Approximation from shift-invariant subspaces of $L_ 2(\mathbf {R}^ d)$
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- by Carl de Boor, Ronald A. DeVore and Amos Ron PDF
- Trans. Amer. Math. Soc. 341 (1994), 787-806 Request permission
Abstract:
A complete characterization is given of closed shift-invariant subspaces of ${L_2}({\mathbb {R}^d})$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- Carl de Boor, The polynomials in the linear span of integer translates of a compactly supported function, Constr. Approx. 3 (1987), no. 2, 199–208. MR 889555, DOI 10.1007/BF01890564
- Carl de Boor, Quasiinterpolants and approximation power of multivariate splines, Computation of curves and surfaces (Puerto de la Cruz, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 307, Kluwer Acad. Publ., Dordrecht, 1990, pp. 313–345. MR 1064965
- M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6 (1990), no. 3, 225–255. MR 1054754, DOI 10.1007/BF01890410
- 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
- C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), no. 2, 775–788. MR 688977, DOI 10.1090/S0002-9947-1983-0688977-5
- 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
- C. de Boor and K. Höllig, $B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99–115. MR 729403, DOI 10.1007/BF02786872
- C. de Boor and K. Höllig, Approximation order from bivariate $C^{1}$-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), no. 4, 649–655. MR 687634, DOI 10.1090/S0002-9939-1983-0687634-4
- C. de Boor and R.-Q. Jia, Controlled approximation and a characterization of the local approximation order, Proc. Amer. Math. Soc. 95 (1985), no. 4, 547–553. MR 810161, DOI 10.1090/S0002-9939-1985-0810161-X
- Carl de Boor and Amos Ron, The exponentials in the span of the multi-integer translates of a compactly supported function; quasi-interpolation and approximation order, J. London Math. Soc. (2) 45 (1992), no. 3, 519–535. MR 1180260, DOI 10.1112/jlms/s2-45.3.519
- Carl de Boor and Amos Ron, Fourier analysis of the approximation power of principal shift-invariant spaces, Constr. Approx. 8 (1992), no. 4, 427–462. MR 1194028, DOI 10.1007/BF01203462
- Charles K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. MR 1033490, DOI 10.1137/1.9781611970173
- W. A. Light and E. W. Cheney, Quasi-interpolation with translates of a function having noncompact support, Constr. Approx. 8 (1992), no. 1, 35–48. MR 1142692, DOI 10.1007/BF01208904
- N. Dyn, I. R. H. Jackson, D. Levin, and A. Ron, On multivariate approximation by integer translates of a basis function, Israel J. Math. 78 (1992), no. 1, 95–130. MR 1194962, DOI 10.1007/BF02801574
- Wolfgang Dahmen and Charles A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234. MR 709352, DOI 10.1016/0024-3795(83)80015-9
- Wolfgang Dahmen and Charles A. Micchelli, On the approximation order from certain multivariate spline spaces, J. Austral. Math. Soc. Ser. B 26 (1984), no. 2, 233–246. MR 765640, DOI 10.1017/S033427000000446X
- N. Dyn and A. Ron, Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems, Trans. Amer. Math. Soc. 319 (1990), no. 1, 381–403. MR 956032, DOI 10.1090/S0002-9947-1990-0956032-6
- Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- E. J. Halton and W. A. Light, On local and controlled approximation order, J. Approx. Theory 72 (1993), no. 3, 268–277. MR 1209967, DOI 10.1006/jath.1993.1021
- Rong Qing Jia, Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh, Trans. Amer. Math. Soc. 295 (1986), no. 1, 199–212. MR 831196, DOI 10.1090/S0002-9947-1986-0831196-2
- Rong Qing Jia, A counterexample to a result concerning controlled approximation, Proc. Amer. Math. Soc. 97 (1986), no. 4, 647–654. MR 845982, DOI 10.1090/S0002-9939-1986-0845982-1
- Rong Qing Jia and Junjiang Lei, Approximation by multi-integer translates of functions having global support, J. Approx. Theory 72 (1993), no. 1, 2–23. MR 1198369, DOI 10.1006/jath.1993.1002
- I. R. H. Jackson, An order of convergence for some radial basis functions, IMA J. Numer. Anal. 9 (1989), no. 4, 567–587. MR 1030648, DOI 10.1093/imanum/9.4.567
- William A. Light, Recent developments in the Strang-Fix theory for approximation orders, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 285–292. MR 1123748
- Junjiang Lei and Rong Qing Jia, Approximation by piecewise exponentials, SIAM J. Math. Anal. 22 (1991), no. 6, 1776–1789. MR 1129411, DOI 10.1137/0522111
- W. R. Madych, Error estimates for interpolation by generalized splines, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 297–306. MR 1123750
- W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, J. Approx. Theory 60 (1990), no. 2, 141–156. MR 1033167, DOI 10.1016/0021-9045(90)90079-6
- W. R. Madych and S. A. Nelson, Multivariate interpolation and conditionally positive definite functions. II, Math. Comp. 54 (1990), no. 189, 211–230. MR 993931, DOI 10.1090/S0025-5718-1990-0993931-7
- M. J. D. Powell, The theory of radial basis function approximation in 1990, Advances in numerical analysis, Vol. II (Lancaster, 1990) Oxford Sci. Publ., Oxford Univ. Press, New York, 1992, pp. 105–210. MR 1172121
- Amos Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1991), no. 1, 73–90. MR 1110099, DOI 10.4064/sm-98-1-73-90 C. Rabut, Polyharmonic cardinal $B$-splines, Parts A and B, preprint, 1989.
- A. Ron and N. Sivakumar, The approximation order of box spline spaces, Proc. Amer. Math. Soc. 117 (1993), no. 2, 473–482. MR 1110553, DOI 10.1090/S0002-9939-1993-1110553-2 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. G. Strang and G. Fix, A Fourier analysis of the finite element variational method, C.I.M.E. II, Ciclo 1971, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), 1973, pp. 793-840.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 341 (1994), 787-806
- MSC: Primary 41A25; Secondary 41A63, 42B10, 46E20
- DOI: https://doi.org/10.1090/S0002-9947-1994-1195508-X
- MathSciNet review: 1195508