Multiscale kernels

Advances in Computational Mathematics

Volumn 25, Issue 4, 2006, Pages 357-380

  Opfer, Roland ^a  

^a UNIVERSITY OF GÖTTINGEN   (Germany)

Author keywords

Data compression; Multiresolution; Radial basis functions; Refineable functions; Scattered data approximation; Wavelets

Indexed keywords

EID: 33646732403     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-7622-3     Document Type: Article

References (28)

1. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337-404.
2. R.K. Beatson and L. Greengard, A short course on fast multipole methods, in: Wavelets, Multilevel Methods and Elliptic PDEs, eds. M. Ainsworth, J. Levesley, W. Light and M. Marietta (Oxford Univ. Press, Oxford, 1997) pp. 1-37.
3. M. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge Univ. Press, Cambridge, 2003).
4. A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Trans. Amer. Math. Soc. 453 (1991) 186.
5. W. Cheney and W. Light, A Course in Approximation Theory, Brooks/Cole Series in Advanced Mathematics (USA, 2000).
6. C.K. Chui, An Introduction to Wavelets, Wavelet Analysis and its Applications (1) (Academic Press, London, 1992).
7. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992).
8. C. de Boor, A Practical Guide to Splines, revised ed. (Springer, New York, 2001).
9. d), Trans. Amer. Math. Soc. 341 (1994) 787-806.
10. C. de Boor, K. Höllig and S. Riemenschneider, Box Splines (Springer, New York, 1993).
11. R.A. DeVore, Wavelets, in: Acta Numerica 1992, Vol. 1 (Cambridge Univ. Press, Cambridge, 1992) pp. 1-56.
12. N. Dyn, Minimal interpolation and approximation in Hubert spaces, SIAM J. Numer. Anal. 8 (1971) 583-597.
13. T. Evgeniou, M. Pontil and T. Poggio, Regularization networks and support vector machines, Adv. Comput. Math. 13 (2000) 1-50.
14. G.H. Golub and C.F. Van Loan, Matrix Computations (Johns Hopkins Univ. Press, Baltimore, MD, 1996).
15. M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards 49 (1952) 409-436.
16. L.-T. Luh, Characterizations of native spaces, Dissertation, Universität Göttingen (1998).
17. S. Mallat, A theory for multiresolution signal decompostion: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11 (1989) 674-693.
18. H. Meschkowski, Hilbertsche Räume mit Kernfunktion (Springer, Berlin, 1962).
19. Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Vol. 37 (Cambridge Univ. Press, Cambridge, 1992).
20. C.A. Micchelli, Using the refinement equation for the construction of pre-wavelets, Numer. Algorithms 1 (1991) 75-116.
21. W. Pogorzelski, Integral Equations and their Applications, Vol. 1 (Pergamon Press, Oxford, 1966).
22. R. Schaback, Native Hubert spaces for radial basis functions I, in: New Developments in Approximation Theory, eds. M.D. Buhmann, D.H. Mache, M. Felten and M.W. Müller, International Series of Numerical Mathematics (Birkhäuser, Basel, 1999) pp. 255-282.
23. R. Schaback, A unified theory of radial basis functions (native Hubert spaces for radial basis functions II), J. Comput. Appl. Math. 121 (2000) 165-177.
24. R. Schaback and H. Wendland, Characterization and construction of radial basis functions, in: Multivariate Approximation and Applications, eds. N. Dyn, D. Leviatanand, D. Levin and A. Pinkus (Cambridge Univ. Press, Cambridge, 2001) pp. 1-24.
25. R. Schaback and H. Wendland, Approximation by positive definite kernels, for: IDOMat2001 Proceedings (invited lecture) (2002).
26. R. Schaback and H. Wendland, Inverse and saturation theorems for radial basis function interpolation, Math. Comp. 71 (2002) 669-681.
27. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995) 389-396.
28. H. Wendland, Scattered data modelling by radial and related functions, Habilitationsschrift, University of Göttingen (2002).