Approximation from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})

de Boor, Carl; DeVore, Ronald A.; Ron, Amos

doi:10.1090/S0002-9947-1994-1195508-X

Approximation from shift-invariant subspaces of $L\sb 2(\bold R\sp d)$

Authors: Carl de Boor, Ronald A. DeVore and Amos Ron
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
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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.

[A] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
[B1] 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, https://doi.org/10.1007/BF01890564
[B2] 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
[Bu1] M. D. Buhmann, Multivariate cardinal interpolation with radial-basis functions, Constr. Approx. 6 (1990), no. 3, 225–255. MR 1054754, https://doi.org/10.1007/BF01890410
[Bu2] M. D. Buhmann, On quasi-interpolation with radial basis functions, J. Approx. Theory 72 (1993), no. 1, 103–130. MR 1198376, https://doi.org/10.1006/jath.1993.1009
[BD] C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), no. 2, 775–788. MR 688977, https://doi.org/10.1090/S0002-9947-1983-0688977-5
[BuD] 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
[BH1] C. de Boor and K. Höllig, 𝐵-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99–115. MR 729403, https://doi.org/10.1007/BF02786872
[BH2] C. de Boor and K. Höllig, Approximation order from bivariate 𝐶¹-cubics: a counterexample, Proc. Amer. Math. Soc. 87 (1983), no. 4, 649–655. MR 687634, https://doi.org/10.1090/S0002-9939-1983-0687634-4
[BJ] 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, https://doi.org/10.1090/S0002-9939-1985-0810161-X
[BR1] 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, https://doi.org/10.1112/jlms/s2-45.3.519
[BR2] 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, https://doi.org/10.1007/BF01203462
[C] 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
[CL] 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, https://doi.org/10.1007/BF01208904
[DJLR] 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, https://doi.org/10.1007/BF02801574
[DM1] Wolfgang Dahmen and Charles A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234. MR 709352, https://doi.org/10.1016/0024-3795(83)80015-9
[DM2] 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, https://doi.org/10.1017/S033427000000446X
[DR] 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, https://doi.org/10.1090/S0002-9947-1990-0956032-6
[H] Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
[HL] E. J. Halton and W. A. Light, On local and controlled approximation order, J. Approx. Theory 72 (1993), no. 3, 268–277. MR 1209967, https://doi.org/10.1006/jath.1993.1021
[J1] 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, https://doi.org/10.1090/S0002-9947-1986-0831196-2
[J2] Rong Qing Jia, A counterexample to a result concerning controlled approximation, Proc. Amer. Math. Soc. 97 (1986), no. 4, 647–654. MR 845982, https://doi.org/10.1090/S0002-9939-1986-0845982-1
[JL] 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, https://doi.org/10.1006/jath.1993.1002
[Ja] 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, https://doi.org/10.1093/imanum/9.4.567
[L] 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
[LJ] Junjiang Lei and Rong Qing Jia, Approximation by piecewise exponentials, SIAM J. Math. Anal. 22 (1991), no. 6, 1776–1789. MR 1129411, https://doi.org/10.1137/0522111
[M] 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
[MN1] W. R. Madych and S. A. Nelson, Polyharmonic cardinal splines, J. Approx. Theory 60 (1990), no. 2, 141–156. MR 1033167, https://doi.org/10.1016/0021-9045(90)90079-6
[MN2] 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, https://doi.org/10.1090/S0025-5718-1990-0993931-7
[P] 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
[R] Amos Ron, A characterization of the approximation order of multivariate spline spaces, Studia Math. 98 (1991), no. 1, 73–90. MR 1110099, https://doi.org/10.4064/sm-98-1-73-90
[Ra] C. Rabut, Polyharmonic cardinal -splines, Parts A and B, preprint, 1989.
[RS] A. Ron and N. Sivakumar, The approximation order of box spline spaces, Proc. Amer. Math. Soc. 117 (1993), no. 2, 473–482. MR 1110553, https://doi.org/10.1090/S0002-9939-1993-1110553-2
[S] 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.
[SF] 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.