multiquadric interpolation*, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 51–58. MR**1123718** - Charles K. Chui and Jian-zhong Wang,
*A cardinal spline approach to wavelets*, Proc. Amer. Math. Soc.**113**(1991), no. 3, 785–793. MR**1077784**, DOI 10.1090/S0002-9939-1991-1077784-X
C. K. Chui, J.D. Ward, and J. Stöckler, - Charles A. Micchelli, Christophe Rabut, and Florencio I. Utreras,
*Using the refinement equation for the construction of pre-wavelets. III. Elliptic splines*, Numer. Algorithms**1**(1991), no. 4, 331–351. MR**1139461**, DOI 10.1007/BF02142379 - W. Haussmann and K. Jetter (eds.),
*Multivariate approximation and interpolation*, International Series of Numerical Mathematics, vol. 94, Birkhäuser Verlag, Basel, 1990. MR**1111923**, DOI 10.1007/978-3-0348-5685-0 - M. J. D. Powell,
*Tabulation of thin plate splines on a very fine two-dimensional grid*, Numerical methods in approximation theory, Vol. 9 (Oberwolfach, 1991) Internat. Ser. Numer. Math., vol. 105, Birkhäuser, Basel, 1992, pp. 221–244. MR**1269364**, DOI 10.1007/978-3-0348-8619-2_{1}3 - Larry L. Schumaker,
*Spline functions: basic theory*, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR**606200**

*Analytic wavelets generated by radial functions*, manuscript, 1994. G. H. Hardy, J. E. Littlewood, and G. Pólya,

*Inequalities*, Cambridge Univ. Press, Cambridge, 1934.

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp.
**64**(1995), 1611-1625 - MSC: Primary 42C15
- DOI: https://doi.org/10.1090/S0025-5718-1995-1308448-4
- MathSciNet review: 1308448