Approximation from shift-invariant subspaces of L2(ℝd)

Transactions of the American Mathematical Society

Volumn 341, Issue 2, 1994, Pages 787-806

  De Boor, Carl ^a   Devore, Ronald A ^a   Ron, Amos ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Approximation order; Orthogonal projection; Radial basis functions; Shift invariant spaces; Strang Fix conditions

Indexed keywords

EID: 84966255902     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-1994-1195508-X     Document Type: Article

References (36)

1. R. A. Adams, Sobolev spaces, Academic Press, New York, 1975.
2. C. de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), 199-208.
3. C. de Boor, Quasiinterpolants and approximation power of multivariate splines, Computation of Curves and Surfaces (M. Gasca and C. A. Micchelli, eds.), Kluwer Academic Publishers, Dordrecht, Netherlands, 1990, pp. 313-345.
4. M. D. Buhmann, Multivariate interpolation with radial basis functions, Constr. Approx. 6 (1990), 225-256.
5. M. D. Buhmann, On quasi-interpolation with radial basis functions, J. Approx. Theory 72 (1993), 103-130.
6. C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), 775-788.
7. M. D. Buhmann and N. Dyn, Error estimates for multiquadric interpolation, Curves and Surfaces (P. J. Laurent, A. Le Méhauté and L. L. Schumaker, eds.), Academic Press, New York, 1990, pp. 1-4.
8. C. de Boor and K. Höllig, B-splines from parallelepipeds, J. Analyse Math. 42 (1982-1983), 99-115.
9. 1-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), 649-655.
10. C. de Boor and R. Q. Jia, Controlled approximation and a characterization of the local approximation order, Proc. Amer. Math. Soc. 95 (1985), 547-553.
11. C. de Boor and A. Ron, The exponentials in the span of the integer translates of a compactly supported function: approximation orders and quasi-interpolation, J. London Math. Soc. 45 (1992), 519-535.
12. C. de Boor and A. Ron, Fourier analysis of approximation orders from principal shift-invariant spaces, Constr. Approx. 8 (1992), 427-462.
13. C. K. Chui, Multivariate splines, CBMS-NSF Regional Conference Ser. Appl. Math., no. 54, SIAM, Philadelphia, Pa., 1988.
14. E. W. Cheney and W. A. Light, Quasi-interpolation with basis functions having non-compact support, Constr. Approx. 8 (1992), 35-48.
15. N. Dyn, I. R. H. Jackson, D. Levin, and A. Ron, On multivariate approximation by the integer translates of a basis function, Israel J. Math. 78 (1992), 95-130.
16. W. Dahmen and C. A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/3 (1983), 217-234.
17. W. Dahmen and C. A. Micchelli, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser. B 26 (1984), 233-246.
18. N. Dyn and A. Ron, Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems, Trans. Amer. Math. Soc. 319 (1990), 381-404.
19. H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964.
20. E. J. Halton and W. A. Light, On local and controlled approximation order, J. Approx. Theory 72 (1993), 268-277.
21. R. Q. Jia, Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc. 295 (1986), 199-212.
22. R. Q. Jia, A counterexample to a result concerning controlled approximation, Proc. Amer. Math. Soc. 97 (1986), 647-654.
23. R. Q. Jia and J. J. Lei, Approximation by multiinteger translates of functions having global support, J. Approx. Theory 72 (1993), 2-23.
24. I. R. H. Jackson, An order of convergence for some radial basis functions, IMA J. Numer. Anal. 9 (1989), 567-587.
25. W. A. Light, Recent developments in the Strang-Fix theory for approximation orders, Curves and Surfaces (P. J. Laurent, A. le Méhauté, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 285-292.
26. J. J. Lei and R. Q. Jia, Approximation by piecewise exponentials, SIAM J. Math. Anal. 22 (1991), 1776-1789.
27. W. R. Madych, Error estimates for interpolation by generalized splines, Curves and Surfaces (P. J. Laurent, A. le Méhauté, and L. L. Schumaker, eds.), Academic Press, New York, 1991, pp. 297-306.
28. W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines. I, J. Approx. Theory 40 (1990), 141-156.
29. W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive functions. II, Math. Comp. 54 (1990), 211-230.
30. M. J. D. Powell, The theory of radial basis function approximation in 1990, Advances in Numerical Analysis, Vol II: Wavelets, Subdivision Algorithms and Radial Basis Functions (W. A. Light, ed.), Oxford Univ. Press, 1992, pp. 105-210.
31. A. Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1) (1991), 73-90.
32. C. Rabut, Polyharmonic cardinal B-splines, Parts A and B, preprint, 1989.
33. A. Ron and N. Sivakumar, The approximation order of box spline spaces, Proc. Amer. Math. Soc. 117 (1993), 473-482.
34. I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Parts A and B, Quart. Appl. Math. 4 (1946), 45-99; 112-141.
35. G. Strang and G. Fix, A Fourier analysis of the finite element variational method, C.I.M.E.
36. II, Ciclo 1971, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), 1973, pp. 793-840.