Approximate approximations from scattered data

Journal of Approximation Theory

Volumn 145, Issue 2, 2007, Pages 141-170

  Lanzara, F ^a   Maz'ya, V ^b,c,d   Schmidt, G ^e  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b LINKÖPING UNIVERSITY   (Sweden)

^c OHIO STATE UNIVERSITY   (United States)

^d UNIVERSITY OF LIVERPOOL   (United Kingdom)

^e WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

Cubature of integral operators; Error estimates; Multivariate approximation; Scattered data quasi interpolation

Indexed keywords

EID: 33947213514     PISSN: 00219045     EISSN: 10960430     Source Type: Journal    
DOI: 10.1016/j.jat.2006.08.003     Document Type: Article

References (26)

1. Babuška I., Banerjee U., and Osborn J.E. On the approximability and the selection of particle shape functions. Numer. Math. 96 (2004) 601-640
2. Buhmann M.D., Dyn N., and Levin D. On quasi interpolation by radial basis functions with scattered data. Constr. Approx. 11 (1995) 239-254
3. d). Trans. AMS 341 (1994) 787-806
4. de Boor C., and Ron A. Fourier analysis of the approximation power of principal shift-invariant spaces. Constr. Approx. 8 (1992) 427-462
5. Dyn N., and Ron A. Radial basis function approximation: from gridded centers to scattered centers. Proc. London Math. Soc. 71 (1995) 76-108
6. Farwig R. Multivariate interpolation of arbitrarily spaced data by moving least squares methods. J. Comput. Appl. Math. 16 (1986) 79-93
7. Ivanov T., Maz'ya V., and Schmidt G. Boundary layer approximate approximations for the cubature of potentials in domains. Adv. Comput. Math. 10 (1999) 311-342
8. Jetter K., and Zhou D.X. Order of linear approximation from shift-invariant spaces. Constr. Approx. 11 (1995) 423-438
9. Karlin V., and Maz'ya V. Time-marching algorithms for non local evolution equations based upon approximate approximations. SIAM J. Sci. Comput. 18 (1997) 736-752
10. Levin D. The approximations power of moving least-squares. Math. Comput. 67 (1998) 1517-1531
11. Maz'ya V. A new approximation method and its applications to the calculation of volume potentials. Boundary point method. 3. DFG-Kolloqium des DFG-Forschungsschwerpunktes Randelementmethoden (1991)
12. Maz'ya V. Approximate approximations. In: Whiteman J.R. (Ed). The Mathematics of Finite Elements and Applications. Highlights 1993 (1994), Wiley & Sons, Chichester
13. Maz'ya V., and Karlin V. Semi-analytic time marching algorithms for semi-linear parabolic equations. BIT 34 (1994) 129-147
14. Maz'ya V., and Schmidt G. Approximate approximations and the cubature of potentials. Rend. Mat. Acc. Lincei 6 (1995) 161-184
15. Maz'ya V., and Schmidt G. On approximate approximation using Gaussian kernels. IMA J. Numer. Anal. 16 (1996) 13-29
16. Maz'ya V., and Schmidt G. Approximate wavelets and the approximation of pseudodifferential operators. App. Comput. Harm. Anal. 6 (1999) 287-313
17. Maz'ya V., and Schmidt G. On quasi-interpolation with non-uniformly distributed centers on domains and manifolds. J. Approx. Theory 110 (2001) 125-145
18. Maz'ya V., Schmidt G., and Wendland W. On the computation of multi-dimensional single layer harmonic potentials via approximate approximations. Calcolo 40 (2003) 33-53
19. Mhaskar H.N. Approximation theory and neural networks. In: Jain P.K., et al. (Ed). Wavelet and Allied Topics (2001), Narosa Publishing House, New Delhi 247-289
20. H.N. Mhaskar, On the degree of approximation in multivariate weighted approximation, in: M.D. Buhmann, D.H. Mache (Eds.), Advanced Problems in Constructive Approximation, ISNM, vol. 142, Birkhäuser, Basel, 2002, pp. 129-141.
21. Powell M.J.D. The theory of radial basis functions in 1990. In: Light W. (Ed). Advances in numerical analysis. vol. 2: Wavelets, Subdivision Algorithms, and Radial Basis Functions (1992), Clarendon Press, Oxford 105-210
22. Schaback R., and Wu Z.M. Construction techniques for highly accurate quasi-interpolation operators. J. Approx. Theory 91 (1997) 320-331
23. G. Schmidt, On approximate approximations and their applications, in: The Maz'ya anniversary collection, v.1, Operator Theory: Adv. Appl. 109 (1999) 111-138.
24. Stein E.M., and Weiss G. Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton University press, Princeton, NJ
25. Watson G.N. A Treatise on the Theory of Bessel Functions (1922), Cambridge University Press, Cambridge
26. Wu Z.M., and Liu J.P. Generalized Strang-Fix condition for scattered data quasi-interpolation. Adv. Comput. Math. 23 (2005) 201-214