Radial basis functions

Acta Numerica

Volumn 9, Issue , 2000, Pages 1-38

  Buhmann, M D ^a  

^a JUSTUS LIEBIG UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85095815155     PISSN: 09624929     EISSN: 14740508     Source Type: Journal    
DOI: 10.1017/S0962492900000015     Document Type: Article

References (70)

1. M. Abramowitz and I. Stegun, eds (1972), Handbook of Mathematical Functions, National Bureau of Standards, Washington.
2. R. W. Anderson, S. Das and E. L. Keller (1998), 'Estimation of spatiotemporal neural activity using radial basis function networks', J. Comput. Neuroscience 5, 421-441.
3. N. Arad, N. Dyn and D. Reisfeld (1994), 'Image warping by radial basis functions: applications to facial expressions', Graphical Models and Image Processing 56, 161-172.
4. R. Askey (1973), 'Radial characteristic functions', MRC Report 1262, University of Wisconsin, Madison.
5. I. Barrodale and C. A. Zala (1997), 'MJDP-BCS industrial liaison: applications to defence science', in Approximation and Optimization: Tributes to M. J. D. Powell (M. D. Buhmann and A. Iserles, eds), Cambridge University Press, Cambridge, pp. 31-46.
6. I. Barrodale and C. A. Zala (1999), 'Warping aerial photographs to orthomaps using thin plate splines', Adv. Comput. Math. 11, 211-227.
7. B. J. C. Baxter (1991), 'Conditionally positive functions and p-norm distance matrices', Constr. Approx. 7, 427-440.
8. R. K. Beatson and W. A. Light (1992), 'Quasi-interpolation in the absence of polynomial reproduction', in Numerical Methods of Approximation Theory (D. Braess and L. L. Schumaker, eds), Birkhäuser, Basel, pp. 21-39.
9. R. K. Beatson and W. A. Light (1997), 'Fast evaluation of radial basis functions: methods for 2-dimensional polyharmonic splines', IMA J. Numer. Anal. 17, 343-372.
10. R. K. Beatson and G. N. Newsam (1992), 'Fast evaluation of radial basis functions: I', Comput. Math. Appl. 24, 7-19.
11. R. K. Beatson, G. Goodsell and M. J. D. Powell (1995), 'On multigrid techniques for thin plate spline interpolation in two dimensions', in Vol. 32 of Lectures in Applied Mathematics, pp. 77-97.
12. R. K. Beatson, J. B. Cherrie and C. T. Mouat (1999), 'Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration', Adv. Comput. Math. 11, 253-270.
13. A. Bejancu (1997), 'The uniform convergence of multivariate natural splines', DAMTP Technical Report, University of Cambridge.
14. A. Bejancu (1999), 'Local accuracy for radial basis function interpolation on finite uniform grids', J. Approx. Theory 99 242-257.
15. M. D. Buhmann (1990a), 'Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics', Constr. Approx. 6, 21-34.
16. M. D. Buhmann (1990b), 'Multivariate cardinal-interpolation with radial-basis functions', Constr. Approx. 6, 225-255.
17. M. D. Buhmann (1998), 'Radial functions on compact support', Proceedings of the Edinburgh Mathematical Society 41, 33-46.
18. M. D. Buhmann (2000), 'A new class of radial basis functions with compact support', to appear in Math. Comput.
19. M. D. Buhmann and C. A. Micchelli (1991), 'Multiply monotone functions for cardinal interpolation', Adv. Appl. Math. 12, 359-386.
20. M. D. Buhmann and C. A. Micchelli (1992), 'On radial basis approximations on periodic grids', Mathematical Proceedings of the Cambridge Philosophical Society 112, 317-334.
21. J. Duchon (1976), 'Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces', Rev. Française Automat. Informat. Rech. Opér. Anal. Numer. 10, 5-12.
22. J. Duchon (1978), 'Sur l'erreur d'interpolation des fonctions de plusieurs variables pars les Dm-splines', Rev. Française Automat. Informat. Rech. Opér. Anal. Numer. 12, 325-334.
23. J. Duchon (1979), 'Splines minimizing rotation-invariant semi-norms in Sobolev spaces', in Constructive Theory of Functions of Several Variables (W. Schempp and K. Zeller, eds), Springer, Berlin/Heidelberg, pp. 85-100.
24. N. Dyn, F. J. Narcowich and J. D. Ward (1997), 'A framework for interpolation and approximation on Riemannian manifolds', in Approximation and Optimization: Tributes to M. J. D. Powell (M. D. Buhmann and A. Iserles, eds), Cambridge University Press, Cambridge, pp. 133-144.
25. R. Eckhorn (1999), 'Neural mechanisms of scene segmentation: recordings from the visual cortex suggest basic circuits for linking field models', IEEE Trans. Neural Net. 10, 1-16.
26. R. Estrada (1998), 'Regularization of distributions', Int. J. Math. Math. Sci. 21, 625-636.
27. G. Fasshauer (1999), 'Solving diérential equations with radial basis functions: multilevel methods and smoothing', Adv. Comput. Math. 11, 139-159.
28. A. C. Faul and M. J. D. Powell (1999), 'Proof of convergence of an iterative technique for thin plate spline interpolation in two dimensions', Adv. Comput. Math. 11, 183-192.
29. M. Floater and A. Iske (1996), 'Multistep scattered data interpolation using compactly supported radial basis functions', J. Comput. Appl. Math. 73, 65-78.
30. C. Franke and R. Schaback (1998), 'Solving partial diérential equations by collocation using radial basis functions', Comput. Math. Appl. 93, 72-83.
31. W. Freeden (1981), 'On spherical spline interpolation and approximation', Math. Meth. Appl. Sci. 3, 551-575.
32. W. Freeden and P. Hermann (1986), 'Uniform approximation by harmonic splines', Math. Z. 193, 265-275.
33. W. Freeden, M. Schreiner and R. Franke (1995), 'A survey on spherical spline approximation', Technical Report 95-157, University of Kaiserslautern.
34. F. Girosi (1992), 'Some extensions of radial basis functions and their applications in artificial intelligence', Comput. Math. Appl. 24, 61-80.
35. M. von Golitschek and W. A. Light (2000), 'Interpolation by polynomials and radial basis functions on spheres', to appear in Constr. Approx.
36. L. Greengard and V. Rokhlin (1987), 'A fast algorithm for particle simulations', J. Comput. Phys. 73, 325-348.
37. R. L. Hardy (1990), 'Theory and applications of the multiquadric-biharmonic method', Comput. Math. Appl. 19, 163-208.
38. S. Hochreiter and J. Schmidhuber (1999), 'Feature extraction through Lococode', Neural Computation 11, 679-714.
39. K. Jetter, J. Stöckler and J. D. Ward (1999), 'Error estimates for scattered data interpolation on spheres', Math. Comput. 68, 733-747.
40. M. J. Johnson (1997), 'An upper bound on the approximation power of principal shift-invariant spaces', Constr. Approx. 13, 155-176.
41. M. J. Johnson (1998a), 'A bound on the approximation order of surface splines', Constr. Approx. 14, 429-438.
42. M. J. Johnson (1998b), 'On the error in surface spline interpolation of a compactly supported function', manuscript, University of Kuwait.
43. M. J. Johnson (1999), 'Approximation in Lp(Rd) from spaces spanned by the perturbed integer translates of a radial function', manuscript, University of Kuwait.
44. D. S. Jones (1982), The Theory of Generalised Functions, Cambridge University Press, Cambridge.
45. J. Levesley, W. A. Light, D. Ragozin and X. Sun (1999), 'A simple approach to the variational theory for interpolation on spheres', in New Developments in Approximation Theory (M. D. Buhmann, M. Felten, D. Mache and M. W.Müller, eds), Birkhäuser, Basel, pp. 117-143.
46. W. R. Madych and S. A. Nelson (1990), 'Polyharmonic cardinal splines', J. Approx. Theory 60, 141-156.
47. O. V. Matveev (1997), 'On a method for interpolating functions on chaotic nets', Math. Notes 62, 339-349. Translated from Mat. Zametki (1997) 62, 404-417.
48. C. A. Micchelli (1986), 'Interpolation of scattered data: distance matrices and conditionally positive definite functions', Constr. Approx. 1, 11-22.
49. J. K. Misiewicz and D. St. P. Richards (1994), 'Positivity of integrals of Bessel functions', SIAM J. Math. Anal. 25, 596-601.
50. F. J. Narcowich, R. Schaback and J. D. Ward (1999), Multilevel interpolation and approximation, Appl. Comput. Harm. Analysis 7, 243-261.
51. R. Pollandt (1997), 'Solving nonlinear equations of mechanics with the boundary element method and radial basis functions', Internat. J. Numer. Methods Engrg 40, 61-73.
52. M. J. D. Powell (1992a), 'The theory of radial basis function approximation in 1990', in Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Functions (W. A. Light, ed.), Oxford University Press, Oxford, pp. 105-210.
53. M. J. D. Powell (1992b), 'Tabulation of thin plate splines on a very fine two-dimensional grid', in Numerical Methods of Approximation Theory (D. Braess and L. L. Schumaker, eds), Birkhäuser, Basel, pp. 221-244.
54. M. J. D. Powell (1993), 'Truncated Laurent expansions for the fast evaluation of thin plate splines', Numer. Alg. 5, 99-120.
55. M. J. D. Powell (1994), 'The uniform convergence of thin-plate spline interpolation in two dimensions', Numer. Math. 67, 107-128.
56. M. J. D. Powell (1999), 'Recent research at Cambridge on radial basis functions', in New Developments in Approximation Theory (M. D. Buhmann, M. Felten, D. Mache and M. W. Müller, eds), Birkhäuser, Basel, pp. 215-232.
57. A. Ron and X. Sun (1996), 'Strictly positive definite functions on spheres in Euclidean spaces', Math. Comput. 65, 1513-1530.
58. S. Saitoh (1988), Theory of Reproducing Kernels and its Applications, Longman, Harlow.
59. R. Schaback and H. Wendland (1998), 'Inverse and saturation theorems for radial basis function interpolation', Technical Report, University of Göttingen.
60. R. Schaback and Z. Wu (1995), 'Operators on radial basis functions', J. Comput. Appl. Math. 73, 257-270.
61. I. J. Schoenberg (1942), 'Positive definite functions on spheres', Duke Math. J. 9, 96-108.
62. R. Sibson and G. Stone (1991), 'Computation of thin plate splines', SIAM J. Sci. Statist. Comput. 12, 1304-1313.
63. E. M. Stein and G. Weiss (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton.
64. H. Wendland (1995), 'Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree', Adv. Comput. Math. 4, 389-396.
65. H. Wendland (1998), 'Error estimates for interpolation by radial basis functions of minimal degree', J. Approx. Theory 93, 258-272.
66. H. Wendland (1997), 'Sobolev-type error estimates for interpolation by radial basis functions', in Surface Fitting and Multiresolution Methods (A. LeMéhauté and L. L. Schumaker, eds), Vanderbilt University Press, Nashville, pp. 337-344.
67. Z. Wu (1992), 'Hermite-Birkhoff interpolation of scattered data by radial basis functions', Approx. Theory Appl. 8, 1-10.
68. Z. Wu (1995), 'Multivariate compactly supported positive definite radial functions', Adv. Comput. Math. 4, 283-292.
69. Z. Wu and R. Schaback (1993), 'Local error estimates for radial basis function interpolation of scattered data', IMA J. Numer. Anal. 13, 13-27.
70. Y. Xu and E. W. Cheney (1992), 'Strictly positive definite functions on spheres', Proc. Amer. Math. Soc. 116, 977-981.